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We were given the following exercise. Let $\qquad \displaystyle f(n) = \begin{cases} 1 & 0^n \text{ occurs in the decimal representation of } \pi \\ 0 & \text{else}\end{cases}$ Prove that $f$ is computable. How is this possible? As far as I know, we do not know wether $\pi$ contains every sequence of digits (or which...
(I'm a student with some mathematical background and I'd like to know how to count the number of a specific kind of binary trees.) Looking at Wikipedia page for Binary Trees, I've noticed this assertion that the number of rooted binary trees of size $n$ would be this Catalan Number: $$C_n = \dfrac{1}{n+1}{2n \choose n}...
A deterministic finite automaton (DFA) is a state machine model capable of accepting all and only regular languages. DFAs can be (and usually are) defined in such a way that each state must provide some transition for all elements of the input alphabet; in other words, the transition function $\delta : Q \times \Sigma ...
Let $\qquad L = \{a^n \mid \exists_{p \geq n}\ p\,,\ p+2 \text{ are prime}\}.$ Is $L$ regular? This question looked suspicious at the first glance and I've realized that it is connected with the twin prime conjecture. My problem is that the conjecture has not been resolved yet, so I am not sure how can I proceed with d...
It is not uncommon to see students starting their PhDs with only a limited background in mathematics and the formal aspects of computer science. Obviously it will be very difficult for such students to become theoretical computer scientists, but it would be good if they could become savvy with using formal methods and ...
This is a follow-up question of this one. In a previous question about exotic state machines, Alex ten Brink and Raphael addressed the computational capabilities of a peculiar kind of state machine: min-heap automata. They were able to show that the set of languages accepted by such machines ($HAL$) is neither a subset...
This is a follow-up question of this one. In a previous question about exotic state machines, Alex ten Brink and Raphael addressed the computational capabilities of a peculiar kind of state machine: min-heap automata. They were able to show that the set of languages accepted by such machines ($HAL$) is neither a subset...
This is a follow-up question of this one. In a previous question about exotic state machines, Alex ten Brink and Raphael addressed the computational capabilities of a peculiar kind of state machine: min-heap automata. They were able to show that the set of languages accepted by such machines ($HAL$) is neither a subset...
The following DFA is a lexical analyzer which is supposed to recognize comments. The lexical analyzer will ignore the comment and goes back to the state one. I'm told that there's something wrong with it but I can't figure it out. What's the problem? FWIW, those tiny signs are stars which are necessary for C-style com...
How can it be decidable whether $\pi$ has some sequence of digits? inspired me to ask whether the following innocent-looking variation is computable: $$f(n) = \begin{cases} 1 & \text{if $\bar n$ occurs in the decimal representation of $\pi$} \\ 0 & \text{otherwise} \\ \end{cases}$$ where $\bar n$ is the decimal...
Given an instance of SAT, I would like to be able to estimate how difficult it will be to solve the instance. One way is to run existing solvers, but that kind of defeats the purpose of estimating difficulty. A second way might be looking a the ratio of clauses to variables, as is done for phase transitions in random-S...
Consider directed graphs. We call a node $v$ superstar if and only if no other node can be reached from it, but all other nodes have an edge to $v$. Formally: $\qquad \displaystyle $v$ \text{ superstar } :\Longleftrightarrow \mathrm{outdeg}(v) = 0 \land \mathrm{indeg}(v) = n-1$ with $n$ the number of nodes in the graph...
Minimum bandwidth problem is to a find an ordering of graph nodes on integer line that minimizes the largest distance between any two adjacent nodes. The decision problem is NP-complete even for binary trees. Complexity Results for Bandwidth Minimization. Garey, Graham, Johnson and Knuth, SIAM J. Appl. Math., Vol. 34,...
I've always thought vaguely that the answer to the above question was affirmative along the following lines. Gödel's incompleteness theorem and the undecidability of the halting problem both being negative results about decidability and established by diagonal arguments (and in the 1930's), so they must somehow be two ...
