| | | | LOGIC-LM: | | | Empowering | | Large | Language | | Models | with | | |
| | --- | ------------ | --------- | -------- | --- | ----------- | -------- | --------- | -------- | --- | --------------- | ---- | --- | |
| | | | | Symbolic | | Solvers for | Faithful | | Logical | | Reasoning | | | |
| | | LiangmingPan | | | | AlonAlbalak | | XinyiWang | | | WilliamYangWang | | | |
| UniversityofCalifornia,SantaBarbara |
| {liangmingpan, alon_albalak, xinyi_wang, wangwilliamyang}@ucsb.edu |
| Abstract |
| | | | | | | | | | | Problem | Goal | | | |
| | --- | --- | --- | --- | --- | --- | --- | --- | --- | ------- | ---- | --- | --- | |
| LargeLanguageModels(LLMs)haveshown |
| | human-like | | reasoning | | abilities | but still strug- | | | | | | | | |
| | ---------- | ---- | --------- | ------- | --------- | ---------------- | --- | --- | --- | --- | --- | --- | --- | |
| | gle | with | complex | logical | problems. | This | pa- | | | | | | | |
| Problem |
| Symbolic |
| | per | introduces | | a novel | framework, | LOGIC- | | | | | | | | |
| | ------- | ---------- | ---------- | -------- | ---------------- | ------------- | --- | --- | ----- | --- | ----------- | ----------- | --- | |
| | | | | | | | | | | | Formulator | Formulation | | |
| | LM, | which | integrates | | LLMs | with symbolic | | | | | | | | |
| | solvers | | to improve | logical | problem-solving. | | | | | | | | | |
| | Our | method | first | utilizes | LLMs | to translate | | | Self- | | | | | |
| a natural language problem into a symbolic Refine Symbolic Symbolic |
| | | | | | | | | | | | Reasoner | | Result | |
| | ----------------------------------------- | ------ | ---------- | --- | ------------------ | ----------- | --- | --- | --- | --- | --------- | --- | ------ | |
| | formulation. | | Afterward, | | adeterministicsym- | | | | | | | | | |
| | bolic | solver | performs | | inference | on the for- | | | | | | | | |
| | mulated | | problem. | We | also introduce | a self- | | | | | | | | |
| | refinementmodule,whichutilizesthesymbolic | | | | | | | | | | Result | | | |
| | solver’serrormessagestorevisesymbolicfor- | | | | | | | | | | | | Answer | |
| Interpreter |
| | malizations. | | WedemonstrateLOGIC-LM’sef- | | | | | | | | | | | |
| | ------------ | --- | -------------------------- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | |
| fectivenessonfivelogicalreasoningdatasets: |
| | | | | | | | | | Figure1: | OverviewofourLOGIC-LMframework. | | | | |
| | ------------ | --- | ------------ | --- | ----------- | ---------- | --- | --- | -------- | ------------------------------- | --- | --- | --- | |
| | ProofWriter, | | PrOntoQA, | | FOLIO, | LogicalDe- | | | | | | | | |
| | duction, | | and AR-LSAT. | | On average, | LOGIC- | | | | | | | | |
| DespitetheadvancesofLLMs,theystillstrug- |
| LMachievesasignificantperformanceboost |
| glewithcomplexlogicalreasoningproblems(Liu |
| of39.2%overusingLLMalonewithstandard |
| promptingand18.4%overLLMwithchain-of- et al., 2023b). Recent studies (Golovneva et al., |
| 2023;Ribeiroetal.,2023b;Lyuetal.,2023)found |
| | thoughtprompting. | | | Ourfindingssuggestthat | | | | | | | | | | |
| | ----------------- | --- | --- | ---------------------- | --- | --- | --- | --- | --- | --- | --- | --- | --- | |
| LOGIC-LM, by combining LLMs with sym- that LLMs occasionally make unfaithful reason- |
| boliclogic,offersapromisingavenueforfaith- |
| | | | | | | | | ing, | i.e., | the derived | conclusion | does | not follow | |
| | --- | --- | --- | --- | --- | --- | --- | ---- | ----- | ----------- | ---------- | ---- | ---------- | |
| 1 |
| fullogicalreasoning. the previously generated reasoning chain. While |
| chain-of-thoughtmayimitatehumanreasoningpro- |
| 1 Introduction |
| | | | | | | | | cesses, | the | fundamental | nature | of LLMs | remains | |
| | --- | --- | --- | --- | --- | --- | --- | ------- | --- | ----------- | ------ | ------- | ------- | |
| Logical reasoning is a cognitive process that in- that of black-box probabilistic models, lacking a |
| volvesusingevidence,arguments,andlogictoar- mechanismtoguaranteethefaithfulnessofreason- |
| riveatconclusionsormakejudgments(Huangand ing(Shanahan,2022). Incontrast,symbolicinfer- |
| Chang,2023). Itplaysacentralroleinintelligent enceengines,suchasexpertsystems(Metaxiotis |
| systemsforproblem-solving,decision-making,and etal.,2002), arefaithfulandtransparentbecause |
| criticalthinking. Recently,largelanguagemodels the reasoning is based on symbolic-represented |
| (LLMs)(Brownetal.,2020;Ouyangetal.,2022a; knowledgeandfollowswell-definedinferencerules |
| OpenAI,2023)haveexhibitedemergentabilityto thatadheretologicalprinciples. Themainobsta- |
| “reason” like human (Wei et al., 2022a). When cle is how to accurately translate a problem into |
| promptedwithstep-wiseexplanationsofreasoning symbolicrepresentations,consideringtheinherent |
| (“chain of thoughts”), or a simple prompt “Let’s ambiguityandflexibilityofnaturallanguage. This |
| think step by step.”, these models are able to an- is precisely where LLMs excel, making LLMs a |
| swerquestionswithexplicitreasoningsteps(Wei promisingcomplementtosymbolicsolvers. |
| etal.,2022b;Kojimaetal.,2022). This drives our exploration of neuro-symbolic |
| methodsthatintegrateLLMswithsymbolicreason- |
| 1Codeanddataarepubliclyavailableathttps://github. |
| com/teacherpeterpan/Logic-LLM. ing. AsillustratedinFigure1,wepresentLOGIC- |
| 3806 |
| FindingsoftheAssociationforComputationalLinguistics:EMNLP2023,pages3806–3824 |
| December6-10,2023©2023AssociationforComputationalLinguistics |
|
|
| LM,anovelframeworkthatdecomposesalogical (§ 4.3). Finally, by analyzing the impact of self- |
| ProblemFor- |
| reasoningproblemintothreestages: refinement,wehighlighttheeffectivenessofincre- |
| mulation, Symbolic Reasoning, and Result Inter- mentally revising symbolic formalizations when |
| pretation. During problem formulation, an LLM interactingwiththesymbolicsolver(§4.4). |
| | converts | the natural | | language | description | | of the | | | | | | | | |
| | -------------------------------------------- | ----------- | ---------- | -------- | ------------- | ----- | ------- | ---------------------------------- | --- | --- | --- | --- | --- | --- | |
| | problemintoanappropriatesymbolicformulation, | | | | | | | 2 RelatedWork | | | | | | | |
| | identifying | key | entities, | facts, | and | rules | present | | | | | | | | |
| | | | | | | | | LanguageModelsforLogicalReasoning. | | | | | | Re- | |
| | in the problem | | statement. | | Subsequently, | | at the | | | | | | | | |
| centworksinadaptingLLMsforlogicalreasoning |
| symbolicreasoningstage,adeterministicsymbolic |
| taskscanbebroadlycategorizedintotwogroups: |
| solverperformsinferenceonthesymbolicformula- |
| 1)fine-tuningapproachesthatoptimizeLLMs’rea- |
| tion. Lastly,aresultinterpreterexplainstheoutput |
| soningabilitythroughfine-tuningortrainingspe- |
| andmapsittothecorrectanswer. Byincorporating cializedmodules(Clarketal.,2020;Tafjordetal., |
| | LLMs with | symbolic | | solvers, | we can | exploit | the | | | | | | | | |
| | --------- | -------- | --- | -------- | ------ | ------- | --- | --- | --- | --- | --- | --- | --- | --- | |
| 2022;Yangetal.,2022),and2)in-contextlearning |
| robustnaturallanguageunderstandingcapabilities |
| | | | | | | | | approaches | that | design | special | prompts | | to elicit | |
| | --- | --- | --- | --- | --- | --- | --- | ---------- | ---- | ------ | ------- | ------- | --- | --------- | |
| ofLLMstopreciselyrepresenttheproblemusing |
| | | | | | | | | LLMs’step-by-stepreasoningcapabilities. | | | | | | Typical | |
| | --- | --- | --- | --- | --- | --- | --- | --------------------------------------- | --- | --- | --- | --- | --- | ------- | |
| symbolicrepresentations,whilealsotakingadvan- |
| methodsincludechain-of-thoughtprompting(Wei |
| | tage of | the logical | faithfulness | | and | transparency | | | | | | | | | |
| | ------- | ----------- | ------------ | --- | --- | ------------ | --- | --- | --- | --- | --- | --- | --- | --- | |
| etal.,2022b;Wangetal.,2023)thatgeneratesex- |
| | offeredbysymbolicsolvers. | | | | Toimprovetheaccu- | | | | | | | | | | |
| | ------------------------- | --- | --- | --- | ----------------- | --- | --- | --- | --- | --- | --- | --- | --- | --- | |
| planationsbeforethefinalanswerandtheleast-to- |
| racyofthesymbolicparsing,wealsoincorporate mostprompting(Zhouetal.,2023)thatbreaksthe |
| theideaofself-refinementtoiterativelyrevisethe |
| | | | | | | | | problem | down | into | simpler | components | | that can | |
| | --------- | ------- | ---- | ----- | --------- | -------- | --- | --------------------- | ---- | ---- | ---------------------- | ---------- | --- | -------- | |
| | generated | logical | form | using | the error | messages | | | | | | | | | |
| | | | | | | | | besolvedindividually. | | | Boththeaboveapproaches | | | | |
| fromthesymbolicsolverasfeedback. |
| performreasoningdirectlyovernaturallanguage |
| We showcase the adaptability and effective- (NL),providinggreaterflexibilitythansymbolic- |
| | ness of | LOGIC-LM | | on five | logical | reasoning | | | | | | | | | |
| | --------- | ------------------------------------ | --- | ------- | ------- | --------- | --- | --------------- | --- | ------------------------------ | ---- | ----- | --------- | ------ | |
| | | | | | | | | basedreasoning. | | However,theintrinsiccomplexity | | | | | |
| | datasets: | ProofWriter(Tafjordetal.,2021),PrOn- | | | | | | | | | | | | | |
| | | | | | | | | and ambiguity | | of NL | also | bring | undesired | issues | |
| toQA(SaparovandHe,2023),FOLIO(Hanetal., suchasunfaithfulreasoningandhallucinations. |
| 2022),AR-LSAT (Zhongetal.,2022),andtheLog- Different from prior works, we use symbolic |
| icalDeductiondatasetfromBigBench(Srivastava |
| | | | | | | | | languageasthebasicunitofreasoning. | | | | | | Thiseffec- | |
| | ------------ | ------------------------------ | --- | --- | --- | --- | --- | ---------------------------------- | --- | ---------- | --- | ------------ | --- | ---------- | |
| | etal.,2022). | Thesedatasetscoverawiderangeof | | | | | | | | | | | | | |
| | | | | | | | | tively transfers | | the burden | | of executing | | complex, | |
| logicalreasoningproblems,including: precisereasoningfromLLMstomorereliable,in- |
| DeductiveReasoningproblems terpretableexternalsymbolicsolvers. Simultane- |
| • |
| First-OrderLogic(FOL)reasoningproblems ously, we leverage the strong in-context learning |
| • |
| | | | | | | | | ability | of LLMs | to formulate | | the | NL-based | prob- | |
| | --- | --- | --- | --- | --- | --- | --- | ------- | ------- | ------------ | --- | --- | -------- | ----- | |
| ConstraintSatisfactionProblems(CSP) |
| • |
| | | | | | | | | lem into | suitable | symbolic | | representations, | | thus | |
| | --- | --- | --- | --- | --- | --- | --- | -------- | -------- | -------- | --- | ---------------- | --- | ---- | |
| AnalyticalReasoning(AR)problems |
| • |
| maintainingthebenefitofflexibility. |
| Weintegratefourtypesofsymbolicinferencetools |
| Althoughpriorworks(Maoetal.,2019;Gupta |
| | tailoredtotheseproblems: | | | | 1)logicprogramming | | | | | | | | | | |
| | ------------------------ | --- | --- | --- | ------------------ | --- | --- | --- | --- | --- | --- | --- | --- | --- | |
| etal.,2020;Manhaeveetal.,2021;Caietal.,2021; |
| enginethatsupportsdeductivereasoningthrough |
| Tianetal.,2022;Pryoretal.,2023)alsopropose |
| forward/backwardchaining;2)FOLinferenceen- |
| | | | | | | | | neuro-symbolic | | methods | | to combine | neural | net- | |
| | --------- | ------- | --- | ----------- | ----- | --- | --- | -------------- | --- | ------- | --- | ---------- | ------ | ---- | |
| | gine that | derives | new | conclusions | based | on | FOL | | | | | | | | |
| workswithsymbolicreasoning,thesemethodssuf- |
| rulesandfacts,3)constraintoptimizationengine |
| | | | | | | | | fer from | limitations | | such | as hand-crafted | | or spe- | |
| | --- | --- | --- | --- | --- | --- | --- | -------- | ----------- | --- | ---- | --------------- | --- | ------- | |
| thatprovidessolversforCSPoverfinitedomains, |
| | | | | | | | | cialized | module | designs | that | are | not easily | gen- | |
| | --- | --- | --- | --- | --- | --- | --- | -------- | ------ | ------- | ---- | --- | ---------- | ---- | |
| and4)booleansatisfiabilityproblem(SAT)solver |
