Logical Closure Properties of Propositional Proof Systems

Transcription

1 of Logical of Propositional Institute of Theoretical Computer Science Leibniz University Hannover Germany Theory and Applications of Models of Computation 2008

2 Outline of

3 Propositional of Definition (Cook, Reckhow 79) A propositional proof system is a polynomial time computable function P with rng(p) = TAUT. A string π with P(π) = ϕ is called a P-proof of ϕ. Motivation Proofs can be easily checked. Examples truth-table method, resolution,

4 Frege Systems of F use: axiom schemes: ϕ ϕ, ϕ ϕ ψ,... ϕ ϕ ψ rules: (modus ponens) ψ A Frege proof of a formula ϕ is a sequence (ϕ 1,...,ϕ n = ϕ) of propositional formulas such that for i = 1,...,n: ϕ i is a substitution instance of an axiom, or ϕ i was derived by modus ponens from ϕ j, ϕ k with j, k < i.

5 Extensions of Frege Systems of Extended Frege EF Abbreviations for complex formulas: p ϕ, where p is a new propositional variable. with substitution SF Substitution rule: ϕ σ(ϕ) for arbitrary substitutions σ Extensions of EF Let Φ be a polynomial-time computable set of tautologies. EF + Φ: Φ as axiom schemes

6 Between of Definition (Cook, Reckhow 79) A proof system Q p-simulates a proof system P df (P p Q) there exists a poly-time function f such that P(π) = Q(f(π)) for all π. P and Q are p-equivalent (P p Q) P p Q and Q p P. df

7 The Simulation Order of Optimal Proof System? ZFC EF p SF Frege not polynomially bounded AC 0 -Frege Cutting Planes PCR of Resolution Davis Putnam Polynomial Calculus Nullstellensatz Truth Table

8 of of Definition A proof system P is closed under modus ponens there exists a poly-time procedure with Input: P-proofs π 1,...,π k of ϕ 1,...,ϕ k a P-proof π k+1 of ϕ 1 ϕ k+1 Output: a P-proof of ϕ k+1. Definition df P is closed under substitutions there exists a poly-time procedure with Input: a P-proof of ϕ and a substitution instance σ(ϕ) of ϕ Output: a P-proof of σ(ϕ). df

9 Robust Properties of Definition A property E is robust if it is preserved inside a degree, i.e., if P fulfills E and P p Q, then also Q fulfills E. Remark 1. Closure under modus ponens and substitutions are robust. 2. They are also independent from each other.

10 An Example of Proposition F and extended EF are closed under modus ponens and substitutions.

11 An Example of Proposition F and extended EF are closed under modus ponens and substitutions. Proof. Modus ponens is available as a rule in F and EF.

12 An Example of Proposition F and extended EF are closed under modus ponens and substitutions. Proof. Modus ponens is available as a rule in F and EF. Substitutions in F : Let ϕ 1,...,ϕ k be an F -proof. Then σ(ϕ 1 ),...,σ(ϕ k ) is an F -proof of σ(ϕ k ).

13 An Example of Proposition F and extended EF are closed under modus ponens and substitutions. Proof. Modus ponens is available as a rule in F and EF. Substitutions in F : Let ϕ 1,...,ϕ k be an F -proof. Then σ(ϕ 1 ),...,σ(ϕ k ) is an F -proof of σ(ϕ k ). Substitutions in EF : Use EF p SF and robustness.

14 : of The language of arithmetic uses the symbols 0, S, +,,... Σ b 1-formulas are formulas in prenex normal form with only bounded -quantifiers, i.e. ( x t(y))ψ(x, y). Σ b 1-formulas describe NP-sets. Π b 1-formulas: ( x t(y))ψ(x, y) conp-sets

15 Translating Π b 1-Formulas into Propositional Formulas Definition (Cook 75, Krajíček, Pudlák 90) Let ϕ Π b 1. Then there are propositional formulas ϕ n, n N such that: ϕ n can be constructed in polynomial time from 1 n. ϕ n is a tautology N = ϕ(a) for all a N of length n of

16 The Property of Definition The reflection principle of a propositional proof system P is defined by the arithmetic formula where RFN(P) = ( π)( ϕ)prf P (π,ϕ) Taut(ϕ) Prf P is a Σ b 1-formula formalizing P-proofs Taut is a Π b 1-formula for propositional tautologies.