In a previous question there was a definition of weight balanced trees and a question regarding red-black trees. This question is to ask the same question, but for AVL trees. The question is, given the definition of $\mu$-balanced trees as in the other question, Is there some $\mu \gt 0$ such that all big enough AVL...
There is an easy polynomial algorithm to decide whether there is a path between two nodes in a directed graph (just do a routine graph traversal with, say, depth-first-search). However it seems that, surprisingly, the problem gets much harder if instead of testing for the existence we want want to count the number of p...
Definition from wikipedia: A graph is an ordered pair $G = (V, E)$ comprising a set $V$ of nodes together with a set $E$ of edges, which are two-element subsets of $V$. The set of all finite graphs (modulo isomorphism: we don't want nodes to have identities) is countable and could be enumerated. But what would be an ...
I was searching for interesting and easy to state open problems in computability (understandable by undergraduate students taking their first course in computability) to give examples of open problems (and obviously I want the students to be able to understand the problem without needing too much new definitions and al...
Consider the following protocol, meant to authenticate $A$ (Alice) to $B$ (Bob) and vice versa. $$ \begin{align} A \to B: &\quad \text{“I'm Alice”}, R_A \\ B \to A: &\quad E(R_A, K) \\ A \to B: &\quad E(\langle R_A+1, P_A\rangle, K) \\ \end{align} $$ $R$ is a random nonce. $K$ is a pre-shared symmetric key. $...
Rewriting system is a set of rules in the form of $A \leftrightarrow B$. If we apply that rule to a string $w$ we replace any substring $A$ in $w$ with a substring $B$ and vice versa. Given a starting string $AAABB$ can we derive $BAAB$ in the system with the following rules: $A \leftrightarrow BA$ $BABA \leftrighta...
Access control models are typically very coarse-grained, allowing one access to a resource (possibly with some combination of read/write/execute permission) or exclude such access. Some models of database security allow access to be granted on a per row basis (though I don't have a reference handy). Have fine-grained a...
I've heard the motto interaction is more powerful than algorithms from Peter Wegner. The basis of the idea is that a (classical) Turing Machine cannot handle interaction, that is, communication (input/output) with the outside world/environment. How can this be so? How can something be more powerful than a Turing Machi...
We know post-order, post L(x) => [x] post N(x,l,r) => (post l) ++ (post r) ++ [x] and pre-order pre L(x) => [x] pre N(x,l,r) => [x] ++ (pre l) ++ (pre r) and in-order traversal resp. sequentialisation. in L(x) => [x] in N(x,l,r) => (in l) ++ [x] ++ (in r) One can easily see that neither describes a given...
We're in a shared memory concurrency model where all reads and writes to integer variables are atomic. do: $S_1$ in parallel with: $S_2$ means to execute $S_1$ and $S_2$ in separate threads, concurrently. atomically($E$) means to evaluate $E$ atomically, i.e. all other threads are stopped during the execution of...
A drink dispenser requires the user to insert a coin ($\bar c$), then press one of three buttons: $\bar d_{\text{tea}}$ requests a cup of tea $e_{\text{tea}}$, ditto for coffee, and $\bar r$ requests a refund (i.e. the machine gives back the coin: $\bar b$). This dispenser can be modeled by the following CCS process: $...
Intro graphics courses usually have a project that asks you to build a ray tracer to render a scene. Many graphics students entering grad school say that they want to work on ray tracing. And yet it seems that ray tracing is a dead field in venues like SIGGRAPH etc. Is ray tracing really the best way to render a scene...
I'm pretty fluent in C/C++, and can make my way around the various scripting languages (awk/sed/perl). I've started using python a lot more because it combines some of the nifty aspects of C++ with the scripting capabilities of awk/sed/perl. But why are there so many different programming languages ? I'm guessing all t...
If you have to recommend books for someone who wants to learn more about TCS at the introductory level such as automata theory, algorithmics, complexity theory, etc, what book(s) would you recommend for those who are interested and want to learn more about TCS, but not had any exposure to it?