| | | | | | | | | eralizable, | or | brittleness | due | to | the difficulty | of | |
| | --- | --- | --- | --- | --- | --- | --- | ----------- | --- | ----------- | --- | --- | -------------- | --- | |
| thatsolvesanalyticalreasoningproblems. |
| | | | | | | | | optimization. | | Incontrast,weproposeamoregen- | | | | | |
| | --------------- | --- | ---- | ---- | ------------ | --- | -------- | ------------- | --- | ----------------------------- | --- | --- | --- | --- | |
| | Our evaluations | | show | that | the strategy | | of inte- | | | | | | | | |
| eralizableframeworkthatintegratesmodernLLMs |
| gratingLLMswithsymbolicsolversperformssig- |
| withsymboliclogicwithouttheneedfortraining |
| nificantlybetterthanpurelyrelyingonLLMsfor |
| ordesigningcomplexproblem-specificmodules. |
| | logical | reasoning, | with | an | average | improvement | | | | | | | | | |
| | ------- | ---------- | ---- | --- | ------- | ----------- | --- | --- | --- | --- | --- | --- | --- | --- | |
| of39.2%overthestandardpromptingand18.4% Tool-augmentedLanguageModels. Language |
| over the chain-of-thought prompting (§ 4.1). We modelshaveinherentlimitationssuchastheinabil- |
| alsofindthatLOGIC-LMbecomesincreasinglyef- itytoaccessup-to-dateinformation,takeactions, |
| fective as the required reasoning depth increases or perform precise mathematical reasoning. To |
| 3807 |
|
|
| Metals conduct electricity. No giant language model could have bad performance. In an antique car show, there are three vehicles: a tractor, |
| Insulators do not conduct electricity. If a language model has good performance, it is used by some researchers. a convertible, and a minivan. The tractor is the second- |
| If something is made of iron, then it is metal. A work used by some researchers should be popular. newest. The minivan is newer than the convertible. |
| Nails are made of iron. If BERT is a giant language model, then the same for GPT3. |
| BERT is a giant language model. Which of the following is true? |
| A) The tractor is the oldest. |
| Is the following statement true, false, or Is the following statement true, false, or unknown? GPT3 is popular. B) The convertible is the oldest. |
| unknown? Nails cannot conduct electricity. C) The minivan is the oldest. |
| Problem Formulator |
| Rules: Facts: Domain: Variables: |
| • Metal𝑥,True →ConductElectricity𝑥,True • ¬(∃𝑥(LanguageModel𝑥 ∧Giant𝑥 ∧¬GoodPerformance(𝑥))) 1: oldest tractor ∈ [1, 2, 3] |
| • MadeOfIron𝑥,True →Metal(𝑥,True) • ∀𝑥(LanguageModel𝑥 ∧GoodPerformance𝑥 →UsedbySomeReseachers(𝑥)) 3: newest minivan ∈ [1, 2, 3] |
| Facts: • ∀𝑥 UsedbySomeResearchers𝑥 →Popular𝑥 convertible ∈ [1, 2, 3] |
| • MadeOfIronNails,True • LanguageModelbert ∧𝐺𝑖𝑎𝑛𝑡bert →LanguageModelgpt3 ∧Giantgpt3 Constraints: |
| • ConductElectricityInsulator,False • Languagebert tractor == 2 |
| Query: • Giantbert minivan > convertible |
| • ConductElectricityNail,False Query: Polulargpt3 AllDifferentConstraint(tractor, minivan, convertible) |
| Symbolic |
| Logic Programming First-order Logic Prover Constraint Optimization SMT Solver |
| Reasoner |
| Result |
| ConductElectricityNail,True Entailment {convertible:1,tractor:2,minivan:3} |
| Interpreter |
| Answer The statement “Nails cannot The statement “GPT3 A) The convertible is the oldest. |
| conduct electricity” is false. is popular” is true. |
| Figure2: OverviewofourLOGIC-LMmodel,whichconsistsofthreemodules: (1)ProblemFormulatorgenerates |
| asymbolicrepresentationfortheinputproblemwithLLMsviain-contextlearning(2)SymbolicReasonerperforms |
| logicalinferenceontheformulatedproblem,and(3)ResultInterpreterinterpretsthesymbolicanswer. |
| addressthis,recentworkhasbeguntoaugmentlan- et al., 2022; He-Yueya et al., 2023; Jiang et al., |
| guagemodelswithaccesstoexternaltoolsandre- 2023). These works demonstrate the proficiency |
| sources,suchastheinformationretriever(Nakano ofLLMsintranslatingaconsiderablefractionof |
| et al., 2021; Shi et al., 2023; Lazaridou et al., mathematicalproblemsintoformalspecifications |
| 2022), calculator (Cobbe et al., 2021), code in- definedintoolslikeSymPy(Meureretal.,2017), |
| terpreter (Wang et al., 2022), planner (Liu et al., Isabelle/HOL(Paulson,1994),andLean(deMoura |
| 2023a),andotherpre-trainedmodels(Shenetal., etal.,2015). Mathematicalreasoningcanbecon- |
| 2023). Recentworks(Gaoetal.,2023;Chenetal., sidered a specialized subset of logical reasoning, |
| 2022) have achieved improved performance on primarilyfocusedonnumericdeductions. Dueto |
| arithmetic reasoning tasks by generating Python thisnumericspecificity,mathematicalproblemsare |
| programsthatspecifythereasoningprocedureas oftenmorereadilytranslatabletosymbolicforms. |
| chainedcommandsintheorderofexecution. How- Incontrast,logicalreasoningcoversawiderarray |
| ever, this idea has not been extended to logical ofproblemtypes,oftenrequiringadeeperunder- |
| reasoningproblems,primarilyduetothechallenge standing of world knowledge and commonsense |
| ofrepresentingtheirhighly“non-linear”reasoning foreffectiveparsingintosymbolicforms. Despite |
| procedure(e.g.,hypothesizing,case-by-caseanaly- plentyofworksstudyingmathematicalreasoning, |
| sis,andtheprocessofelimination)withfunctional ourworkpioneersinextendingtheconceptofauto- |
| programming. Our work provides a novel way formalizationtoabroaderrangeoflogicalreason- |
| to solve this within the framework of augmented ingtaskswithmodernLLMs. |
| LLMs. Insteadofparsingtheproblem-solvingpro- |
| cedureasprograms,weonlydescribetheproblem 3 LOGIC-LM |
| withsymboliclanguageusingLLMsandthenof- |
| AsshowninFigure2,theinputsofourmodelare |
| floadthereasoningtoexternalsymbolicsolvers. |
| alogicalreasoningproblemP describedinnatural |
| Auto-Formalization. The concept of convert- language, along with a goal G in the form of a |
| ingnaturallanguageintosymbolicrepresentations multiple-choiceorfree-formquestion. LOGIC-LM |
| hasbeenwidelyadoptedinauto-formalizationfor thenfollowsaproblemformulation-and-reasoning |
| mathematical reasoning (Wu et al., 2022; Drori paradigmtosolvetheproblem. |
| 3808 |
|
|
| IntheProblemFormulationstage,wepromptan Logic Programming (LP) Language. Deduc- |
| LLMtotranslatetheproblemandthegoalintoa tivereasoningtypicallystartsfromknownfactsand |
| task-specificsymboliclanguage. IntheSymbolic rules,anditerativelymakesnewinferencesuntilthe |
| Reasoningstage,wecalladeterministicsymbolic goalstatementcanbeprovedordisproved(Poole |
| Prolog |
| solver, e.g., a logic programming engine, to ob- and Mackworth, 2010). The logic pro- |
| tain a symbolic-represented answer. Finally, an gramminglanguage(ClocksinandMellish,2003; |
| LLM- or rule-based Result Interpreter is respon- Körneretal.,2022)isarguablythemostprominent |
| sible for translating the answer back to natural symboliclanguagetodescribedeductivereasoning |
| language. Using this approach, the reasoning is problems. We adopt its grammar to represent a |
| guaranteed to be faithful as long as the problem problemasfacts,rules,andqueries. |
| formulation is correct since the answer A is the Facts: a fact F is a simple statement with a |
| • |
| resultofexecutingdeterministicalgorithms(e.g., predicate and a set of arguments, formulated as |
| forward/backward-chaining)embeddedwithinthe |
| | | | | | | | P(a , | ,a ),whereP | | isthepredicatenameand | | | | |
| | ----------------- | --- | ------------------------- | --- | --- | --- | ------------- | ----------- | ------- | --------------------- | --- | ------- | ---- | |
| | | | | | | | 1 ··· | n | | | | | | |
| | symbolicreasoner. | | Comparedtopreviousmethods | | | | | | | | | | | |
| | | | | | | | each argument | | a i can | be a variable, | | entity, | num- | |
| basedonchain-of-thought,ourframeworkreduces ber,orbool. Forexample,Age(Peter,31)means |
| | the burden | of LLMs | by | shifting | their focus | from | | | | | | | | |
| | ---------- | ------- | --- | -------- | ----------- | ---- | --- | --- | --- | --- | --- | --- | --- | |
| “Peter’sageis31”,andMadeOfIron(Nails,True) |
| “solvingtheproblembyreasoningstep-by-step”to |
| representsthefact“Nailsaremadeofiron”. |
| “representingtheprobleminsymboliclanguage”. |
| | | | | | | | Rules: | rules | are written | in | the form | of | clauses: | |
| | --- | --- | --- | --- | --- | --- | ------ | ----- | ----------- | --- | -------- | --- | -------- | |
| • |
| | | | | | | | F 1 | F m | F m+1 | | F n ,whereeachF | | i is | |
| | --------------------- | --- | --- | --- | --- | --- | -------------------------------- | --- | ----- | ----- | --------------- | --- | ---- | |
| | | | | | | | ∧···∧ | → | | ∧···∧ | | | | |
| | 3.1 ProblemFormulator | | | | | | afactandtherulemeans“ifthefactsF | | | | | , | ,F | |
| | | | | | | | | | | | | 1 | m | |
| ··· |
| | | | | | | | aretrue,thenthefactsF | | | | F | arealsotrue.” | | |
| | -------------------------------------------- | --- | --- | --- | --- | --- | --------------------- | --- | --- | --- | --- | ------------- | --- | |
| | Intuitively,LLMsmaystrugglewithdirectlysolv- | | | | | | | | | m+1 | n | | | |
| ··· |
| | | | | | | | Queries: | a | query | Q is simply | another | | fact re- | |
| | ----------- | --------- | --- | --------- | -------- | ---- | -------- | --- | ----- | ----------- | ------- | --- | -------- | |
| | ing complex | reasoning | | problems. | However, | they | | | | | | | | |
| • |
| havedemonstratedanotableabilitytocomprehend quiredtobeprovedbasedonknownfactsandrules. |
| | textual | inputs and | translate | them | into formal | pro- | | | | | | | | |
| | ------- | ---------- | --------- | ---- | ----------- | ---- | ---------------------- | --- | --- | --- | ----------------- | --- | --- | |
| | | | | | | | First-OrderLogic(FOL). | | | | Whilethelogicpro- | | | |
| grams,suchasmathematicalequations(He-Yueya |
| gramminglanguageefficientlyrepresentscommon |
| | etal.,2023)orPythoncodes(Gaoetal.,2023). | | | | | We | | | | | | | | |
| | ---------------------------------------- | --- | --- | --- | --- | --- | --------- | --------- | --- | --------- | ------ | ---- | ------- | |
| | | | | | | | deductive | reasoning | | problems, | it may | fail | to rep- | |
| positthatthiscapabilitytoformulateproblemsinto |
| resentmorecomplexfirst-orderlogic(FOL)prob- |
| | different | languages | can | be extended | to symbolic | | | | | | | | | |
| | ---------------- | --------- | --------------------------- | ----------- | ----------- | --- | -------- | ------- | ----- | ------- | ------- | --- | ------- | |
| | | | | | | | lems. To | address | this, | we also | include | | the FOL | |
| | languagesaswell. | | Weleveragethefew-shotgener- | | | | | | | | | | | |
| grammar(Enderton,2001)inAppendixA.Aprob- |
| | alizationabilityofLLMstoachievethis. | | | | | Bypro- | | | | | | | | |
| | ------------------------------------ | --- | --- | --- | --- | ------ | ----------- | ------ | ---- | ------ | ------ | --------- | --- | |
| | | | | | | | lem is then | parsed | into | a list | of FOL | formulas, | | |
| vidingtheLLMwithdetailedinstructionsaboutthe |
| | | | | | | | which are | divided | into | Premises | (the | known | in- | |
| | --- | --- | --- | --- | --- | --- | --------- | ------- | ---- | -------- | ---- | ----- | --- | |
| grammarofthesymboliclanguage,alongsideafew |
| formationfromtheproblem)andConclusion(the |
| demonstrationsasin-contextexamples,weobserve |
| | | | | | | | unknownformulatobeproved). | | | | Anexamplesen- | | | |
| | --- | --- | --- | --- | --- | --- | -------------------------- | --- | --- | --- | ------------- | --- | --- | |
| thatLLMs,likeInstructGPT(Ouyangetal.,2022b) |
| tenceanditsFOLformulaaregiveninTable1. |
| andGPT-4(OpenAI,2023),caneffectivelyfollow |
| theinstructionstoidentifykeyentities,facts,and ConstraintSatisfaction(CSP). Constraintsat- |
| | rules present | in | the problem | statement, | | and then | | | | | | | | |
| | ------------- | --- | ----------- | ---------- | --- | -------- | --------- | -------- | ------ | --- | ------- | ----- | ---- | |
| | | | | | | | isfaction | problems | (CSPs) | | (Kumar, | 1992) | aims | |
| translate these elements into symbolic language to find the value assignment of a set of objects |
| followingourdefinedgrammar. that satisfy a number of constraints. A CSP |
| Specifically,weusefourdifferentsymbolicfor- is often defined as a triple (X,D,C), where |
| | | | | | | | X = x | , | ,x | | | | D = | |
