Technology Category
Knowledge Representation and Reasoning: Computational Complexity of ReasoningConstraint Satisfaction and Optimization: Satisfiability Modulo TheoriesSearch and Optimization: Sampling/Simulation-based Search
Characterizing p-Simulation Between Theories
🤖 AI Summary
This paper investigates necessary and sufficient conditions for p-simulation between axiomatic theories and its implications for central questions in computational complexity.
Method: Employing interpretability techniques from computability theory, proof theory, and model theory—combined with Π₁-theorem preservation analysis and non-relativizing methods—the authors establish precise connections between efficient interpretability and p-simulation.
Contribution/Results: They prove that if a theory S efficiently interprets S + φ, then S p-simulates S + φ, and the two notions are equivalent within S; further, no c.e. theory can p-simulate all theories. These results entail P ≠ NP ≠ coNP and yield a unified conjectural framework subsuming the Feige Hypothesis, existence of one-way functions, and circuit lower bounds. The work reveals structural limitations of p-simulation and provides novel meta-logical tools for separating complexity classes.
Technology Category
Application Category
📝 Abstract
This paper characterizes when one axiomatic theory, as a proof system for tautologies, $p$-simulates another, by showing: (i)~if c.e. theory $mathcal{S}$ efficiently interprets $mathcal{S}{+}φ$, then $mathcal{S}$ $p$-simulates $mathcal{S}{+}φ$ (Jeřábek in Pudlák17 proved simulation), since the interpretation maps an $mathcal{S}{+}φ$-proof whose lines are all theorems into an $mathcal{S}$-proof; (ii)~$mathcal{S}$ proves ``$mathcal{S}$ efficiently interprets $mathcal{S}{+}φ$'' iff $mathcal{S}$ proves ``$mathcal{S}$ $p$-simulates $mathcal{S}{+}φ$'' (if so, $mathcal{S}$ already proves the $Π_1$ theorems of $mathcal{S}{+}φ$); and (iii)~no $mathcal{S}$ $p$-simulates all theories. Result (iii) implies $ extbf{P}{
eq} extbf{NP}{
eq} extbf{coNP}$, using the nonrelativizing fact ``no c.e. theory interprets all c.e. theories'' (false for $mathcal{S}$ with predicate for true sentences). To explore whether this framework resolves other open questions, the paper formulates conjectures stronger than ``no optimal proof system exists'' that imply Feige's Hypothesis, the existence of one-way functions, and circuit lower bounds.
Problem
Research questions and friction points this paper is trying to address.
Characterizes p-simulation between axiomatic theories.
Links efficient interpretation to p-simulation in proofs.
Explores implications for P vs NP and proof systems.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Efficient interpretation enables p-simulation between theories
Proof system equivalence under efficient interpretation
No theory p-simulates all possible theories
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