Fully connected (at least layer to layer with more than 2 hidden layers) backprop networks are universal learners. Unfortunately, they are often slow to learn and tend to over-fit or have awkward generalizations. From fooling around with these networks, I have observed that pruning some of the edges (so that their wei...
To my knowledge, there are three major process algebras that have inspired a vast range of research into formal models of concurrency. These are: CCS and $\pi$-calculus both by Robin Milner CSP by Tony Hoare and ACP by Jan Bergstra and Jan Willem Klop All three seem to have to this day a quite active following and va...
Windows NT used a point-to-point protocol where a client can communicate "securely" with a server by using a stream cipher to encrypt an array of messages with some key $k$. The server also encrypts its response with the same key $k$. But how is it aware of this key? More general: if Alice and Bob use some encryption/d...
I read in many places that some problems are difficult to approximate (it is NP-hard to approximate them). But approximation is not a decision problem: the answer is a real number and not Yes or No. Also for each desired approximation factor, there are many answers that are correct and many that are wrong, and this c...
From the answers to (When) is hash table lookup O(1)?, I gather that hash tables have $O(1)$ worst-case behavior, at least amortized, when the data satisfies certain statistical conditions, and there are techniques to help make these conditions broad. However, from a programmer's perspective, I don't know in advance wh...
Consider the following protocol, meant to authenticate $A$ (Alice) to $B$ (Bob) and vice versa. $$ \begin{align} A \to B: &\quad \text{“I'm Alice”}, R_A \\ B \to A: &\quad E(\langle 1, R_A\rangle, K) \\ A \to B: &\quad E(\langle 2, R_A+1, P_A\rangle, K) \\ \end{align} $$ $R$ is a random nonce. $K$ is a pre-shared...
I recently read that it is possible to have arrays which need not be initialized, i.e. it is possible to use them without having to spend any time trying to set each member to the default value. i.e. you can start using the array as if it has been initialized by the default value without having to initialize it. (Sorry...
The Eagles are a rock supergroup from the 70s and 80s, responsible for such classics as Hotel California. They have two quite distinctive sounds, one where guitarist Joe Walsh is present (for example, in Life in the Fast Lane) and one where he is absent. The latter songs have a markedly more sombre/boring feel. I'm cur...
A typical generational garbage collector keeps recently allocated data in a separate memory region. In typical programs, a lot of data is short-lived, so collecting young garbage (a minor GC cycle) frequently and collecting old garbage infrequently is a good compromise between memory overhead and time spent doing GC. I...
There are a great many data structures that implement the priority-queue interface: Insert: insert an element into the structure Get-Min: return the smallest element in the structure Extract-Min: remove the smallest element in the structure Common data structures implementing this interface are (min)heaps. Usually, t...
I've heard of (structural) induction. It allows you to build up finite structures from smaller ones and gives you proof principles for reasoning about such structures. The idea is clear enough. But what about coinduction? How does it work? How can one say anything conclusive about an infinite structure? There are (at...
Most of us learned programming using "textual" programming languages like Basic, C/C++, and Java. I believe it is more natural and efficient for humans to think visually. Visual programming allows developers to write programs by manipulating graphical elements. I guess using visual programming should improve the qualit...
A polynomial-time Turing machine algorithm is considered efficient if its run-time, in the worst-case, is bounded by a polynomial function in the input size. I'm aware of the strong Church-Turing thesis: Any reasonable model of computation can be efficiently simulated on Turing machines However, I'm not aware of soli...
Given a labelled transition system $(S,\Lambda,\to)$, where $S$ is a set of states, $\Lambda$ is a set of labels, and $\to\subseteq S\times\Lambda\times S$ is a ternary relation. As usual, write $p \stackrel\alpha\rightarrow q$ for $(p,\alpha,q)\in\to$. The labelled transition $p\stackrel\alpha\to q$ denotes that the s...
This might border on computational cognitive science, but I am curious as to how the process followed by common pathfinding algorithms (such as A*) compares to the process humans use in different pathfinding situations (given the same information). Are these processes similar?