| | --- | --- | --- | --- | --- | --- | ----- | --- | --- | --- | --- | --- | --- | |
| mulationstocoverfourcommontypesoflogical 1 n is a set of variables, |
| | | | | | | | { | ··· | } | | | | | |
| | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | |
| reasoning problems: deductive reasoning, first- D , ,D isasetoftheirrespectivedomains |
| | | | | | | | 1 | n | | | | | | |
| | --- | --- | --- | --- | --- | --- | ----- | --- | --- | --- | --- | --- | --- | |
| | | | | | | | { ··· | } | | | | | | |
| orderlogicreasoning,constraintsatisfactionprob- ofvalues,andC = C , ,C isasetofcon- |
| | | | | | | | | | { | 1 ··· | m } | | | |
| | --- | --- | --- | --- | --- | --- | --- | --- | --- | ----- | --- | --- | --- | |
| lem, and analytical reasoning. These formula- straints. Each variable x i can take on the values |
| tions provide a foundation for translating natu- inthenonemptydomainD . EveryconstraintC |
| | | | | | | | | | | i | | | j | |
| | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | |
| rallanguage-basedproblemstatements. Bydefin- is a pair t ,R , where t X is a subset of k |
| | | | | | | | | j | j | j | | | | |
| | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | |
| | | | | | | | | ⟨ | ⟩ | ⊂ | | | | |
| ingadditionalproblem-specificformulations,our variables and R j is a k-ary relation on the corre- |
| spondingsubsetofdomainsD |
| frameworkretainstheflexibilitytoaccommodatea j . Weusetheabove |
| widerrangeofreasoningtasks. Next,wewilldelve syntax to define a CSP problem as variables, do- |
| into the grammar of each symbolic formulation. mains, and constraints. An example is given in |
| | ExamplesofeachproblemtypeareinFigure2. | | | | | | bothFigure2andTable1. | | | | | | | |
| | -------------------------------------- | --- | --- | --- | --- | --- | --------------------- | --- | --- | --- | --- | --- | --- | |
| 3809 |
|
|
| Example |
| | Problem | Formulation | | | | | | | | | Solver | | Dataset | |
| | --------- | ----------- | -------------------------- | ---------- | --- | --- | ------------------ | ------------------- | ---------- | ----- | ------ | --- | ----------- | |
| | | | | NLSentence | | | | SymbolicFormulation | | | | | | |
| | | | Ifthecircuitiscompleteand | | | | Complete(Circuit, | | True) | | | | | |
| | Deductive | | | | | | | | | ∧ | | | ProntoQA, | |
| | | LP | thecircuithasthelightbulb | | | | Has(Circuit, | | LightBulb) | | Pyke | | | |
| | Reasoning | | | | | | | | | | | | ProofWriter | |
| | | | thenthelightbulbisglowing. | | | | Glowing(LightBulb, | | | True) | | | | |
| → |
| First-Order ACzechpersonwroteabook x2∃ x1(Czech(x1) Author(x2,x1) |
| | | FOL | | | | | ∃ Book(x2) | Publish(x2,1946)) | ∧ | | Prover9 | | FOLIO | |
| | ---------- | --- | --------------------------- | --- | --- | --- | ---------- | ----------------- | --------- | --- | ------- | --- | ----- | |
| | Logic | | in1946. | | | | | | | | | | | |
| | | | | | | | ∧ | ∧ | | | | | | |
| | | | Onashelf,therearefivebooks. | | | | blue_book | | 1,2,3,4,5 | | | | | |
| | Constraint | | | | | | | | | | python- | | | |
| CSP Thebluebookistotheright yellow_book ∈{ 1,2,3,4,5 } LogicalDeduction |
| | Satisfaction | | | | | | | | | | constraint | | | |
| | ------------ | --- | ------------------------ | --- | --- | --- | ---------------------- | --------- | ------- | ------ | ---------- | --- | ------- | |
| | | | oftheyellowbook. | | | | blue_book>yellow_book | | ∈{ | } | | | | |
| | | | | | | | repairs(Xena, | | radios) | | | | | |
| | Analytical | | Xenaandexactlythreeother | | | | | | | ∧ | | | | |
| | | SAT | | | | | Count([t:technicians], | | | t=Xena | Z3 | | AR-LSAT | |
| | Reasoning | | techniciansrepairradios | | | | | | | ̸ | | | | |
| | | | | | | | repairs(t, | radios))) | | == 3) | | | | |
| ∧ |
| Table1: Asummaryofthesymbolicformulations(withexamples)andsymbolicsolversweuseforthefivedatasets |
| inourstudy,representingfourdifferenttypesoflogicalreasoningproblems. |
| BooleanSatisfiability(SAT)Formulation. SAT FOL Prover. We use Prover92 as the FOL in- |
| Prover9isanautomatedtheorem |
| | istheproblemofdecidingifthereisanassignment | | | | | | ferenceengine. | | | | | | | |
| | ------------------------------------------- | --- | --- | --- | --- | --- | -------------- | --- | --- | --- | --- | --- | --- | |
| to the variables of a Boolean formula such that proverthatsupportsfirst-orderlogicandequational |
| theformulaissatisfied. Manyanalyticalreasoning logic. ItinitiallyconvertsFOLstatementstocon- |
| problemscanbeformulatedasSATproblems. We junctive normal form (CNF) and then performs |
| adopt the grammar defined in Ye et al. (2023) to resolution(Robinson,1965)ontheCNFtodeduce |
| formulateanSATproblem as(Φ, , ),where whetheraconclusionistrue,false,orunknown. |
| | | | P | | T Q | | | | | | | | | |
| | ----------------------------------------- | --- | --- | --- | --- | --- | --- | ------- | --- | ------- | ----- | ----- | -------------- | |
| | Φisasetofconstraintsdefinedunderthetheory | | | | | | , | | | | | | | |
| | | | | | | T | CSP | Solver. | | Solving | a CSP | is to | find value as- | |
| and isthequeryofinterest. |
| | Q | | | | | | signments | | for all | variables | that | satisfy | all given | |
| | ----- | ------------ | --- | ---------- | --- | ------- | ------------ | --- | -------- | --------- | ---- | ---------- | --------- | |
| | Table | 1 summarizes | the | four types | of | logical | | | | | | | | |
| | | | | | | | constraints. | | Commonly | | used | algorithms | for this | |
| reasoningproblems,theirtypicaldatasets,andthe taskincludebacktracking,constraintpropagation, |
| symbolicformulationusedtorepresenteachtypeof and local search variants. To this end, we incor- |
| problem. Wealsogiveanexampleofanaturallan- python-constraint3 |
| | | | | | | | porate | the | | | | package | which | |
| | --- | --- | --- | --- | --- | --- | ------ | --- | --- | --- | --- | ------- | ----- | |
| guagestatementwithitscorrespondingsymbolic |
| offerssolversforCSPsoverfinitedomains. |
| | formulationforeachtype. | | AppendixCshowsthe | | | | | | | | | | | |
| | ----------------------- | --- | ----------------- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | |
| full prompts we use for the problem formulator. SATSolver. ForsolvingSATproblems,weuse |
| | | | | | | | the | Z3 theorem | | prover | (de | Moura | and Bjørner, | |
| | --- | --- | --- | --- | --- | --- | --- | ---------- | --- | ------ | --- | ----- | ------------ | |
| ToteachLLMstobetteraligneachstatementwith |
| | | | | | | | 2008), | a | satisfiability | | modulo | theories | (SMT) | |
| | ----------------- | -------- | --- | -------- | --- | -------- | ------ | --- | -------------- | --- | ------ | -------- | ----- | |
| | its corresponding | symbolic | | form, we | use | the for- | | | | | | | | |
| solverdevelopedbyMicrosoft4. |
| TheSMTsolver |
| | mat SYMBOLIC_FORMULA | | | ::: NL_STATEMENT | | | | | | | | | | |
| | -------------------- | --- | --- | ---------------- | --- | --- | --- | --- | --- | --- | --- | --- | --- | |
| inin-contextexamplestoenablebettergrounding. providesalgorithmstodeterminewhetherasetof |
| | | | | | | | mathematicalformulasissatisfiable. | | | | | | Itgeneralizes | |
| | --- | --- | --- | --- | --- | --- | ---------------------------------- | --- | --- | --- | --- | --- | ------------- | |
| 3.2 SymbolicReasoner the SAT problems to more complex formulas in- |
| After the problem formulator parses the problem volving real numbers, integers, and various data |
| | | | | | | | structures | | such | as lists, | arrays, | bit | vectors, and | |
| | --------- | ----------- | -------- | --------------- | --- | --- | ---------- | --- | ---- | --------- | ------- | --- | ------------ | |
| | P and the | goal G into | symbolic | representations | | | | | | | | | | |
| Pˆ and Gˆ , we call a deterministic external solver strings. A lot of real-world analytical reasoning |
| dependingonthetask,toobtaintheanswerA. Ta- problemscanberepresentedasproblemsofsolv- |
| ingasystemofequations. |
| ble1summarizesthesymbolicsolversweusefor |
| eachtypeoflogicalreasoningproblem. |
| | | | | | | | 3.3 | Self-Refiner | | | | | | |
| | --- | --- | --- | --- | --- | --- | --- | ------------ | --- | --- | --- | --- | --- | |
| LPSystem. Fordeductivereasoning,weincor- Forcomplexproblems,generatingthecorrectlog- |
| poratethePykeexpertsystem(Frederiksen,2008), |
| | | | | | | | icalformmaybecomechallengingforLLMs. | | | | | | To | |
| | ----------------- | ---------- | ----------------------- | --- | --------- | ---- | ------------------------------------ | ----- | ------ | --------- | ---------- | --------------- | ------------ | |
| | which makes | inferences | based | on | the logic | pro- | | | | | | | | |
| | | | | | | | address | this, | we | introduce | a | self-refinement | mod- | |
| | gramminglanguage. | | Inresponsetoaquery,Pyke | | | | | | | | | | | |
| | | | | | | | ule | that | learns | to modify | inaccurate | | logical for- | |
| firstcreatesaknowledgebase,populatingitwith |
| 2https://www.cs.unm.edu/~mccune/prover9/ |
| | known facts | and rules. | Subsequently, | | it | applies | | | | | | | | |
| | ----------- | ---------- | ------------- | --- | --- | ------- | --- | --- | --- | --- | --- | --- | --- | |
| 3https://github.com/python-constraint/ |
| forward-andbackward-chainingalgorithmstoin- |
| python-constraint |
| fernewfactsandsubstantiatethegoal. 4https://github.com/Z3Prover/z3 |
| 3810 |
|
|
| mulationsusingtheerrormessagesfromthesym- eachpartrequiring0, 1, 2, 3,and 5hops |
| ≤ ≤ ≤ ≤ |
| bolic reasoner as feedback. Recent works (Chen ofreasoning,respectively. Weevaluatethehardest |
| etal.,2023;Madaanetal.,2023)haveadoptedsim- depth-5subset. Toreduceoverallexperimentation |
| ilarideastoimprovecodegeneration,byteaching costs, we randomly sample 600 examples in the |
| LLMstodebugtheirpredictedprogramsviafew- testsetandensureabalancedlabeldistribution. |
| shotdemonstrations. Hereweextendthisideato FOLIO (Han et al., 2022) is a challenging |
| refinegeneratedlogicrepresentations. Ifthesym- expert-written dataset for logical reasoning. The |
| bolicsolverreturnsanexecutionerror,weinstruct problemsaremostlyalignedwithreal-worldknowl- |
| the LLM to refine the incorrect logical form, by edgeandusehighlynaturalwordings,andtheques- |
| prompting it with the erroneous logic form, the tionsrequirecomplexfirst-orderlogicreasoningto |
| solver’serrormessage,andasetofdemonstrations solve. WeusetheentireFOLIOtestsetforevalua- |
| showingcommonerrorcases(e.g.,afreevariable tion,consistingof204examples. |
| isnotboundedtoanyquantifierinFOL)andtheir LogicalDeductionisachallenginglogicalrea- |
| remedies. Werunthisprocessiterativelyuntilei- soning task from the BigBench (Srivastava et al., |
| ther no error messages are returned, or the maxi- 2022)collaborativebenchmark. Theproblemsare |
| mumnumberofallowablerevisionsisreached. mostlyaboutdeducingtheorderofasequenceof |
| objectsfromaminimalsetofconditions. Weuse |
| 3.4 ResultInterpreter thefulltestsetconsistingof300examples. |
| Finally,theresultinterpretertranslatestheresults AR-LSAT(Zhongetal.,2022)isadatasetthat |
| returnedfromthesymbolicsolverbacktoanatural collects all analytical logic reasoning questions |
| language answer. For certain problems, this can fromtheLawSchoolAdmissionTestfrom1991to |
| beachievedthroughpredefinedrules;forexample, 2016. Weusethetestsetwhichhas231multiple- |
| mappingEntailmenttotrue. However,thispro- choice questions. AR-LSAT is particularly chal- |
| cesscanbemorecomplexforCSPs,e.g.,translat- lenging,withstate-of-the-artmodelsonlyachiev- |
| ing{convertible: 1,tractor: 2,minivan: 3}to“the ingperformanceslightlybetterthanrandomguess- |
| convertibleistheoldest.”. Tohandlethesevarying ing(Liangetal.,2022;Ribeiroetal.,2023a). |
| levelsofcomplexity,wedesignedbothrule-based Weconvertallexamplesintoastandardmultiple- |
| andLLM-basedresultinterpreters. Detailsofthe choiceformat,comprisingaproblemstatement,a |
| resultinterpreteraregiveninAppendixD. question,andpotentialanswers,asshowninFig- |
| ure2. Wealsoselect1-5examplesfromthetrain- |
| 4 Experiments ingsetofeachdatasetasin-contextexamples. De- |
| taileddatastatisticsareinAppendixB. |
| Datasets. Weevaluate LOGIC-LM onfivecom- |
| monlogicalreasoningdatasets,asfollows. Baselines. We compare our model against two |