17 The Property of Definition The reflection principle of a propositional proof system P is defined by the arithmetic formula where RFN(P) = ( π)( ϕ)prf P (π,ϕ) Taut(ϕ) Prf P is a Σ b 1-formula formalizing P-proofs Taut is a Π b 1-formula for propositional tautologies. Definition A propositional proof system P has the reflection property if there exists a polynomial-time algorithm that on input 1 n outputs a P-proof of RFN(P) n.

18 Robustness of of Proposition Let P p Q. Then P has reflection iff Q has reflection.

19 Robustness of of Proposition Let P p Q. Then P has reflection iff Q has reflection. Proof. Assume that Q has reflection and let g : P p Q.

20 Robustness of of Proposition Let P p Q. Then P has reflection iff Q has reflection. Proof. Assume that Q has reflection and let g : P p Q. Then Q g computes P.

21 Robustness of of Proposition Let P p Q. Then P has reflection iff Q has reflection. Proof. Assume that Q has reflection and let g : P p Q. Then Q g computes P. Q proves RFN(P) with respect to the Turing machine Q g, because Q proves rng(q) TAUT, and therefore also rng(q g) TAUT.

22 Robustness of of Proposition Let P p Q. Then P has reflection iff Q has reflection. Proof. Assume that Q has reflection and let g : P p Q. Then Q g computes P. Q proves RFN(P) with respect to the Turing machine Q g, because Q proves rng(q) TAUT, and therefore also rng(q g) TAUT. P p Q = P proves RFN(P) with respect to Q g.

23 Which Have? of

24 Which Have? of Strong proof systems have reflection, weak systems probably not.

25 Which Have? of Strong proof systems have reflection, weak systems probably not. Theorem (Krajíček, Pudlák 89) EF has the reflection property. Theorem (Atserias, Bonet 02) Resolution requires exponential-size proofs for its reflection principle (in the canonical formulation).

26 Characterization of Extensions of EF of Main Theorem For all proof systems P p EF the following conditions are equivalent: 1. P is p-equivalent to a proof system EF + ϕ with a true Π b 1-formula ϕ. 2. P is p-equivalent to a system EF + Φ with some poly-time set Φ of true Π b 1 -formulas. 3. P has the reflection property and is closed under modus ponens and substitutions.

27 Characterization of Extensions of EF of Main Theorem For all proof systems P p EF the following conditions are equivalent: 1. P is p-equivalent to a proof system EF + ϕ with a true Π b 1-formula ϕ. 2. P is p-equivalent to a system EF + Φ with some poly-time set Φ of true Π b 1 -formulas. 3. P has the reflection property and is closed under modus ponens and substitutions. Proof. 1 2: clear

28 Characterization of Extensions of EF of Main Theorem For all proof systems P p EF the following conditions are equivalent: 1. P is p-equivalent to a proof system EF + ϕ with a true Π b 1-formula ϕ. 2. P is p-equivalent to a system EF + Φ with some poly-time set Φ of true Π b 1 -formulas. 3. P has the reflection property and is closed under modus ponens and substitutions. Proof. 1 2: clear 2 3: EF + Φ has these closure properties, and robustness transfers them to P.

29 Ingredients of the Proof 3 1 of Deduction theorem for EF There exists a poly-time procedure that takes as input an EF -proof of a formula ψ from a finite set of tautologies Φ as extra assumptions, and produces an EF -proof of the implication ( ϕ Φ ϕ) ψ. Proposition If the proof system P p EF has reflection and P is closed under under substitutions and modus ponens, then EF + RFN(P) p P. Proposition For every proof system P we have P p EF + RFN(P).

