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Polynomial circuit simulation of a verifier

Lax429075.CircuitMachine · concepts/Lax429075/CircuitMachine.lean · lax-429075

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    Natural Language Statement

    Lemma

    For a fixed polynomial time verifier and polynomial certificate bound, construct a circuit whose free inputs represent the certificate. The circuit is satisfiable exactly when some bounded certificate is accepted. Its encoded gate clauses are produced in polynomial time. The machine simulation and time bound follow from a finite stack program that emits the initial layer, the transition layers, and the final acceptance constraint.

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    Evidence

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    In the paper

    • page 3 of this submission's paper

    Lean source view on GitHub

    1import Lax429075.Tseitin
    2import Lax429075.Encoding
    3import Lax434930.NondeterministicPolynomialTime
    4
    5/-!
    6---
    7title: Polynomial circuit simulation of a verifier
    8type: lemma
    9---
    10For a fixed polynomial time verifier and polynomial certificate bound,
    11construct a circuit whose free inputs represent the certificate. The circuit
    12is satisfiable exactly when some bounded certificate is accepted. Its encoded
    13gate clauses are produced in polynomial time. The machine simulation and
    14time bound follow from a finite stack program that emits the initial layer,
    15the transition layers, and the final acceptance constraint.
    16-/
    17
    18namespace Lax429075.CircuitMachine
    19
    20open Lax434930.PolynomialTime Lax434930.Certificates Circuits Encoding
    21
    22axiom compile (V : Language) (hV : V ∈ P) (p : Polynomial ℕ) :
    23 ∃ circuits : Word → Circuit,
    24 Nonempty (Turing.TM2ComputableInPolyTime id id
    25 (fun x => encodeCNF (Tseitin.encode (circuits x)))) ∧
    26 ∀ x, Satisfiable (circuits x) ↔
    27 ∃ y : Word, y.length ≤ p.eval x.length ∧ pair x y ∈ V
    28
    29end Lax429075.CircuitMachine
    30
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