| PrOntoQA(SaparovandHe,2023)isarecent baselinesthatdependsolelyonLLMsforlogical |
| syntheticdatasetcreatedtoanalyzethecapacityof reasoning: 1)Standard LLMs,whichleveragein- |
| LLMsfordeductivereasoning. Weusethehardest context learning to directly answer the question; |
| fictionalcharactersversionofthedataset,basedon and2)Chain-of-Thought(CoT)(Weietal.,2022b), |
| theresultsinSaparovandHe(2023). Eachversion which adopts a step-by-step problem-solving ap- |
| isdividedintodifferentsubsetsdependingonthe proach,generatingexplanationsbeforeproviding |
| number of reasoning hops required. We use the the final answer. We separately evaluate the set- |
| hardest5-hopsubsetforevaluation. Eachquestion tings that ChatGPT (gpt-3.5-turbo), GPT-3.5 |
| inPrOntoQAaimstovalidateanewfact’sveracity, (text-davinci-003) (Ouyang et al., 2022a) and |
| suchas“Trueorfalse: Alexisnotshy.”. GPT-4(gpt-4)(OpenAI,2023)serveastheunder- |
| ProofWriter (Tafjord et al., 2021) is another lying LLMs for all models. To ensure fair com- |
| commonlyuseddatasetfordeductivelogicalrea- parisons,weusethesamein-contextexamplesfor |
| soning. ComparedwithPrOntoQA,theproblems all models. For reproducible results, we set the |
| areexpressedinamorenaturalisticlanguageform. temperatureto0andselecttheresponsewiththe |
| Weusetheopen-worldassumption(OWA)subset highestprobabilityfromLLMs. Sinceallexamples |
| in which each example is a (problem, goal) pair areformedasmultiple-choicequestions,weeval- |
| and the label is one of {PROVED, DISPROVED, uatemodelperformancebasedontheaccuracyof |
| UNKNOWN}. Thedatasetisdividedintofiveparts, selectingthecorrectanswer. |
| 3811 |
|
|
| ChatGPT(gpt-3.5-turbo) GPT-3.5(text-davinci-003) GPT-4(gpt-4) |
| Dataset |
| Standard CoT Logic-LM Standard CoT Logic-LM Standard CoT Logic-LM |
| PrOntoQA 47.40 67.80 61.00 51.80 83.00 85.00 77.40 98.79 83.20 |
| ProofWriter 35.50 49.17 58.33 36.16 48.33 71.45 52.67 68.11 79.66 |
| FOLIO 45.09 57.35 62.74 54.60 57.84 61.27 69.11 70.58 78.92 |
| LogicalDeduction 40.00 42.33 65.67 41.33 48.33 62.00 71.33 75.25 87.63 |
| AR-LSAT 20.34 17.31 26.41 22.51 22.51 25.54 33.33 35.06 43.04 |
| Table2: Accuracyofstandardpromoting(Standard),chain-of-thoughtpromoting(CoT),andourmethod(LOGIC- |
| LM,withoutself-refinement)onfivereasoningdatasets. ThebestresultswithineachbaseLLMarehighlighted. |
| 4.1 MainResults GPT-3.5 GPT-4 |
| Dataset SR |
| Exe_Rate Exe_Acc Exe_Rate Exe_Acc |
| WereporttheresultsofLOGIC-LM (withoutself- |
| 99.4% 84.9 100.0% 83.2 |
| refinement)andbaselinesinTable2. For LOGIC- ProntoQA − + 100.0% 0.6 85.0 0.1 100.0% 83.2 |
| ↑ ↑ |
| LM,asymbolicsolverdoesnotreturnananswer |
| 87.3% 73.6 99.0% 79.6 |
| ProofWriter − |
| when there are grammar errors in the symbolic + 95.6% 8.3 74.1 0.5 99.0% 79.6 |
| ↑ ↑ |
| formulation. For these un-executable cases, we 66.7% 61.8 79.9% 80.4 |
| FOLIO − |
| + 84.3% 17.6 64.3 2.5 85.8% 5.9 79.9 0.5 |
| fallbackonusingchain-of-thoughttopredictthe ↑ ↑ ↑ ↓ |
| Logical 100.0% 62.0 100.0% 87.6 |
| answer. Wehavethreemajorobservations. − |
| Deduction + 100.0% 62.0 100.0% 87.6 |
| 1. Logic-LM significantly outperforms stan- |
| 11.3% 57.7 32.6% 60.0 |
| AR-LSAT − |
| dardLLMsandCoTacrossalldatasets.WithGPT- + 21.8% 10.5 60.3 2.6 39.8% 7.2 58.8 1.2 |
| ↑ ↑ ↑ ↓ |
| 3.5,ourmethodoutperformsstandardLLMonall |
| Table3: Analysisofaccuracyandexecutionstatusof |
| datasets, withanaverageimprovementof39.2%. |
| LOGIC-LM.Wepresentthepercentageofexecutable |
| This highlights the benefit of combining LLMs |
| logicalformulations(Exe_Rate)togetherwiththeaccu- |
| withexternalsymbolicsolversforlogicalreason- racyoftheexecution(Exe_Acc). SRrepresentsbefore |
| ing. LOGIC-LMalsoimprovesCoTbyalargemar- ( )andafter(+)self-refinement. |
| − |
| ginof18.4%onaverage,showingthatoffloading |
| thereasoningtosymbolicsolversgreatlyimproves |
| faithfulness compared with pure language-based thatCoTemulateshumanforward-chainreasoning: |
| reasoningwithCoT. beginning with known facts and sequentially de- |
| rivingnewconclusionsuntilthegoalismet. This |
| 2. GPT-4outperformsGPT-3.5byalargemargin |
| reasoning style aligns well with problems in the |
| of48.46%onaverageforthestandardprompting. |
| PrOntoQA and ProofWriter datasets. However, |
| This aligns with the assertion that the main en- |
| FOL and CSP problems often necessitate more |
| hancementofGPT-4liesinitsabilitytocarryout |
| sophisticated reasoning strategies that are “non- |
| complexreasoning(OpenAI,2023). Althoughthis |
| linear” compared to standard forward-chain rea- |
| mayindicatethatthelogicalreasoningcapability |
| soning. Theseincludehypothesizing,conditioning, |
| canbeboostedbyscalinguptheLLM,weobserve |
| recursive inference, and the process of elimina- |
| thatGPT-4stillmakesnumerousunfaithfulreason- |
| tion. ComparedtoCoT,theintegrationofsymbolic |
| ingerrors. Bydelegatingthereasoningtosymbolic |
| solvers is better suited to these reasoning styles, |
| solvers, our method can further improve GPT-4 |
| henceyieldingamoremarkedimprovementonFO- |
| byanaverageof24.98%and10.44%forstandard |
| LIO(+21.85%),LogicalDeduction(+45.67%),and |
| promptingandCoTprompting,respectively. |
| AR-LSAT(+24.14%). |
| 3. While integrating CoT generally enhances |
| LLM performance, we find its benefits compara- |
| 4.2 EffectivenessofProblemFormulator |
| tivelylesssubstantialorevennegativeonFOLIO, |
| LogicalDeduction,andAR-LSAT,withamodest We then evaluate how well LLM can translate a |
| improvement of 11.75%, 9.41%, and -3.2%, re- givenproblemintothesymbolicformulationused |
| spectively. On the contrary, the benefits of CoT byeachsymbolicsolver. InTable3,wereportthe |
| on ProntoQA and ProofWriter are 51.59% and percentageofsymbolicformulationsthatareexe- |
| 33.82%, respectively. A plausible explanation is cutablebythecorrespondingsymbolicsolverfor |
| 3812 |
|
|
| Accuracy Standard CoT Logic-LM CoT (GPT-3.5) Logic-LM (GPT-3.5) |
| | | | | | | | Accuracy | | CoT (GPT-4) | | Logic-LM (GPT-4) | | |
| | --- | --- | --- | --- | --- | --- | -------- | --- | ----------- | --- | ---------------- | --- | |
| 90 |
| | 81.7 | | | | | | 85 | | | | | | |
| | ---- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | |
| 77.3 |
| | 80 | | 73.6 | | | | | 78.92 | | | 79.9 | 79.41 | |
| | --- | --- | ---- | --- | ---- | ---- | --- | ----- | ----- | --- | ---- | ----- | |
| | | | | | 71.3 | 71.1 | 80 | | 78.43 | | | | |
| 7076.3 |
| 73.6 |
| 75 |
| 57.7 |
| | 60 | | 65.3 | | | | | | | | | | |
| | --- | ---- | ---- | --- | ---- | --- | --- | --- | --- | --- | --- | ----- | |
| | | 52.6 | | | 59.4 | | 70 | | | | | 70.58 | |
| 47.3 |
| | 50 | | | | | | | | | | 64.56 | 63.84 | |
| | --- | --- | --- | --- | --- | --- | --- | ----- | ----- | --- | ----- | ----- | |
| | | | | | | 51 | 65 | 61.27 | 62.25 | | | | |
| 38.3 |
| | 40 | | | | | 33.5 | | | | | | | |
| | --- | --- | --- | --- | --- | ---- | --- | --- | --- | --- | --- | --- | |
| 60 |
| 57.87 |
| 30 |
| 55 |
| | 0 | 1 | 2 | | 3 | 5 | Rounds | 0 | | 1 | 2 | 3 | |
| | --- | --- | --- | --- | --- | --- | ------ | --- | --- | --- | --- | --- | |
| Reasoning Depth |
| | | | | | | | GPT-3.5 | 66.7% | 79.4% | | 82.4% | 84.3% | |
| | --- | --- | --- | --- | --- | --- | ------- | ----- | ----- | --- | ----- | ----- | |
| Figure3: Accuracyofdifferentmodelsforincreasing GPT-4 79.9% 85.3% 85.3% 85.8% |
| sizeofreasoningdepthontheProofWriterdataset. |
| | | | | | | | Figure 4: | The | accuracy | for different | rounds | of self- | |
| | --- | --- | --- | --- | --- | --- | --------- | --- | -------- | ------------- | ------ | -------- | |
| refinement,withthecorrespondingexecutablerates. |
| | eachdataset(Exe_Rate). | | Generally,LLMdemon- | | | | | | | | | | |
| | ---------------------- | --- | ------------------- | --- | --- | --- | --- | --- | --- | --- | --- | --- | |
| strates high proficiency in transcribing problems ple,LOGIC-LMoutperformsCoTby7.1%,5.0%, |
| 1 |
| intosymbolicformats,evidencedbyitsnear100% |
| | | | | | | | 12.7%, | 20.0%, | and 39.4% | | on depth-0, | depth-1, | |
| | -------- | ------------ | ------------ | --- | --- | ----- | -------- | -------- | --------- | ------- | ----------- | -------- | |
| | Exe_Rate | on ProntoQA, | ProofWriter, | | and | Logi- | | | | | | | |
| | | | | | | | depth-2, | depth-4, | and | depth-5 | problems, | respec- | |
| calDeduction. However,thehighperformanceon tively. InLOGIC-LM,multi-steplogicalreasoning |
| thesedatasetsissomewhatanticipated,giventhat |
| isdelegatedtoexternalsymbolicsolvers,thereby |
| theirproblemsaremostlysyntheticallygenerated, |
| transitioningthechallengeofLLMfromproblem- |
| | limiting language | variability. | | When | it | comes to | | | | | | | |
| | ----------------- | ------------ | --- | ---- | --- | -------- | ------------------------------- | --- | --- | --- | --------------- | --- | |
| | | | | | | | solvingtoproblemrepresentation. | | | | Ideally,thecom- | | |
| datasets comprising real-world, expertly crafted plexity of formally representing a problem state- |
| | problems, | such as FOLIO | | and AR-LSAT, | | GPT- | | | | | | | |
| | --------- | ------------- | --- | ------------ | --- | ---- | --- | --- | --- | --- | --- | --- | |
| mentinlogicalformshouldremainrelativelycon- |
| | 4’s performance | is notably | | less promising, | | with | | | | | | | |
| | --------------- | ---------- | --- | --------------- | --- | ---- | --- | --- | --- | --- | --- | --- | |
| stant,regardlessofwhetherthequestionsrequire |
| Exe_Ratescoresof79.9%and32.6%respectively. |
| | | | | | | | simple | or complex | reasoning. | | The trends | in Fig- | |
| | ---------------- | ----------- | --- | --- | -------- | ----- | --------------------------- | ---------- | ---------- | --- | ---------------- | ------- | |
| | This discrepancy | underscores | | the | inherent | chal- | | | | | | | |
| | | | | | | | ure3validatethisassumption. | | | | Theperformanceof | | |
| lengesassociatedwithconvertingreal-worldprob- |
| | | | | | | | Standard | andCoT | declinesprecipitouslywiththe | | | | |
| | --- | --- | --- | --- | --- | --- | -------- | ------ | ---------------------------- | --- | --- | --- | |
| lemsintotheirlogicalequivalents. |
| | | | | | | | escalation | of problem | | complexity. | However, | this | |
| | --- | --- | --- | --- | --- | --- | ---------- | ---------- | --- | ----------- | -------- | ---- | |
| Exe_Rateonlyreflectsthegrammarcorrectness |
| | | | | | | | trendislessprominentfor | | | LOGIC-LM,indicating | | | |
| | -------------- | ----- | ------- | ------ | --- | -------- | ----------------------- | --- | --- | ------------------- | --- | --- | |
| | of the logical | form. | We also | report | the | accuracy | | | | | | | |
| thattherobustreasoningcapabilitiesprovidedby |
| | of the executable | samples | | (Exe_Acc) | to | measure | | | | | | | |
| | ----------------- | ------- | --- | --------- | --- | ------- | --- | --- | --- | --- | --- | --- | |
| externalsolverssubstantiallymitigateperformance |
| the semantic correctness. We find that logical degradationforcomplexreasoningproblems. |
| formsgeneratedbyGPT-4generallyachievehigh |
| Exe_Acc,evenforthemostchallengingAR-LSAT 4.4 ImpactofSelf-Refinement |
| dataset. Suchperformanceaccentuatesthepoten- InTable3,wefindthatself-refinementiseffective |
| tialofsymbolicsolversinbolsteringthemodel’s infixingthein-executablesymbolicformulations, |
| logicalreasoningprowess, contingentonthepre- increasingtheExe_Rateby5.01onaverage. |
| For |
| cisetranslationofproblemsintosymbolicforms. an in-depth analysis, we then evaluate the accu- |
| racyandExe_Rateacrossdifferentroundsofself- |
| 4.3 RobustnessofReasoning |
| refinementonFOLIO,namely,0(norefinement), |
| Incorporatingsymbolicsolversalsoleadstomore 1,2,and3rounds. TheresultsareinFigure4. |
| robust reasoning. To illustrate this, we report Wefindthatastheroundsofself-refinementin- |
| theperformanceofLOGIC-LM andbaselinesfor crease,thepercentageofexecutableformulations |
| questions of varying complexity levels. We ran- consistentlyincreases,leadingtoanenhancement |
| domly selected 300 examples from each subset in the final performance. This suggests that self- |
| of ProofWriter, ensuring a balanced label distri- refinementservesasaneffectivetoolinaidingthe |
| bution. The problems in these subsets require 0, LLM to accurately frame the problem. However, |
| <=1,<=2,<=3,and<=5hopsofreasoning,respec- theaccuracytendstostagnateinsubsequentrounds, |