Let $\Sigma$ be the set of terminal and $N$ the set of non-terminal symbols of some context-free grammar $G$. Say I have a string $a \in (\Sigma \cup N)^+$ such that $x a y \in \mathcal{S}(G)$ where $x,y\in (\Sigma \cup N)^*$ and $\mathcal{S}(G)$ are the sentential forms of $G$. Given $G$, I'd like to determine a set $...
I have an assignment where I need to make use a binary search tree and alter it to self order itself such that items that are accessed the most (have a higher priority) are at the top of the tree, the root being the most accessed node. The professor gave me the BST and node struct to work with, but trying to get my hea...
While discussing some intro level topics today, including the use of genetic algorithms; I was told that research has really slowed in this field. The reason given was that most people are focusing on machine learning and data mining. Update: Is this accurate? And if so, what advantages does ML/DM have when compared w...
What are some good online resources that will help me better understand BigO notation, running time & invariants? I'm looking for lectures, interactive examples if possible.
You have one coin. You may flip it as many times as you want. You want to generate a random number $r$ such that $a \leq r < b$ where $r,a,b\in \mathbb{Z}^+$. Distribution of the numbers should be uniform. It is easy if $b -a = 2^n$: r = a + binary2dec(flip n times write 0 for heads and 1 for tails) What if $b-a \...
Suppose I have two strings. Call them $A$ and $B$. Neither string has any repeated characters. How can I find the shortest sequence of insert, move, and delete operation that turns $A$ into $B$, where: insert(char, offset) inserts char at the given offset in the string move(from_offset, to_offset) moves the character ...
Suppose we have a balanced binary tree, which represents a recursive partitioning of a set of $N$ points into nested subsets. Each node of the tree represents a subset, with the following properties: subsets represented by two children nodes of the same parent are disjoint, and their union is equal to the subset repre...
I understand the theory behind Bayesian networks, and am wondering what it takes to build one in practice. Let's say for this example, that I have a Bayesian (directed) network of 100 discrete random variables; each variable can take one of up to 10 values. Do I store all the nodes in a DAG, and for each node store its...
I believe I have a reasonable grasp of complexities like $\mathcal{O}(1)$, $\Theta(n)$ and $\Theta(n^2)$. In terms of a list, $\mathcal{O}(1)$ is a constant lookup, so it's just getting the head of the list. $\Theta(n)$ is where I'd walk the entire list once, and $\Theta(n^2)$ is walking the list once for each element ...
Simply put, if one were to build a quantum computing device with the power of, say, 20 qubits, could such a computer be used to make any kind of modern hashing algorithm useless? Would it even be possible to harness the power of quantum computing in a traditional computing application?
Can every linear grammar be converted to a linear Greibach normal form, a form in which all productions look like $A \rightarrow ax$ where $a \in T$ and $x \in V \cup \{\lambda\}$? ($T$ is the set of terminals, $V$ is the set of non-terminals, $\lambda$ is the empty sequence.)
I know that since ~2004, Moore's law stopped working for CPU clock speed. I'm looking for a graph showing this, but am unable to find it: most charts out there show the transistor count or the capacity per year. Where can I find some data showing the CPU frequency of computers (anything is fine, personal computers, ser...
This is a puzzle about measuring network latency that I created. I believe the solution is that it's impossible, but friends disagree. I'm looking for convincing explanations either way. (Though it is posed as a puzzle I think it fits on this web site because of its applicability to the design and experience of communi...
This is a follow up to my earlier questions on coinduction and bisimulation. A relation $R \subseteq S \times S$ on the states of an LTS is a bisimulation iff $\forall (p,q)\in R,$ $$ \begin{array}{l} \text{ if } p \stackrel\alpha\rightarrow p' \text{ then } \exists q', \; q \stackrel\alpha\rightarrow q' \t...
How to improve the worst case scenario for a depth first search on an Euler graph, starting at some point and ending at that same point? I need to do the whole search but it is not fast enough for large amounts of data. I have tried bidirectional search but I can not keep the result numerically ordered. Therefore I won...
