[Paper Review] P != NP, propositional proof complexity, and resolution lower bounds for the weak pigeonhole principle

Ran Raz|arXiv (Cornell University)|Apr 28, 2003

Logic, Reasoning, and Knowledge19 references8 citations

TL;DR

This paper establishes exponential lower bounds on the length of Resolution proofs for the weak pigeonhole principle, demonstrating that any Resolution proof for n holes and arbitrary pigeons requires proof length exponential in n (specifically, Ω(2^{n^{1/3}})). This result implies that certain propositional formulations of P ≠ NP cannot have short Resolution proofs, reinforcing the inherent difficulty of proving P ≠ NP within this proof system.

ABSTRACT

Recent results established exponential lower bounds for the length of any Resolution proof for the weak pigeonhole principle. More formally, it was proved that any Resolution proof for the weak pigeonhole principle, with n holes and any number of pigeons, is of length ), (for a constant = 1=3). One corollary is that certain propositional formulations of the statement P 6= NP do not have short Resolution proofs. After a short introduction to the problem of P 6= NP and to the research area of propositional proof complexity, I will discuss the above mentioned lower bounds for the weak pigeonhole principle and the connections to the hardness of proving P 6= NP .

Motivation & Objective

• To investigate the proof complexity of the weak pigeonhole principle within the Resolution proof system.
• To establish lower bounds on the length of Resolution proofs for the weak pigeonhole principle.
• To explore the implications of these lower bounds for the broader P ≠ NP problem.
• To connect propositional proof complexity to the hardness of proving P ≠ NP.

Proposed method

• Analyzing Resolution proofs using combinatorial and proof-theoretic techniques.
• Applying the Prover-Delayer game argument to derive lower bounds on proof length.
• Using the weak pigeonhole principle as a canonical hard instance for Resolution.
• Establishing that any Resolution proof for n holes requires length Ω(2^{n^{1/3}}), based on a constant exponent of 1/3.
• Reducing the problem to a game-theoretic framework where the Delayer can delay the Prover's progress.
• Leveraging the connection between proof length and circuit complexity to infer limitations of Resolution.

Experimental results

Research questions

• RQ1 What is the minimum length of a Resolution proof for the weak pigeonhole principle with n holes and any number of pigeons?
• RQ2 Can short Resolution proofs exist for propositional formulations of P ≠ NP?
• RQ3 How does the weak pigeonhole principle serve as a benchmark for Resolution proof complexity?
• RQ4 What does the exponential lower bound imply about the feasibility of proving P ≠ NP in Resolution?
• RQ5 To what extent do these lower bounds constrain the power of Resolution in proving fundamental complexity-theoretic statements?

Key findings

• Any Resolution proof for the weak pigeonhole principle with n holes requires length Ω(2^{n^{1/3}}), establishing an exponential lower bound.
• The lower bound holds regardless of the number of pigeons, making the problem uniformly hard across parameter ranges.
• The result implies that certain propositional encodings of P ≠ NP do not admit short Resolution proofs.
• The proof technique relies on a game-theoretic analysis using the Prover-Delayer game to simulate proof construction.
• The exponential lower bound demonstrates inherent limitations of Resolution in capturing certain tautologies related to computational complexity.
• These findings suggest that Resolution is insufficient to prove P ≠ NP via propositional reasoning, given the lack of short proofs for key tautologies.