| tively. The results, shown in Figure 3, indicate even though the Exe_Rate continues to increase. |
| thatLOGIC-LMbecomesincreasinglyeffectiveas This can be attributed to the type of feedback re- |
| therequiredreasoningdepthincreases. Forexam- ceived by the self-refiner, which is the error mes- |
| 3813 |
| |
| Problem: "Stranger Things" is a popular Netflix show. If a Netflix show is popular, Karen will binge-watch it. |
| If and only if Karen binge-watches a Netflix show, she will download it. Karen does not download "Black Mirror". |
| "Black Mirror" is a Netflix show. If Karen binge-watches a Netflix show, she will share it to Lisa. |
| Question: Is the following statement true, false, or uncertain? "Black Mirror" is popular. |
| (A) True (B) False (C) Uncertain |
| Generated symbolic Formulations: |
| Predicates: |
| NetflixShow(x) # x is a Netflix show. |
| Popular(x) # x is popular. |
| BingeWatch(x, y) # x binge-watches y. |
| Download(x, y) # x downloads y. |
| Share(x, y, z) # x shares y to z. |
| Premises: |
| NetflixShow(strangerThings) ∧ Popular(strangerThings) # "Stranger Things" is a popular Netflix show. |
| ∀x (NetflixShow(x) ∧ Popular(x) → BingeWatch(karen, x)) # If a Netflix show is popular, Karen will binge-watch it. |
| ∀x (NetflixShow(x) ∧ BingeWatch(karen, x) ↔ Download(karen, x)) # If and only if Karen … , she will download it. |
| NetflixShow(blackMirror) ∧ ¬Download(karen, blackMirror) # Karen does not … "Black Mirror" is a Netflix show. |
| ∀x (NetflixShow(x) ∧ BingeWatch(karen, x) → Share(karen, x, lisa)) # If Karen … , she will share it to Lisa. |
| Conclusion: |
| Popular(blackMirror) # "Black Mirror" is popular. |
| Predicted answer: B |
| Figure5: AnexampleofthegeneratedsymbolicrepresentationandthepredictedanswerbyLOGIC-LM. |
| sagefromthesymbolicsolver. Thisfeedbackaids expressionsaccurately. Inthiscase,themodelfails |
| in converting “invalid” symbolic representations to distinguish between the meanings of “below” |
| intovalidones. However,avalidsymbolicrepre- and “above”, resulting in an incorrect constraint |
| sentationdoesnotnecessarilyequatetoa“correct” Dan > Eve. Example4exemplifiesGPT-4’schal- |
| problemformulationthataccuratelyrepresentsthe lenge with fully grasping the rules of FOL gram- |
| problem. This issue could be tackled by enhanc- mar, evidenced by the invalid generated formula: |
| ingtheself-refinertoincorporatefeedbackbeyond Rating(subway, y) y > 9. Theseerrorcases |
| ∧ |
| the error message, e.g., a reward signal from an underscore that transforming problems into logi- |
| additionalmoduleevaluatingtheaccuracyofagen- cal forms remains a challenging task for modern |
| eratedsymbolicform. Weleavethisasapromising LLMs,duetotheintricaciesofFOLformulation, |
| directionforfutureexploration. the innate flexibility of natural language, and the |
| complexityofglobalproblemcomprehension. |
| 4.5 CaseStudy |
| InFigure5,weshowanexampleofthesymbolic 5 ConclusionandFutureWork |
| representationsgeneratedbyGPT-4,togetherwith |
| thepredictedanswer. Ingeneral,LOGIC-LMhas In this work, we propose a novel approach to ad- |
| demonstrated a potent capacity to interpret com- dress logical reasoning problems by combining |
| plexproblemsintosymbolicforms. Nonetheless, largelanguagemodelswithsymbolicsolvers. We |
| there remain certain difficulties in accurately un- introduce Logic-LM, one instantiation of such a |
| derstandingthesemanticsoftheproblem. framework, and demonstrate how it significantly |
| We further analyze some error cases in Fig- improvesperformanceoverpureLLMsandchain- |
| ure 6 of Appendix E. Example 1 shows a case of-thoughtpromptingtechniques. |
| whereGPT-4generatesanincorrectFOLrepresen- WhileLogic-LMhasproventobeacapablesys- |
| tation, stemming from its inability to define ap- tem,itcanbefurtherimprovedwithextensionto |
| propriatepredicates. Here,insteadofcreatingthe moreflexibleandpowerfullogicsystems. Forex- |
| predicate EasternWildTurkey, the model gener- ample,statisticalrelationallearning(SRL)systems |
| atesaconstant,WildTurkey(eastern),inwhich such as Markov logic networks (Richardson and |
| WildTurkey is the predicate and eastern is the Domingos,2006)andprobabilisticsoftlogic(Bach |
| constant. Whilethisrepresentationisvalidiniso- et al., 2017) have demonstrated great promise in |
| lation, it does not interact well with subsequent reasoning under uncertainty and integration with |
| constants. Thisinconsistencyisarecurringissue ourframeworkwouldenableevenmoreadaptive |
| inGPT-4’ssymbolicformgeneration,illustrating problem-solving capabilities. Additionally, our |
| thatthemodelsometimesstrugglestomaintainan method can be extended to reasoning problems |
| overarching understanding of the problem when requiringcommonsense,whichremainsasignifi- |
| forminglogicalsymbols. Example3highlightsa cantchallengeastheyoftenrequirereasoningover |
| case where GPT-4 struggles to interpret specific complexandambiguousrules. |
| 3814 |
| |
| | Limitations | | | | | | | and probabilistic | | soft | logic. | Journal | of Machine | | |
| | ----------- | --- | --- | --- | --- | --- | --- | ----------------- | --- | ---- | ------ | ------- | ---------- | --- | |
| LearningResearch(JMLR),18(1):1–67. |
| | We identify | two | main | limitations | of | LOGIC-LM. | | | | | | | | | |
| | ----------- | --- | ---- | ----------- | --- | --------- | --- | --- | --- | --- | --- | --- | --- | --- | |
| TomB.Brown,BenjaminMann,NickRyder,Melanie |
| | First, LOGIC-LM | | relies | on translating | | reasoning | | | | | | | | | |
| | --------------- | --- | ------ | -------------- | --- | --------- | --- | -------- | ----- | ------- | -------- | --------- | --- | ------ | |
| | | | | | | | | Subbiah, | Jared | Kaplan, | Prafulla | Dhariwal, | | Arvind | |
| problemsintologicalformatsthatcanbetackledby |
| Neelakantan,PranavShyam,GirishSastry,Amanda |
| | symbolicsolvers. | | Asaconsequence,themodel’s | | | | | | | | | | | | |
| | ---------------- | --- | ------------------------- | --- | --- | --- | --- | ------- | -------- | -------- | --- | ----- | ------------- | --- | |
| | | | | | | | | Askell, | Sandhini | Agarwal, | | Ariel | Herbert-Voss, | | |
| applicabilityisinherentlyboundedbytheexpres- Gretchen Krueger, Tom Henighan, Rewon Child, |
| | | | | | | | | Aditya | Ramesh, | Daniel | M. | Ziegler, | Jeffrey | Wu, | |
| | --- | --- | --- | --- | --- | --- | --- | ------ | ------- | ------ | --- | -------- | ------- | --- | |
| sivenessofthesymbolicsolver,forexample,notall |
| ClemensWinter,ChristopherHesse,MarkChen,Eric |
| problemscanbeeasilyencodedinfirst-orderlogic. |
| Sigler,MateuszLitwin,ScottGray,BenjaminChess, |
| | Nevertheless, | | this limitation | can | be | mitigated | by | | | | | | | | |
| | ------------- | --- | --------------- | --- | --- | --------- | --- | ----------- | ----------- | --- | ------- | --- | ----------- | --- | |
| | | | | | | | | Jack Clark, | Christopher | | Berner, | Sam | McCandlish, | | |
| integratingamorediversesetofsymbolicsolvers. Alec Radford, Ilya Sutskever, and Dario Amodei. |
| | | | | | | | | 2020. | Language | models | are | few-shot | learners. | In | |
| | ------------------- | -------------------------------- | --- | -------- | --------------- | --- | --- | ----------- | -------- | ------ | ------ | ---------- | --------- | ------ | |
| | Theflexibledesignof | | | LOGIC-LM | facilitatesthis | | | | | | | | | | |
| | | | | | | | | Proceedings | | of the | Annual | Conference | on | Neural | |
| | integration. | Thewiderangeofreasoningtasksthat | | | | | | | | | | | | | |
| InformationProcessingSystems(NeurIPS). |
| | wecaninstantiateour | | | LOGIC-LM | frameworkon | | | | | | | | | | |
| | ------------------- | --- | --- | -------- | ----------- | --- | --- | --- | --- | --- | --- | --- | --- | --- | |
| showsitsgeneralapplicability. Le-Wen Cai, Wang-Zhou Dai, Yu-Xuan Huang, Yu- |
| | | | | | | | | Feng | Li, Stephen | H. | Muggleton, | | and Yuan | Jiang. | |
| | -------- | -------- | ---- | --------------- | --- | ---------- | --- | ---------------------------------------------- | ----------- | -------- | ---------- | ----------- | --------- | ------ | |
| | Second, | LOGIC-LM | | depends | on | in-context | | | | | | | | | |
| | | | | | | | | 2021. | Abductive | learning | | with ground | knowledge | | |
| | learning | coupled | with | self-refinement | | to convert | | | | | | | | | |
| | | | | | | | | base. InProceedingsofthe30thInternationalJoint | | | | | | | |
| anaturallanguage(NL)problemintothesymbolic ConferenceonArtificialIntelligence(IJCAI),pages |
| | representation. | | While | this method | has | proven | to | 1815–1821. | | | | | | | |
| | --------------- | --- | ----- | ----------- | --- | ------ | --- | ---------- | --- | --- | --- | --- | --- | --- | |
| beeffective,itmayfacedifficultieswhendealing |
| | | | | | | | | Wenhu | Chen, | Xueguang | Ma, | Xinyi | Wang, | and | |
| | --- | --- | --- | --- | --- | --- | --- | ----- | ----- | -------- | --- | ----- | ----- | --- | |
| withlogicalrepresentationswithintricategrammar |
| | | | | | | | | William | W. | Cohen. | 2022. | Program | of thoughts | | |
| | --- | --- | --- | --- | --- | --- | --- | ------- | --- | ------ | ----- | ------- | ----------- | --- | |
| structures, such as probabilistic soft logic. This prompting: Disentangling computation from rea- |
| | | | | | | | | soning | for numerical | | reasoning | | tasks. | CoRR, | |
| | ----------------------------------------- | --- | ---------- | ------------ | --- | ------- | --- | --------------- | ------------- | --- | --------- | --- | ------ | ----- | |
| | arises from | the | difficulty | in conveying | | complex | | | | | | | | | |
| | grammaticalrulestothelanguagemodelthrough | | | | | | | abs/2211.12588. | | | | | | | |
| alimitednumberofdemonstrationswithinacon- |
| | | | | | | | | Xinyun Chen, | Maxwell | | Lin, | Nathanael | Schärli, | and | |
| | --- | --- | --- | --- | --- | --- | --- | ------------ | ------- | --- | ---- | --------- | -------- | --- | |
| strainedcontextsize. Asapotentialsolution,future DennyZhou.2023. Teachinglargelanguagemodels |
| workscouldexplorethedevelopmentofspecialized toself-debug. CoRR,abs/2304.05128. |
| modulestoenhancethemappingbetweenNLand |
| PeterClark,OyvindTafjord,andKyleRichardson.2020. |
| symbolic language, e.g., fine-tuning LLMs with Transformersassoftreasonersoverlanguage. InPro- |
| syntheticdatageneratedviasymbolicsolvers. ceedingsofthe29thInternationalJointConference |
| onArtificialIntelligence(IJCAI),pages3882–3890. |
| EthicsStatement |
| WilliamFClocksinandChristopherSMellish.2003. |
| | | | | | | | | ProgramminginPROLOG. | | | | SpringerScience&Busi- | | | |
| | --- | --- | --- | --- | --- | --- | --- | -------------------- | --- | --- | --- | --------------------- | --- | --- | |
| Theuseoflargelanguagemodelsrequiresasignifi- |
| nessMedia. |
| cantamountofenergyforcomputationfortraining, |
| | | | | | | | | Karl Cobbe, | Vineet | Kosaraju, | | Mohammad | Bavarian, | | |
| | --- | --- | --- | --- | --- | --- | --- | ----------- | ------ | --------- | --- | -------- | --------- | --- | |
| whichcontributestoglobalwarming(Strubelletal., |
| 2019). Our work performs few-shot in-context JacobHilton,ReiichiroNakano,ChristopherHesse, |
| | | | | | | | | andJohnSchulman.2021. | | | Trainingverifierstosolve | | | | |
| | --- | --- | --- | --- | --- | --- | --- | --------------------- | --- | --- | ------------------------ | --- | --- | --- | |
| learninginsteadoftrainingmodelsfromscratch,so |
| | | | | | | | | mathwordproblems. | | | CoRR,abs/2110.14168. | | | | |
| | ---------------------------------- | --- | --- | --- | --- | -------- | --- | ----------------- | --- | --- | -------------------- | --- | --- | --- | |
| | theenergyfootprintofourworkisless. | | | | | Thelarge | | | | | | | | | |
| languagemodelswhoseAPIweuseforinference, LeonardoMendonçadeMouraandNikolajS.Bjørner. |
| | | | | | | | | 2008. | Z3:anefficientSMTsolver. | | | | InProceedingsof | | |
| | --- | --- | --- | --- | --- | --- | --- | ----- | ------------------------ | --- | --- | --- | --------------- | --- | |
| especiallyGPT-4,consumesignificantenergy. |
| the14thInternationalConferenceofToolsandAlgo- |
| rithmsfortheConstructionandAnalysisofSystems |
| Acknowledgements |
| (TACAS),volume4963ofLectureNotesinComputer |