Assume I have two formulae $\Phi$ and $\Psi$ (over the same set of atomic propositions $AP$) in CTL. We have that $\Phi \equiv \Psi$ iff $Sat_{TS}(\Phi) = Sat_{TS}(\Psi)$ for all transition systems $TS$ over $AP$. Given that there are infinitely many transition systems, it's impossible to check them all. I thought abou...
I was given the question Where does the following language fit in the Chomsky hierarchy? Nonnegative solutions $(x,y)$ to the Diophantine equation $3x-y=1$. I understand languages like $L = \{ 0^n1^n \mid n \ge 1\}$, but this language confuses me. What do the words in the language look like? How could I represent it...
I have the following language $\qquad \{0^i 1^j 2^k \mid 0 \leq i \leq j \leq k\}$ I am trying to determine which Chomsky language class it fits into. I can see how it could be made using a context-sensitive grammar so I know it is atleast context-sensitive. It seems like it wouldn't be possible to make with a context-...
A directed graph is said to be unipathic if for any two vertices $u$ and $v$ in the graph $G=(V,E)$, there is at most one simple path from $u$ to $v$. Suppose I am given a unipathic graph $G$ such that each edge has a positive or negative weight, but contains no negative weight cycles. From this I want to find a $O(\|V...
Note this is a question related to study in a CS course at a university, it is NOT homework and can be found here under Fall 2011 exam2. Here are the two questions I'm looking at from a past exam. They seem to be related, the first: Let $\qquad \mathrm{FINITE}_{\mathrm{CFG}} = \{ < \! G \! > \mid G \text{ is a Contex...
In the script I am currently reading on the lambda calculus, beta equivalence is defined as this: The $\beta$-equivalence $\equiv_\beta$ is the smallest equivalence that contains $\rightarrow_\beta$. I have no idea what that means. Can someone explain it in simpler terms? Maybe with an example? I need it for a lemma ...
OK, so here is a question from a past test in my Theory of Computation class: A useless state in a TM is one that is never entered on any input string. Let $$\mathrm{USELESS}_{\mathrm{TM}} = \{\langle M, q \rangle \mid q \text{ is a useless state in }M\}.$$ Prove that $\mathrm{USELESS}_{\mathrm{TM}}$ is undecidabl...
Write $\bar n$ for the decimal expansion of $n$ (with no leading 0). Let $a$ and $b$ be integers, with $a > 0$. Consider the language of the decimal expansions of the multiples of $a$ plus a constant: $$M = \{ \overline{a\,x+b} \mid x\in\mathbb{N} \}$$ Is $M$ regular? context-free? (Contrast with Language of the graph ...
Write $\bar n$ for the decimal expansion of $n$ (with no leading 0). Let : be a symbol distinct from any digit. Let $a$ and $b$ be integers, with $a > 0$. Consider the language of solutions of the Diophantine equation $y=ax+b$: $$L = \{ \bar{x} \mathtt: \bar{y} \mid y = a\,x + b \}$$ Is $L$ regular? context-free? (Cont...
I have used the technique of dynamic programming multiple times however today a friend asked me how I go about defining my sub-problems, I realized I had no way of providing an objective formal answer. How do you formally define a sub-problem for a problem that you would solve using dynamic programming?
I am currently learning the lambda calculus and was wondering about the following two different kinds of writing a lambda term. $\lambda xy.xy$ $\lambda x.\lambda y.xy$ Is there any difference in meaning or the way you apply beta reduction, or are those just two ways to express the same thing? Especially this defi...
In real world applications is there a concrete benefit when using $\mathcal{O}(\log(\log(n))$ instead of $\mathcal{O}(\log(n))$ algorithms ? This is the case when one use for instance van Emde Boas trees instead of more conventional binary search tree implementations. But for example, if we take $n < 10^6$ then in the...