30 Proposition If P p EF has reflection and is closed under modus ponens and substitutions, then EF + RFN(P) p P. of

31 Proposition If P p EF has reflection and is closed under modus ponens and substitutions, then EF + RFN(P) p P. Proof. If EF + RFN(P) proves ϕ, then there are substitution instances ψ 1,...,ψ k of formulas from RFN(P) such that EF proves ϕ from ψ 1,...,ψ k. of

32 Proposition If P p EF has reflection and is closed under modus ponens and substitutions, then EF + RFN(P) p P. Proof. If EF + RFN(P) proves ϕ, then there are substitution instances ψ 1,...,ψ k of formulas from RFN(P) such that EF proves ϕ from ψ 1,...,ψ k. Deduction theorem for EF = poly-size EF -proof of ( k i=1 ψ i) ϕ of

33 Proposition If P p EF has reflection and is closed under modus ponens and substitutions, then EF + RFN(P) p P. Proof. If EF + RFN(P) proves ϕ, then there are substitution instances ψ 1,...,ψ k of formulas from RFN(P) such that EF proves ϕ from ψ 1,...,ψ k. Deduction theorem for EF = poly-size EF -proof of ( k i=1 ψ i) ϕ Transform this proof into a poly-size EF -proof of (ψ 1 (ψ 2...(ψ k 1 (ψ k ϕ))... )) of

34 Proposition If P p EF has reflection and is closed under modus ponens and substitutions, then EF + RFN(P) p P. Proof. If EF + RFN(P) proves ϕ, then there are substitution instances ψ 1,...,ψ k of formulas from RFN(P) such that EF proves ϕ from ψ 1,...,ψ k. Deduction theorem for EF = poly-size EF -proof of ( k i=1 ψ i) ϕ Transform this proof into a poly-size EF -proof of (ψ 1 (ψ 2...(ψ k 1 (ψ k ϕ))... )) P p EF = poly-size P-proof of this formula of

35 Proposition If P p EF has reflection and is closed under modus ponens and substitutions, then EF + RFN(P) p P. Proof. If EF + RFN(P) proves ϕ, then there are substitution instances ψ 1,...,ψ k of formulas from RFN(P) such that EF proves ϕ from ψ 1,...,ψ k. Deduction theorem for EF = poly-size EF -proof of ( k i=1 ψ i) ϕ Transform this proof into a poly-size EF -proof of (ψ 1 (ψ 2...(ψ k 1 (ψ k ϕ))... )) P p EF = poly-size P-proof of this formula P has reflection + substitution = poly-size P-proofs of ψ 1,...,ψ k of

36 Proposition If P p EF has reflection and is closed under modus ponens and substitutions, then EF + RFN(P) p P. Proof. If EF + RFN(P) proves ϕ, then there are substitution instances ψ 1,...,ψ k of formulas from RFN(P) such that EF proves ϕ from ψ 1,...,ψ k. Deduction theorem for EF = poly-size EF -proof of ( k i=1 ψ i) ϕ Transform this proof into a poly-size EF -proof of (ψ 1 (ψ 2...(ψ k 1 (ψ k ϕ))... )) P p EF = poly-size P-proof of this formula P has reflection + substitution = poly-size P-proofs of ψ 1,...,ψ k Modus ponens for P = poly-size P-proof of ϕ of

37 of The Cook-Reckhow framework is possibly too broad for proof systems used in practice: These systems always satisfy some additional properties. A combination of simple closure properties characterizes the degrees of extensions of EF. Bounded arithmetic is a central tool to understand these systems.

38 A General Programme of Identify the central underlying properties of further systems such as resolution, geometric systems, algebraic systems etc. Provide robust definitions of these systems in terms of these properties.