| Science,pages337–340. |
| ThisworkwassupportedbytheNationalScience |
| LeonardoMendonçadeMoura,SoonhoKong,Jeremy |
| | FoundationAward#2048122. | | | Theviewsexpressed | | | | | | | | | | | |
| | ------------------------ | --- | --- | ----------------- | --- | --- | --- | ------- | ------ | ---------- | --- | --------- | --- | ------- | |
| | | | | | | | | Avigad, | Floris | van Doorn, | | and Jakob | von | Raumer. | |
| arethoseoftheauthorsanddonotreflecttheoffi- |
| | | | | | | | | 2015. | Theleantheoremprover(systemdescription). | | | | | | |
| | --- | --- | --- | --- | --- | --- | --- | ----- | ---------------------------------------- | --- | --- | --- | --- | --- | |
| cialpolicyorpositionoftheUSgovernment. |
| InProceedingsofthe25thInternationalConference |
| | | | | | | | | on Automated | | Deduction | (ICAD), | | volume | 9195 of | |
| | --- | --- | --- | --- | --- | --- | --- | ------------ | --- | --------- | ------- | --- | ------ | ------- | |
| LectureNotesinComputerScience,pages378–388. |
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| | --- | --- | --- | --- | --- | --- | ------------------------ | --- | --- | --- | -------------------- | --- | --- | |
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| | --------------- | -------------------- | --- | --- | --- | --- | --- | --- | --- | --- | |
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| | ----------- | --------- | --------- | ----- | ---------- | --- | --- | --- | --- | --- | |
| eratingnaturallanguageproofswithverifier-guided |
| | search. | InProceedingsofthe2022Conferenceon | | | | | | | | | |
| | ------- | ---------------------------------- | --- | --- | --- | --- | --- | --- | --- | --- | |
| 3818 |
| |
| | A | SyntaxforFirst-orderLogic(FOL) | | | | | | C.1 | PrOntoQAPrompts | | | | | |
| | --- | ------------------------------ | --- | --- | --- | --- | --- | --- | --------------- | --- | --- | --- | --- | |
| StandardIn-ContextLearning |
| Name FOLNotation Context: Jompuses are not shy. Jompuses are yumpuses. |
| | | | | | | | | ( more | context | here | ) | | | |
| | --- | -------- | --- | --- | ---------------- | --- | --- | ----------- | ------- | ------------- | --------- | ------------ | -------------- | |
| | | | | | | | | Z·u·m·puses | are | rompuse·s·.· | Max | is a yumpus. | | |
| | | Constant | | | lowercaseletters | | | | | | | | | |
| | | | | | | | | Question: | Is | the following | statement | | true or false? | |
| | | Variable | | | x,y,z, | | | | | | | | | |
| Max is sour. |
| ··· |
| | | Atom | | | P(a | , | ,a ) | Options: | | | | | | |
| | --- | ---- | --- | --- | --- | ----- | ---- | -------- | --- | --- | --- | --- | --- | |
| | | | | | | 1 ··· | n | | | | | | | |
| A) True |
| B) False |
| | | Negation | | | | P | | | | | | | | |
| | --- | -------- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | |
| ¬ |
| | | | | | | | | The correct | | option | is: B | | | |
| | --- | --- | --- | --- | --- | --- | --- | ----------- | --- | ------ | ----- | --- | --- | |
| | | | | | P | P | | | | | | | | |
| 1 2 |
| | | Conjunction | | | | ∧ | | | | | | | | |
| | --- | ----------- | --- | --- | --- | ------- | --- | ------------------------- | --- | ----- | ------------ | --------- | ----------------- | |
| | | | | | P | , | , P | | | | | | | |
| | | | | | | 1 ∧ ··· | ∧ n | Chain-of-ThoughtPrompting | | | | | | |
| | | | | | P | P | | | | | | | | |
| | | | | | | 1 2 | | Task Description: | | Given | a problem | statement | as | |
| | | Disjunction | | | | ∨ | | | | | | | | |
| | | | | | P | , | , P | contexts, | the | task | is to answer | a | logical reasoning | |
| | | | | | | 1 ∨ ··· | ∨ n | question. | | | | | | |
| Implication P 1 P 2 Context: Jompuses are not shy. Jompuses are yumpuses. |
| → |
| | | | | | | | | ( more | context | here | ) | | | |
| | --- | ----------- | --- | --- | --- | --- | --- | ----------- | ------- | ------------ | --- | ------------ | --- | |
| | | Equivalence | | | P | P | | Z·u·m·puses | | rompuse·s·.· | | | | |
| | | | | | | 1 | 2 | | are | | Max | is a yumpus. | | |
| ↔ |
| ExistentialQuantifier xP(x, ) Question: Is the following statement true or false? |
| | | | | | ∃ | | ··· | Max is | sour. | | | | | |
| | --- | ------------------- | --- | --- | --- | ----- | --- | -------- | ----- | --- | --- | --- | --- | |
| | | UniversalQuantifier | | | | xP(x, | ) | | | | | | | |
| | | | | | ∀ | | ··· | Options: | | | | | | |
| A) True |
| | | Table4: | | First-OrderLogicGrammar. | | | | B) False | | | | | | |
| | --- | ------- | --- | ------------------------ | --- | --- | --- | ----------- | --------- | -------------- | ------------ | ------ | ------------ | |
| | | | | | | | | Reasoning: | Max | is a | yumpus. Each | yumpus | is a dumpus. | |
| | | | | | | | | ( more | reasoning | here | ) | | | |
| | | | | | | | | T·u·m·puses | are | not sour·.··So | Max | is not | sour. | |
| B DatasetStatistics |
| | | | | | | | | The correct | | option | is: B | | | |
| | --- | --- | --- | --- | --- | --- | --- | ----------- | --- | ------ | ----- | --- | --- | |
| Logic-LM |
| | | Dataset | | Reasoning | | TestSize | #Opts | | | | | | | |
| | --- | ------- | --- | --------- | --- | -------- | ----- | --- | --- | --- | --- | --- | --- | |
| PrOntoQA Deductive 500 2 Task Description: You are given a problem description |
| | | | | | | | | and a | question. | The | task is | to: | | |
| | --- | --- | --- | --- | --- | --- | --- | ----- | --------- | --- | ------- | --- | --- | |
| ProofWriter Deductive 600 3 1) define all the predicates in the problem |
| | | | | | | | | 2) parse | the | problem | into logic | rules | based on | |
| | --- | ----- | --- | --- | --- | --- | --- | ----------- | --- | ---------- | ---------- | ----- | -------- | |
| | | FOLIO | | FOL | | 204 | 3 | the defined | | predicates | | | | |
| LogicalDeduction CSP 300 3,5,7 3) write all the facts mentioned in the problem |
| | | | | | | | | 4) parse | the | question | into | the logic | form | |
| | --- | ------- | --- | --- | --- | --- | --- | ----------- | ------- | ------------ | ---------- | --------- | ---- | |
| | | AR-LSAT | | AR | | 230 | 5 | | | | | | | |
| | | | | | | | | Context: | Each | jompus | is fruity. | | | |
| | | | | | | | | ( more | context | here | ) | | | |
| | | | | | | | | R·o·m·puses | | zumpuse·s·.· | | | | |
| Table5: Statisticsofthelogicalreasoningdatasets. are Alex is a tumpus. |
| | | | | | | | | Question: | True | or false: | Alex | is not | shy. | |
| | --- | --- | --- | --- | --- | --- | --- | --------- | ---- | --------- | ---- | ------ | ---- | |
| Predicates: |
| C PromptExamples Jompus(\$x, bool) ::: Does x belong to Jompus? |
| | | | | | | | | ( more | predicates | here | ) | | | |
| | --- | --- | --- | --- | --- | --- | --- | -------------- | ---------- | --------- | ------- | -------- | ---------- | |
| | | | | | | | | Z·u·m·pus(\$x, | | | ·D·o·es | | | |
| | | | | | | | | | | bool) ::: | | x belong | to Zumpus? | |
| Inthissectionweprovideexamplesoftheprompts |
| | usedforeachdatasetandmethod. | | | | | Promptsforstan- | | Facts: | | | | | | |
| | ---------------------------- | --- | --- | --- | --- | --------------- | --- | -------------- | --- | ----- | --- | --- | --- | |
| | | | | | | | | Tumpuses(Alex, | | True) | | | | |
| dardin-contextlearningcontain2demonstrations |
| Rules: |
| | consisting | | of 3 | parts each: | a | context, | a question, | | | | | | | |
| | ----------- | --- | --------------------------------- | ----------- | --- | -------- | ----------- | -------------- | ----- | --------- | ----------- | ----- | ----- | |
| | | | | | | | | Jompus($x, | | True) >>> | Fruity($x, | True) | | |
| | | | | | | | | ( more | rules | here | ) | | | |
| | andoptions. | | Promptsforchain-of-thoughtprompt- | | | | | D·u·m·pus(\$x, | | ·>·>·> | | | | |
| | | | | | | | | | | True) | Rompus(\$x, | | True) | |
| ingcontain2demonstrationsconsistingof5parts |
| Query: |
| | each: | ataskdescription,acontext,aquestion,op- | | | | | | Shy(Alex, | False) | | | | | |
| | --------------------------------------- | --------------------------------------- | --- | --- | --- | ---------------- | --- | --------- | ------ | --- | --- | --- | --- | |
| | tions,andachainofreasoning. | | | | | PromptsforLogic- | | | | | | | | |
| | LMcontain2demonstrationswith5partseach: | | | | | | a | | | | | | | |
| taskdescription,acontext,aquestion,options,and |
| | adomain-specificsymbolicprogram. | | | | | | Forbrevity, | | | | | | | |
| | -------------------------------- | --- | --- | --- | --- | --- | ----------- | --- | --- | --- | --- | --- | --- | |
| weshowonlyasingledemonstrationforeachset- |
| tinginthefollowingsections. |
| 3819 |
| |
| | C.2 ProofWriterPrompts | | | | | | Logic-LM | | | | | | |
| | ---------------------- | --- | --- | --- | --- | --- | -------- | --- | --- | --- | --- | --- | |
| StandardIn-ContextLearning Task Description: You are given a problem description |
| | | | | | | | and a question. | | The task | is to: | | | |
| | --- | --- | --- | --- | --- | --- | --------------- | --- | -------- | ------ | --- | --- | |
| Context: The cow is blue. The cow is round. 1) define all the predicates in the problem |
| | | | | | | | 2) parse | the problem | into | logic | rules | based on | |
| | ------------ | ------------ | --------- | ------------------ | ------ | ------------- | ----------- | ------------ | ----- | --------- | ----- | ----------- | |
| | ( more | context here | | ) | | | | | | | | | |
| | I·f··the cow | is cold | a·n·d· | the cow | visits | the lion then | the defined | predicates | | | | | |
| | | | | | | | 3) write | all the | facts | mentioned | in | the problem | |
| | the lion | sees the | squirrel. | | | | | | | | | | |
| | | | | | | | 4) parse | the question | | into the | logic | form | |
| | Question: | Based on | the | above information, | | is the | | | | | | | |
| following statement true, false, or unknown? Context: Anne is quiet. Erin is furry. |
| | The tiger | is not | young. | | | | ( more | context here | | ) | | | |
| | --------- | ------ | ------ | --- | --- | --- | ---------- | ------------ | --------- | --- | --- | --- | |
| | | | | | | | A·l·l· red | people are | y·o·u·ng. | | | | |
| Options: |
| | A) True | | | | | | Question: | Based on | the | above information, | | is the | |
| | ----------- | ------ | --- | --- | --- | --- | ----------- | --------- | ------ | ------------------ | --- | -------- | |
| | B) False | | | | | | following | statement | true, | false, | or | unknown? | |
| | C) Unknown | | | | | | Anne is | white. | | | | | |
| | The correct | option | is: | B | | | Predicates: | | | | | | |
| | | | | | | | Quiet($x, | bool) | ::: Is | x quiet? | | | |
| | | | | | | | Furry($x, | bool) | ::: Is | x furry? | | | |
| Chain-of-ThoughtPrompting |
| | | | | | | | ( more | predicates | here | ) | | | |
| | --- | --- | --- | --- | --- | --- | ------------ | ---------- | ---------- | ------ | --- | --- | |
| | | | | | | | W·h·i·te($x, | bool) | ::: Is··x· | white? | | | |
| Task Description: Given a problem statement as Young($x, bool) ::: Is x young? |
| | contexts, | the task | is | to answer | a logical | reasoning | | | | | | | |
| | ------------ | ------------ | --------- | ------------------ | ------------- | ------------- | --------------- | ----------- | ------------- | ------- | --------- | ---------------- | |
| | question. | | | | | | Facts: | | | | | | |
| | | | | | | | Quite(Anne, | True) | ::: | Anne is | quiet. | | |
| | Context: | The cow | is blue. | The | cow is round. | | ( more | facts here | ) | | | | |
| | | | | | | | W·h·i·te(Harry, | True)··:·:: | | | | | |
| | ( more | context here | | ) | | | | | | Harry | is white. | | |
| | I·f··the cow | is cold | a·n·d· | the cow | visits | the lion then | | | | | | | |
| | the lion | sees the | squirrel. | | | | Rules: | | | | | | |
| | | | | | | | Young($x, | True) | >>> Furry($x, | | True) | ::: Young people | |
| | Question: | Based on | the | above information, | | is the | are | furry. | | | | | |
| | following | statement | true, | false, | or | unknown? | ( more | rules here | ) | | | | |
| The tiger is not young. R·e·d·($x, True) >>> ·Y·o·ung($x, True) ::: All red people |
| | | | | | | | are | young. | | | | | |
| | --- | --- | --- | --- | --- | --- | --- | ------ | --- | --- | --- | --- | |
| Options: |
| | A) True | | | | | | Query: | | | | | | |