We all know that Markov Chains can be used for generating real-looking text (or real-sounding music). I've also heard that Markov Chains has some applications in the image processing, is that true? What are some other uses of MCs in CS?
I'm looking for a data structure that stores a set of strings over a character set $\Sigma$, capable of performing the following operations. We denote $\mathcal{D}(S)$ as the data structure storing the set of strings $S$. Add-Prefix-Set on $\mathcal{D}(S)$: given some set $T$ of (possibly empty) strings, whose size is...
I've came up with a result while reading some automata books, that Turing machines appear to be more powerful than pushdown automata. Since the tape of a Turing machine can always be made to behave like a stack, it'd seem that we can actually claim that TMs are more powerful. Is this true?
Many intractable $NP$-complete problems can be modeled as deciding whether a set of triples, $F=${$t_1, t_2, ..., t_n$} where each triple $t_i$ is a subset of three elements over base set $U=${$a_1, a_2, ..., a_k$}, satisfy some non-trivial property. For example, 3-edge coloring of cubic graphs can be modeled as the pr...
A unipathic graph is a directed graph such that there is at most one simple path from any one vertex to any other vertex. Unipathic graphs can have cycles. For example, a doubly linked list (not a circular one!) is a unipathic graph; if the list has $n$ elements, the graph has $n-1$ cycles of length 2, for a total of $...
Assume we are working in a Turing-complete, referentially-transparent, higher-order language that supports arbitrary dataflow unification. Shouldn't it then be possible to construct the following function (using Haskell-like syntax, because that's what I'm most familiar with)? -- Takes an arbitrary pure function and c...
If I understand correctly, NFA have the same expressive power as regular expressions. Often, reading off equivalent regular expressions from NFA is easy: you translate cycles to stars, junctions as alternatives and so on. But what to do in this case: [source] The overlapping cycles make it hard to see what this autom...
Note that by "two-way pushdown automaton", I mean a pushdown automaton that can move its reading head both ways on the input tape. I recently had the question of determining the computational power of two-way PDAs in the Chomsky hierarchy. I don't entirely understand two-way PDAs, but I can see how with the ability to ...
The Post Correspondence Problem (PCP) is undecidable. The bounded version of the PCP is $\mathrm{NP}$-complete and the marked version of the PCP (the words of one of the two lists are required to differ in the first letter) is in $\mathrm{PSPACE}$ [1]. Are these restricted versions used to prove some complexity result...
How does the password that we enter (to connect to a wireless network) encrypt the data on the wireless network? Through my reading I am not sure if the password that we enter is the same as the passphrase. If that is right then how can the passphrase generate the four WEP keys? I understand how the four keys work in W...
The simplex algorithm walks greedily on the corners of a polytope to find the optimal solution to the linear programming problem. As a result, the answer is always a corner of the polytope. Interior point methods walk the inside of the polytope. As a result, when a whole plane of the polytope is optimal (if the objecti...
In coding theory, 'how good a code is' means how many channel errors can be corrected, or better put, the maximal noise level that the code can deal with. In order to get better codes, the codes are designed using a large alphabet (rather than binary one). And then, the code is good if it can deal with a large rate of ...
I came across an issue with the definition of a (directed) graph in Sipser's Introduction to the theory of computation, 2nd Ed. On pp.10, An undirected graph, or simply a graph, is a set of points with lines connecting some of the points. The points are called nodes or vertices, and the lines are called edges, ... On t...
Cache-oblivious algorithms and data structures are a rather new thing, introduced by Frigo et al. in Cache-oblivious algorithms, 1999. Prokop's thesis from the same year introduces the early ideas as well. The paper by Frigo et al. present some experimental results showing the potential of the theory and of the cache-o...
Let $\Sigma$ be some finite set of characters of fixed size. Let $\alpha$ be some string over $\Sigma$. We say that a nonempty substring $\beta$ of $\alpha$ is a repeat if $\beta = \gamma \gamma$ for some string $\gamma$. Now, my question is whether the following holds: For every $\Sigma$, there exists some $n \in \ma...
