File size: 5,590 Bytes
639016c | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 | """Small independent Fraction enumerations supplement, but do not replace, written proofs."""
import argparse,itertools,json,math
from fractions import Fraction as F
from pathlib import Path
import numpy as np
from infinite_players.evidence_tree import EvidenceTree
from infinite_players.memory_contract import confusability_graph,exact_coloring
def signprob(sequence,mean):
p=F(1)
for x in sequence:p*=(1+mean*x)/2
return p
def lag_audit():
rho=F(3,5);kappa=F(2,5);count=0;total_atoms=0
for n in range(1,5):
for lag in range(4):
inc_n=n+lag-1
sums=[F(0),F(0),F(0)]
for inc in itertools.product([-1,1],repeat=inc_n):
pi=signprob(inc,rho)
for noises in itertools.product([-1,1],repeat=n):
prob=pi*signprob(noises,kappa)
score=sum(noises[t]*math.prod(inc[t:t+lag]) for t in range(n))
sums[0]+=prob;sums[1]+=prob*score;sums[2]+=prob*score**2;total_atoms+=1
mean=n*kappa*rho**lag
variance=n*(1-kappa**2*rho**(2*lag))+2*kappa**2*sum((n-h)*(rho**(2*h)-rho**(2*lag)) for h in range(1,min(n,lag)))
assert sums[0]==1 and sums[1]==mean and sums[2]-sums[1]**2==variance
count+=1
return {"parameter_cases":count,"enumerated_atoms":total_atoms,"result":"exact mean and covariance sum passed"}
def operator_audit():
factors=[[[F(1,3),F(1,5)],[F(1,4),F(1,2)]],[[F(1,7),F(2,5)],[F(1,3),F(1,6)]],[[F(1,4),F(1,8)],[F(1,2),F(1,3)]]]
initial=[F(2,5),F(3,5)];terminal=[F(3,7),F(4,7)]
result=F(0);atoms=0
for z in itertools.product(range(2),repeat=4):
prob=initial[z[0]]*terminal[z[-1]]
for t in range(3):prob*=factors[t][z[t]][z[t+1]]
result+=prob;atoms+=1
a=np.array(factors,dtype=float)[:,None,:,:]
tree=EvidenceTree(a,["a","b","c"],model_id="audit")
error=abs(math.exp(tree.log_likelihoods(np.array(initial,float),np.array(terminal,float))[0])-float(result))
assert error<1e-14
return {"hidden_paths":atoms,"exact_likelihood":str(result),"float_error":error}
def memory_audit():
rows=[];encoders=0
for n in (1,2,3):
records=list(itertools.product([0,1],repeat=n));best=F(0)
for mapping in itertools.product([0,1],repeat=len(records)):
numerator=0
for cell in (0,1):
for j in range(n):
counts=[sum(1 for i,d in enumerate(records) if mapping[i]==cell and d[j]==bit) for bit in (0,1)]
numerator+=max(counts)
success=F(numerator,n*len(records));best=max(best,success);encoders+=1
graph=confusability_graph(records);states,_=exact_coloring(graph)
assert states==2**n
rows.append({"history_bits":n,"exact_memory_states":states,"best_one_bit_average_recovery":str(best)})
assert rows[-1]['best_one_bit_average_recovery']=='3/4'
return {"deterministic_encoders":encoders,"cases":rows}
def shield_audit():
D=[[F(1,2),F(9,10),F(1,10)],[F(1,2),F(1,10),F(9,10)]]
target=[F(1,2),F(1,2)];reward=[F(2,5),F(9,10),F(4,5)]
mix=[F(0),F(1,2),F(1,2)];baseline=[F(1),F(0),F(0)]
assert [sum(D[i][a]*mix[a] for a in range(3)) for i in range(2)]==target
assert sum(reward[a]*mix[a] for a in range(3))==F(17,20)
p=[F(1,3)]*3;null=[F(-2),F(1),F(1)]
assert all(sum(row[a]*null[a] for a in range(3))==0 for row in D)
scale=sum(reward[a]*null[a] for a in range(3))/sum(v*v for v in null)
direction=[scale*v for v in null]
assert sum(reward[a]*direction[a] for a in range(3))==sum(v*v for v in direction)
# All binary semantic words have the same probability under baseline and mixture.
words=0
for horizon in range(1,11):
for word in itertools.product([0,1],repeat=horizon):
p0=F(1);p1=F(1)
for bit in word:
p0*=sum(D[bit][a]*baseline[a] for a in range(3))
p1*=sum(D[bit][a]*mix[a] for a in range(3))
assert p0==p1;words+=1
return {"semantic_words":words,"reward_before":"2/5","reward_after":"17/20","projected_gain_squared_norm":str(sum(v*v for v in direction))}
def composition_audit():
cases=0;max_error=F(0)
for T in range(1,9):
# Persistent identity mixtures with two opposite deterministic semantic paths.
for p in (F(1,5),F(1,2),F(4,5)):
for q in (F(1,4),F(1,2),F(3,4)):
# Common stochastic binary-symmetric continuation can only contract this distance.
probs=[]
for word in itertools.product([0,1],repeat=T):
l0=F(1);l1=F(1)
for bit in word:l0*=F(4,5) if bit==0 else F(1,5);l1*=F(4,5) if bit==1 else F(1,5)
probs.append((p*l0+(1-p)*l1,q*l0+(1-q)*l1))
error=sum(abs(x-y) for x,y in probs)/2
assert error<=abs(p-q);max_error=max(max_error,error);cases+=1
return {"posterior_horizon_cases":cases,"maximum_observed_TV":str(max_error)}
def main():
p=argparse.ArgumentParser();p.add_argument('--output',type=Path,default=Path('results/exact_audits.json'));a=p.parse_args()
result={"lag":lag_audit(),"operator":operator_audit(),"memory":memory_audit(),"shield":shield_audit(),"composition":composition_audit(),"status":"all assertions passed; in-session independent implementations, not external review"}
a.output.parent.mkdir(parents=True,exist_ok=True);a.output.write_text(json.dumps(result,indent=2,sort_keys=True)+"\n");print(json.dumps(result,indent=2))
if __name__=='__main__':main()
|