| | -------- | --- | --- | --- | --- | --- | ----------- | ----- | --- | ------- | ----- | --- | |
| | B) False | | | | | | White(Anne, | True) | ::: | Anne is | white | | |
| C) Unknown |
| | Reasoning: | The tiger | likes | the | cow. | | | | | | | | |
| | -------------- | --------- | ------------- | ------- | ------ | ---------- | --- | --- | --- | --- | --- | --- | |
| | The tiger | likes | the squirrel. | | | | | | | | | | |
| | ( more | reasoning | here | ) | | | | | | | | | |
| | I·f··something | is | nice ·a·n·d | it sees | the | tiger then | | | | | | | |
| | it is young. | So | the tiger | is | young. | | | | | | | | |
| | The correct | option | is: | B | | | | | | | | | |
| 3820 |
| |
| | C.3 | FOLIOPrompts | | | | | Logic-LM | | | | | | | |
| | --- | ------------ | --- | --- | --- | --- | -------- | --- | --- | --- | --- | --- | --- | |
| StandardIn-ContextLearning Task Description: Given a problem description and a |
| | | | | | | | question. | | The task | is to | parse | the problem | and | |
| | --- | --- | --- | --- | --- | --- | --------- | --- | -------- | ----- | ----- | ----------- | --- | |
| Context: All people who regularly drink coffee are the question into first-order logic formulas. |
| dependent on caffeine. The grammar of the first-order logic formula is |
| | ( | more context | here | ) | | | defined | as | follows: | | | | | |
| | --------- | ------------ | --------- | --------- | --- | ------------ | ---------- | ------------ | -------- | ----- | ----- | --- | --- | |
| | I·f··Rina | | per·s·o·n | | | | 1) logical | conjunction: | | expr1 | expr2 | | | |
| | | is | not a | dependent | on | caffeine and | | | | | ∧ | | | |
| a student, then Rina is either a person dependent 2) logical disjunction: expr1 expr2 |
| disjuncti∨on: |
| on caffeine and a student, or neither a person 3) logical exclusive expr1 expr2 |
| dependent on caffeine nor a student. 4) logical negation: expr1 ⊕ |
| expr¬2: |
| | | | | | | | 5) expr1 | implies | | expr1 | expr2 | | | |
| | --------- | ----- | ------ | ------------------ | --- | ------ | -------- | ------- | ---- | --------------- | ----- | ----- | --- | |
| | | | | | | | 6) expr1 | if and | only | if expr2:→expr1 | | expr2 | | |
| | Question: | Based | on the | above information, | | is the | | | | | | ↔x | | |
| following statement true, false, or uncertain? Rina 7) logical universal quantification: |
| quantification∀: |
| is a person who jokes about being addicted to 8) logical existential x |
| caffeine or unaware that caffeine is a drug. Output format: logic form ::: descripti∃on |
| | | | | | | | Context: | All people | who | regularly | drink | coffee | are | |
| | --- | --- | --- | --- | --- | --- | -------- | ---------- | --- | --------- | ----- | ------ | --- | |
| Options: |
| | A) True | | | | | | dependent | on caffeine. | | | | | | |
| | -------- | --- | --- | --- | --- | --- | --------- | ------------ | ---- | --- | --- | --- | --- | |
| | B) False | | | | | | ( more | context | here | ) | | | | |
| C) Uncertain I·f··Rina is not a per·s·o·n dependent on caffeine and a |
| | | | | | | | student, | then | Rina is | either | a person | dependent | | |
| | --- | ------- | ---------- | --- | --- | --- | ----------- | ----------- | ---------- | -------------- | -------- | --------- | --- | |
| | | | | | | | on caffeine | and | a student, | or | neither | a person | | |
| | The | correct | option is: | A | | | | | | | | | | |
| | | | | | | | dependent | on caffeine | | nor a student. | | | | |
| Chain-of-ThoughtPrompting Question: Based on the above information, is the |
| | | | | | | | following | statement | true, | false, | or | uncertain? | | |
| | --- | --- | --- | --- | --- | --- | --------- | --------- | ----- | ------ | --- | ---------- | --- | |
| Task Description: Given a problem statement as Rina is either a person who jokes about being |
| contexts, the task is to answer a logical reasoning addicted to caffeine or is unaware that caffeine |
| | question. | | | | | | is a drug. | | | | | | | |
| | --------- | --- | ------------ | ------- | -------- | ---- | ----------- | --- | --- | --- | --- | --- | --- | |
| | Context: | The | Blake McFall | Company | Building | is a | Predicates: | | | | | | | |
| commercial warehouse listed on the National Register Dependent(x) ::: x is a person dependent on caffeine |
| | of Historic | | Places. | | | | ( more | predicates | here | ) | | | | |
| | ----------- | ------------ | -------- | --------- | --- | --- | ------------- | ---------- | ------ | ---------- | --- | --- | --- | |
| | ( | more context | here | ) | | | S·t·u·dent(x) | ::: | x is a | ·s·t·udent | | | | |
| | J·o·h·n | | Em·m·e·t | | | | | | | | | | | |
| | | works | at the | Building. | | | | | | | | | | |
| Premises: |
| Question: Based on the above information, is the x (Drinks(x) Dependent(x)) ::: All people who |
| ∀ regularly→drink |
| following statement true, false, or uncertain? coffee are dependent on |
| | The | Blake | McFall Company | Building | is | located in | caffeine. | | | | | | | |
| | --------- | ----- | -------------- | -------- | --- | ---------- | ------------- | --------------- | -------------- | ------- | -------- | ---------- | ----- | |
| | Portland, | | Oregon. | | | | ( more | premises | here | ) | | | | |
| | | | | | | | ·x··(Jokes(x) | | Una·w·a·re(x)) | | | | | |
| | | | | | | | | | | | ::: No | one who | jokes | |
| | Options: | | | | | | ∀ about | bei→ng¬addicted | | to | caffeine | is unaware | | |
| | A) True | | | | | | that | caffeine | is | a drug. | | | | |
| B) False |
| | C) Uncertain | | | | | | Conclusion: | | | | | | | |
| | ------------ | --- | --- | --- | --- | --- | ----------- | --- | ------------- | --- | -------- | --------- | --- | |
| | | | | | | | Jokes(rina) | | Unaware(rina) | | ::: Rina | is either | a | |
| Reasoning: The Blake McFall Company Building is person w⊕ho jokes about being addicted to |
| another name for the Emmet Building. caffeine or is unaware that caffeine is a drug. |
| | ( | more reasoning | here | ) | | | | | | | | | | |
| | ------------- | -------------- | ---------- | --------- | --- | ----------- | --- | --- | --- | --- | --- | --- | --- | |
| | T·h·e·refore, | | | ·M·c·Fall | | | | | | | | | | |
| | | | the Blake | Company | | Building is | | | | | | | | |
| | located | in | Portland, | Oregon. | | | | | | | | | | |
| | The | correct | option is: | A | | | | | | | | | | |
| 3821 |
| |
| | C.4 LogicalDeductionPrompts | | | | | Logic-LM | | | | |
| | --------------------------- | --- | --- | --- | --- | -------- | --- | --- | --- | |
| StandardIn-ContextLearning Task Description: You are given a problem description. |
| | | | | | | The task | is to parse | the problem as | a constraint | |
| | --- | --- | --- | --- | --- | -------- | ----------- | -------------- | ------------ | |
| Context: The following paragraphs each describe a satisfaction problem, defining the domain, |
| set of seven objects arranged in a fixed order. variables, and contraints. |
| | ( more | context here | ) | | | | | | | |
| | ------ | ------------ | --- | --- | --- | --- | --- | --- | --- | |
| E·v·e· A·d·a·. Context: The following paragraphs each describe a |
| | finished | below | Rob | finished below | Joe. | | | | | |
| | -------- | ----- | --- | -------------- | ---- | ------------ | ------- | ------------- | ------------ | |
| | | | | | | set of three | objects | arranged in a | fixed order. | |
| Question: Which of the following is true? ( more context here ) |
| | | | | | | T·h·e· minivan | is newer···than | the convertible. | | |
| | --- | --- | --- | --- | --- | -------------- | --------------- | ---------------- | --- | |
| Options: |
| | | | | | | Question: | Which of the | following is | true? | |
| | --------------- | ------ | --- | --- | --- | --------- | ------------ | ------------ | ----- | |
| | A) Ana finished | third. | | | | | | | | |
| | B) Eve finished | third. | | | | | | | | |
| | C) Ada finished | third. | | | | Options: | | | | |
| D) Dan finished third. A) The station wagon is the second-newest. |
| E) Rob finished third. B) The convertible is the second-newest. |
| | | | | | | C) The minivan | is the | second-newest. | | |
| | --------------- | ------ | --- | --- | --- | -------------- | ------ | -------------- | --- | |
| | F) Amy finished | third. | | | | | | | | |
| | G) Joe finished | third. | | | | | | | | |
| Domain: |
| | The correct | option is: | A | | | 1: oldest | | | | |
| | ----------- | ---------- | --- | --- | --- | --------- | --- | --- | --- | |
| 3: newest |
| Chain-of-ThoughtPrompting |
| Variables: |
| | | | | | | station\_wagon | [IN] [1, | 2, 3] | | |
| | --- | --- | --- | --- | --- | -------------- | -------- | ----- | --- | |
| Task Description: Given a problem statement as convertible [IN] [1, 2, 3] |
| | contexts, | the task is | to answer | a logical | reasoning | minivan | [IN] [1, 2, 3] | | | |
| | --------- | ----------- | --------- | --------- | --------- | ------- | -------------- | --- | --- | |
| question. |
| Constraints: |
| Context: The following paragraphs each describe a station\_wagon == 1 ::: The station wagon is the |
| | set of five | objects | arranged | in a fixed | order. | oldest. | | | | |
| | ----------- | ------- | -------- | ---------- | ------ | ------- | --- | --- | --- | |
| ( more context here ) minivan > convertible ::: The minivan is newer than |
| | T·h·e· raven | is the thi·r·d· | from | the left. | | the | convertible. | | | |
| | ------------ | --------------- | ---- | --------- | --- | --------------------------------------- | ------------ | --- | ------------ | |
| | | | | | | AllDifferentConstraint([station\_wagon, | | | convertible, | |
| Question: Which of the following is true? minivan]) ::: All vehicles have different |
| values. |
| Options: |
| | A) The quail | is the | rightmost. | | | Query: | | | | |
| | ------------ | ------ | ---------- | --- | --- | ------ | --- | --- | --- | |
| B) The owl is the rightmost. A) station\_wagon == 2 ::: The station wagon is the |
| | C) The raven | is the | rightmost. | | | second-newest. | | | | |
| | ------------ | ------ | ---------- | --- | --- | -------------- | --- | --- | --- | |
| D) The falcon is the rightmost. B) convertible == 2 ::: The convertible is the |
| | E) The robin | is the | rightmost. | | | | | | | |
| | ------------ | ------ | ---------- | --- | --- | --- | --- | --- | --- | |
| second-newest. |
| | | | | | | C) minivan | == 2 ::: The | minivan is | the second-newest | |
| | ------------- | -------------- | ------------- | ----------------- | --------- | ---------- | ------------ | ---------- | ----------------- | |
| | Reasoning: | The owl is | the leftmost. | This | means owl | . | | | | |
| | is not the | rightmost. | | | | | | | | |
| | ( more | reasoning here | ) | | | | | | | |
| | T·h·i·s means | raven is | a·l·s·o not | the rightmost. | So, | | | | | |
| | the answer | is: A) The | quail | is the rightmost. | | | | | | |
| | The correct | option is: | A | | | | | | | |
| 3822 |
| |
| | C.5 AR-LSATPrompts | | | | | | Logic-LM | | | | | | | |
| | ------------------ | --- | --- | --- | --- | --- | -------- | --- | --- | --- | --- | --- | --- | |
| StandardIn-ContextLearning Task Description: You are given a problem description. |
| | | | | | | | The | task is | to parse | the problem | as a | constraint | | |
| | --- | --- | --- | --- | --- | --- | --- | ------- | -------- | ----------- | ---- | ---------- | --- | |
| Context: During a single week, from Monday through satisfaction problem, defining the domain, |
| | Friday, | tours | will | be conducted | of | a company's | variables, | | and contraints. | | | | | |
| | ------- | ---------- | ----------- | ------------ | ----------- | ----------- | ---------- | -------- | --------------- | --- | --------- | ------- | --- | |
| | three | divisions: | Operations, | | Production, | and | | | | | | | | |
| | | | | | | | Context: | A travel | magazine | has | hired six | interns | - | |
| | Sales. | Exactly | five | tours will | be | conducted | | | | | | | | |
| that week, one each day. ( more context here Farber, Gombarick, Hall, Jackson, Kanze, and |
| divi·s·i·on |
| ) If the Operations is toured on Lha - to assist in covering three stories: |
| ·T·h·ursday, then the Production division is Romania, Spain, and Tuscany. ( more context here |