In a partially ordered set, am I always able to order two arbitrary elements out of the set? Or is it possible that two elements within the set have no order relation to each other? For example if there are three elements $\{a, b, c\}$ and $a \leq b$ and $a \leq c$, does either $b \leq c$ or $c \leq b$ have to hold? I ...
I'm trying to understand what is wrong with the following proof of the following recurrence $$ T(n) = 2\,T\!\left(\left\lfloor\frac{n}{2}\right\rfloor\right)+n $$ $$ T(n) \leq 2\left(c\left\lfloor\frac{n}{2}\right\rfloor\right)+n \leq cn+n = n(c+1) =O(n) $$ The documentation says it's wrong because of the inducti...
The 3-Partition problem asks whether a set of $3n$ integers can be partitioned into $n$ sets of three integers such that each set sums up to some given integer $B$. The Balanced Partition problem asks whether $2n$ integers can be partitioned into two equal cardinality sets such that both sets have the same sum. Both pr...
This is a follow up to the How does Stack Inspection work? that explores the notion in more detail Stack inspection is a mechanism for ensuring security in the context of the JVM and CLR virtual machines when externally downloaded code modules of different levels of trust may be running together. That system libraries ...
This is precursor to my other, more advanced question about Stack Inspection. Stack Inspection is a security mechanism introduced in the JVM to deal with running code originating from locations having different levels of trust. This is question aims at finding a simple description of its functionality. So: How does st...
Let $G$ be a context-free grammar. A string of terminals and nonterminals of $G$ is said to be a sentential form of $G$ if you can obtain it by applying productions of $G$ zero or more times to the start symbol of $S$. Let $\operatorname{SF}(G)$ be the set of sentential forms of $G$. Let $\alpha \in \operatorname{SF}(G...
I'm trying to prove that a binary tree with $n$ nodes has at most $\left\lceil \frac{n}{2} \right\rceil$ leaves. How would I go about doing this with induction? For people who were following in the original question about heaps, it has been moved here.
I am looking for some hints in a question asked by my instructor. So I just figured out this decision problem is $\sf{NP\text{-}complete}$: In a graph $G$, is there a spanning tree in $G$ that contain an exact set of $S=\{x_1, x_2,\ldots, x_n\}$ as leafs. I figured out we can prove that it is $\sf{NP\text{-}complete}$ ...
Logic gates are an abstract device which can be implemented with electromagnetic relays, vacuum tubes, or transistors. These implemenations have been successful in computing in part because of various properties of chainability, durability, and size beyond their basic binary stability. They also work well because ele...
I have a homework assignment that I've been bashing my head against for some time, and I'd appreciate any hints. It is about choosing a known problem, the NP-completeness of which is proven, and constructing a reduction from that problem to the following problem I'll call DGD (directed graph diagnosis). Problem An ins...
I'm looking for mathematical theories that deal with describing formal languages (set of strings) in general and not just grammar hierarchies.
I have been reading this site with a great deal of interest, but I find a lot of it goes over my head. This has made me wish to learn a lot more about algorithms and CS in general. As far as I can tell from my research, there are two main ways of doing this. I can buy a nice thick heavy book and work my way through it...
I am learning Automated Theorem Proving / SMT solvers / Proof Assistants by myself and post a series of questions about the process, starting here. Note that these topics are not easily digested without a background in (mathematical) logics. If you have problems with basic terms, please read up on those, for instance L...
Assume I have a list of functions, for example $\qquad n^{\log \log(n)}, 2^n, n!, n^3, n \ln n, \dots$ How do I sort them asymptotically, i.e. after the relation defined by $\qquad f \leq_O g \iff f \in O(g)$, assuming they are indeed pairwise comparable (see also here)? Using the definition of $O$ seems awkward, and...
I read that the number of coin tosses of a probabilistic Turing machine (PTM) is not a Blum complexity measure. Why? Clarification: Note that since the execution of the machine is not deterministic, one should be careful about defining the number of coin tosses for a PTM $M$ on input $x$ in a way similar to the time co...

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