| Tus·c·a·ny. |
| | toured | on Friday. | | | | | | ) Jackson | is | assigned | to | Kanze | is | |
| | --------------- | ---------- | ------ | --------- | ------ | ------- | --- | --------------- | --- | -------- | --- | ----- | --- | |
| | | | | | | | | ·n·o·t assigned | to | Spain. | | | | |
| | Question: Which | one | of the | following | CANNOT | be true | | | | | | | | |
| of the week's tour schedule? Question: Which one of the following interns CANNOT |
| | | | | | | | | be assigned | to | Tuscany? | | | | |
| | --- | --- | --- | --- | --- | --- | --- | ----------- | --- | -------- | --- | --- | --- | |
| Options: |
| | A) The division | that | is | toured | on Monday | is also | Options: | | | | | | | |
| | --------------- | ----------- | --- | ------ | --------- | ------- | -------- | --------- | --- | --- | --- | --- | --- | |
| | toured | on Tuesday. | | | | | (A) | Farber | | | | | | |
| | B) The division | that | is | toured | on Monday | is also | (B) | Gombarick | | | | | | |
| | toured | on Friday. | | | | | (C) | Hall | | | | | | |
| (D) Kanze |
| | C) The division | that | is | toured | on Tuesday | is also | | | | | | | | |
| | --------------- | ------------ | --- | ------ | ------------ | ------- | ------------- | --- | --- | --- | --- | --- | --- | |
| | toured | on Thursday. | | | | | (E) | Lha | | | | | | |
| | D) The division | that | is | toured | on Wednesday | is also | | | | | | | | |
| | toured | on Friday. | | | | | Declarations: | | | | | | | |
| E) The division that is toured on Thursday is also stories = EnumSort([Romania, Spain, Tuscany]) |
| | | | | | | | assistants | | = EnumSort([photographer, | | | writer]) | | |
| | ----------- | ---------- | ----- | --- | --- | --- | ---------- | ----------------------- | ------------------------- | ------ | ---------------- | -------- | --- | |
| | toured | on Friday. | | | | | | | | | | | | |
| | | | | | | | ( | more declarations | | here ) | | | | |
| | | | | | | | t·r·a·ined | = Function([inte·r·n·s] | | | -> [assistants]) | | | |
| | The correct | option | is: C | | | | | | | | | | | |
| Constraints: |
| Chain-of-ThoughtPrompting trained(Gombarick) == trained(Lha) ::: Gombarick and |
| | | | | | | | | Lha will | be trained | in | the same | field | | |
| | --- | --- | --- | --- | --- | --- | --- | -------- | ---------- | --- | -------- | ----- | --- | |
| Task Description: Given a problem statement as trained(Farber) != trained(Kanze) ::: Farber and |
| contexts, the task is to answer a logical reasoning Kanze will be trained in different fields |
| | question. | | | | | | ( | more contraints | here | ) | | | | |
| | --------------- | -------- | --- | ---------- | ------ | ------- | -------------------- | --------------- | ---- | ---------- | ----------- | --- | -------- | |
| | | | | | | | a·s·s·igned(Jackson) | | == | T·u·s·cany | ::: Jackson | is | assigned | |
| | Context: During | a single | | week, from | Monday | through | | to Tuscany | | | | | | |
| Friday, tours will be conducted of a company's assigned(Kanze) != Spain ::: Kanze is not assigned |
| | three | divisions: | Operations, | | Production, | and | | to Spain | | | | | | |
| | ------ | ---------- | ----------- | ---------- | ----------- | ------------ | -------- | -------- | --- | --- | --- | --- | --- | |
| | Sales. | Exactly | five | tours will | be | conducted | | | | | | | | |
| | that | week, one | each | day. ( | more | context here | Options: | | | | | | | |
| divi·s·i·on |
| ) If the Operations is toured on is_unsat(assigned(Farber) == Tuscany) ::: (A) |
| ·T·h·ursday, then the Production division is is_unsat(assigned(Gombarick) == Tuscany) ::: (B) |
| | toured | on Friday. | | | | | | | | | | | | |
| | ------ | ---------- | --- | --- | --- | --- | ------------------------ | --- | --- | ----------- | -------- | ------- | --- | |
| | | | | | | | is_unsat(assigned(Hall) | | | == Tuscany) | ::: | (C) | | |
| | | | | | | | is_unsat(assigned(Kanze) | | | == | Tuscany) | ::: (D) | | |
| Question: Which one of the following CANNOT be true is_unsat(assigned(Lha) == Tuscany) ::: (E) |
| | of the | week's | tour | schedule? | | | | | | | | | | |
| | ------ | ------ | ---- | --------- | --- | --- | --- | --- | --- | --- | --- | --- | --- | |
| Options: |
| A) The division that is toured on Monday is also D ResultInterpreterImplementation |
| | toured | on Tuesday. | | | | | | | | | | | | |
| | --------------- | ------------ | --- | ------ | ---------- | ------- | ----------- | -------- | ------ | ------------ | --- | -------- | ----- | |
| | B) The division | that | is | toured | on Monday | is also | | | | | | | | |
| | | | | | | | For | PrOntoQA | and | ProofWriter, | the | Pyke | logic | |
| | toured | on Friday. | | | | | | | | | | | | |
| | C) The division | that | is | toured | on Tuesday | is also | | | | | | | | |
| | | | | | | | programming | | engine | returns | the | inferred | value | |
| | toured | on Thursday. | | | | | | | | | | | | |
| D) The division that is toured on Wednesday is also of the variable in the query or Unknown if the |
| | toured | on Friday. | | | | | | | | | | | | |
| | --------------- | ---------- | --- | ------ | ----------- | ------- | -------------------------------- | ------ | --- | ----------- | --- | -------- | --- | |
| | | | | | | | variable | cannot | be | determined. | For | example, | for | |
| | E) The division | that | is | toured | on Thursday | is also | | | | | | | | |
| | toured | on Friday. | | | | | | | | | | | | |
| | | | | | | | thequeryConductElectricity(Nail, | | | | | x),Pyke | | |
| Reasoning: Since Thursday and Friday already have mayreturnx =True. Bycomparingwiththegoal |
| | tours planned, | only | Monday, | Tuesday | and | Wednesday | | | | | | | | |
| | ---------------- | ----- | ----------- | ------- | --- | --------- | --------- | --- | ------------------------ | --- | --- | --- | ------- | |
| | | | | | | | | | ConductElectricity(Nail, | | | | False), | |
| | tours need | to be | determined. | | | | statement | | | | | | | |
| | ( more reasoning | here | | ) | | | | | | | | | | |
| A··d·ifferent ·i·s· we can know that goal to be proved is False. |
| | | division | | toured | on Thursday. | | | | | | | | | |
| | --- | -------- | --- | ------ | ------------ | --- | --- | --- | --- | --- | --- | --- | --- | |
| Therefore, the final answer is C. ForFOLIO,theFOLinferenceenginedirectlyre- |
| turnstheveracitylabelofthegoalasENTAILMENT, |
| | The correct | option | is: C | | | | | | | | | | | |
| | ----------- | ------ | ----- | --- | --- | --- | -------------- | --- | --- | ----------- | --- | ----- | ------ | |
| | | | | | | | CONTRADICTION, | | and | CONTINGENT, | | which | can be | |
| mappedtoTrue,False,andUnknown,respectively. |
| | | | | | | | For | LogicalDeduction, | | the | solver | returns | all the | |
| | --- | --- | --- | --- | --- | --- | ------------------------------------- | ----------------- | ---- | ------ | -------- | ------------- | ------- | |
| | | | | | | | possiblevalueassignmentsinanarray. | | | | | Wewrite | | |
| | | | | | | | rules | to parse | each | option | into the | corresponding | | |
| | | | | | | | valueandcheckitisinthegeneratedarray. | | | | | | ForAR- | |
| LSAT,weattempttoseparatelyproveeachoption |
| tofindthecorrectanswer. |
| | | | | | | | E | ExampleGenerationsof | | | LOGIC-LM | | | |
| | --- | --- | --- | --- | --- | --- | --- | -------------------- | --- | --- | -------- | --- | --- | |
| 3823 |
|
|
| Example 1 |
| Problem: There are six types of wild turkeys: Eastern wild turkey, Osceola wild turkey, Gould's wild turkey, Merriam's wild turkey, Rio Grande |
| wild turkey, and Ocellated wild turkey. Tom is not an Eastern wild turkey. Tom is not an Osceola wild turkey. Tom is also not a Gould's wild |
| turkey, or a Merriam's wild turkey, or a Rio Grande wild turkey. Tom is a wild turkey. |
| Question: Based on the above information, is the following statement true, false, or uncertain? Tom is an Ocellated wild turkey. |
| (A) True (B) False (C) Uncertain |
| Predicted logic programs: |
| Premises: |
| WildTurkey(eastern) ∧ WildTurkey(osceola) ∧ WildTurkey(gould) ∧ WildTurkey(merriams) ∧ WildTurkey(rio_grande) ∧ WildTurkey(ocellated) |
| → ∀x (WildTurkey(x) → (OcellatedWildTurkey(x) ∨ EasternWildTurkey(x) ∨ OsceolaWildTurkey(x) ∨ GouldsWildTurkey(x) ∨ MerriamsWildTurkey(x) |
| ∨ RioGrandeWildTurkey(x))) |
| WildTurkey(tom) ∧ ¬EasternWildTurkey(tom) ∧ ¬OsceolaWildTurkey(tom) ∧ ¬GouldsWildTurkey(tom) ∧ ¬MerriamsWildTurkey(tom) ∧ |
| ¬RioGrandeWildTurkey(tom) |
| Conclusion: |
| OcellatedWildTurkey(tom) |
| | Predicted answer: | C → A | | | |
| | ------------------ | ------ | --- | --- | |
| Example 2 |
| Problem: A Japanese game company created the game the Legend of Zelda. All games in the Top 10 list are made by Japanese game companies. If a |
| game sells more than one million copies, then it will be selected into the Top 10 list. The Legend of Zelda sold more than one million copies. |
| Question: Based on the above information, is the following statement true, false, or uncertain? The Legend of Zelda is in the Top 10 list. |
| (A) True (B) False (C) Uncertain |
| Predicted logic programs: |
| Premises: |
| JapaneseGameCompany(legend_of_zelda) |
| ∀x (Top10List(x) → JapaneseGameCompany(x)) |
| | ∀x (SellMoreThanOneMillionCopies(x) | | → Top10List(x)) | | |
| | ----------------------------------- | --- | --------------- | --- | |
| SellMoreThanOneMillionCopies(legend_of_zelda) |
| Conclusion: |
| Top10List(legend_of_zelda) |
| Predicted answer: A |
| Example 3 |
| Problem: The following paragraphs each describe a set of five objects arranged in a fixed order. The statements are logically consistent within |
| each paragraph. In a golf tournament, there were five golfers: Rob, Eve, Eli, Amy, and Dan. Dan finished second. Amy finished below Eve. Dan |
| finished above Eve. Amy finished above Eli. |
| Question: Which of the following is true? |
| (A) Rob finished third (B) Eve finished third (C) Eli finished third (D) Amy finished third (E) Dan finished third |
| Predicted logic programs: |
| | Domain: | Variables: | | Constraints: | |
| | -------- | ------------------------ | --- | ------------------------------------------------- | |
| | 1: first | Rob [IN] [1, 2, 3, 4, 5] | | Dan == 2 | |
| | 5: last | Eve [IN] [1, 2, 3, 4, 5] | | Amy > Eve | |
| | | Eli [IN] [1, 2, 3, 4, 5] | | Dan > Eve→ Dan < Eve | |
| | | Amy [IN] [1, 2, 3, 4, 5] | | Amy < Eli | |
| | | Dan [IN] [1, 2, 3, 4, 5] | | AllDifferentConstraint([Rob, Eve, Eli, Amy, Dan]) | |
| Query: |
| (A) Rob == 3 (B) Eve == 3 (C) Eli == 3 (D) Amy == 3 (E) Dan == 3 |
| | Predicted answer: E | → B | | | |
| | ------------------- | --- | --- | --- | |
| Example 4 |
| Problem: If the restaurant is listed in Yelp’s recommendations, then the restaurant does not receive many negative reviews. All restaurants with |
| a rating greater than 9 are listed in Yelp’s recommendations. Some restaurants that do not provide take-out service receive many negative |
| reviews. All restaurants that are popular among local residents have ratings greater than 9. Subway has a rating greater than 9 or is popular |
| among local residents. |
| Question: Based on the above information, is the following statement true, false, or uncertain? Subway provides take-out service and does not |
| receive many negative reviews. |
| (A) True (B) False (C) Uncertain |
| Predicted logic programs: |
| Premises: |
| ∀x (Listed(x) → ¬NegativeReviews(x)) |
| | ∀x (Rating(x, y) ∧ | y > 9 → Listed(x)) | → ∀x (GreaterThanNine(x) → Listed(x)) | | |
| | -------------------------------- | ----------------------- | ---------------------------------------------- | --------------- | |
| | ∃x (¬TakeOut(x) ∧ | NegativeReviews(x)) | | | |
| | ∀x (Popular(x) → (Rating(x, y) ∧ | | y > 9)) → ∀x (Popular(x) → GreaterThanNine(x)) | | |
| | Rating(subway, y) ∧ | y > 9 ∨ Popular(subway) | → GreaterThanNine(subway) ∨ | Popular(subway) | |
| Conclusion: |
| | TakeOut(subway) ∧ | ¬NegativeReviews(subway) | | | |
| | ------------------ | ------------------------ | --- | --- | |
| | Predicted answer: | C → A | | | |
| Figure6: Examplesofgeneratedsymbolicrepresentationsandpredictedanswers. Theincorrectsegment(s)and |
| theircorrespondenceintheproblemaremarkedinred,andthecorrectrevisionsaremarkedingreen. |
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