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Boolean Programs and Quantified Propositional Proof System -


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Boolean Programs and Quantified Propositional Proof System -

  1. 1. Boolean Programs andQuantified Propositional ProofSystemsTo appear in the Bulletin of the Section of LogicStephen Cook and Michael Soltys1
  2. 2. A propositional proof is a derivation of apropositional tautology (e.g. ¬P ∨ P) using aset of rules.A nice example of a propositional proofsystem is Gentzen’s PK system, where wehave sequents of propositional formulas ofthe form:A1, . . . , An → B1, . . . , Bm (∗)and the semantics of sequents is given asfollows. We say that a truth assignment τsatisfies the sequent (∗) iff either τ falsifiessome Ai or τ satisfies some Bi.For example, if τ is given by P = 0, Q = 1,R = 0 then τ satisfiesP ∨ Q, ¬R → Q ∧ ¬R2
  3. 3. A sequent is equivalent to the formula(A1 ∧ · · · ∧ An) ⊃ (B1 ∨ · · · ∨ Bm)In other words, the conjunction of the Ai’simplies the disjunction of the Bi’s.We say that a sequent is valid if it is trueunder all truth assignments. For example, thefollowing are valid sequents:A → A→ A, ¬AA, ¬A →→ A ∨ ¬AA, A ⊃ B → BA PK proof is a sequence of sequents, whereall the sequents are derived from previoussequents or are logical axioms of the formA → A, → 1 or 0 →.3
  4. 4. PK Rules of inference:weakΓ → ∆A, Γ → ∆Γ → ∆Γ → ∆, AexchΓ1, A, B, Γ2 → ∆Γ1, B, A, Γ2 → ∆Γ → ∆1, A, B, ∆2Γ → ∆1, B, A, ∆2contΓ, A, A → ∆Γ, A → ∆Γ → ∆, A, AΓ → ∆, A“¬”Γ → ∆, A¬A, Γ → ∆A, Γ → ∆Γ → ∆, ¬A“∧”A, B, Γ → ∆(A ∧ B), Γ → ∆Γ → ∆, A Γ → ∆, BΓ → ∆, (A ∧ B)“∨”A, Γ → ∆ B, Γ → ∆(A ∨ B), Γ → ∆Γ → ∆, A, BΓ → ∆, (A ∨ B)cutΓ → ∆, A A, Γ → ∆Γ → ∆4
  5. 5. Any valid sequent can be derived from initialsequents of the form A → A, → 1 and 0 →,using the above rules (completeness of PK),and only valid sequents can be derived(soundness of PK).As an example we give a proof of one ofDeMorgan’s laws:¬(P ∧ Q) → ¬P ∨ ¬QHere is the derivation:P → P→ P, ¬P→ P, ¬P, ¬QQ → QQ → Q, ¬P→ Q, ¬P, ¬Q→ P ∧ Q, ¬P, ¬Q→ P ∧ Q, ¬P ∨ ¬Q¬(P ∧ Q) → ¬P ∨ ¬Q5
  6. 6. Abstract Definition: A proof system is afeasible (i.e. easily computable) mapf : {0, 1}∗ onto−→ {tautologies}Where {tautologies} is just the set ofpropositional tautologies, and if f(x) = Athen x is the proof of the propositionaltautology A.A proof system is polynomially bounded ifthere exists a polynomial p such that forevery A ∈ {tautologies}, there exists an xsuch that |x| ≤ p(|A|), i.e. every tautology hasa proof which is “proportional” to its size.See “The Complexity of PropositionalProofs”, by Alasdair Urquhart, in TheBulletin of Symbolic Logic, December 1995,for all the details.6
  7. 7. Interest in propositional proof complexityarose from two fields connected withcomputers: automated theorem proving(artificial intelligence) and computationalcomplexity theory.Some Complexity TheoryP is the class of problems that can be solvedwith algorithms that run in polynomial timein the size of the input (e.g. given a truthassignment τ, and a sequent S, does τ satisfyS?).NP is the class of problems whose solutionscan be verified by an algorithm that runs inpolynomial time (e.g. given a sequent S, is itsatisfiable?)co-NP is the class of problems for which acounter-example can be verified by analgorithm that runs in polynomial time (e.g.given a sequent S, is it valid?).7
  8. 8. Theorem: NP = co-NP iff there is apolynomially bounded proof system for thepropositional tautologies.Also P = NP implies NP = co-NP, thus if weprove that there is no polynomially boundedproof system (which is what expertsconjecture), then it will follow thatNP = co-NP, and therefore P = NP.The problemP?= NPhas been listed by Steve Smale in the topthree “Mathematical Problems for the NextCentury” (Math Intelligencer 20, 1998).8
  9. 9. How could we prove that there is nopolynomially bounded proof system? It seemslike a very difficult problem. The standardapproach is to consider stronger and strongerproof systems, and to exhibit superpolynomiallower bounds for each of them.strongest Quantified PKExtended PKPK————————Bounded Depth PKResolutionweakest Truth TablesHowever, we can only prove superpolynomiallower bounds for the weaker systems (theones below the line). The systems above theline have no known lower bounds.9
  10. 10. Quantified PKQuantified propositional PK is formed byintroducing propositional quantifiers:∀xA(x) whose meaning is A(0) ∧ A(1)∃xA(x) whose meaning is A(0) ∨ A(1)The propositional quantifiers do not increasethe expressive power of formulas. Instead,they allow us to shorten some propositionalformulas. For example, the formula(x1,...,xn)∈{0,1}nA(x1, . . . , xn)has size in the order of 2n|A|, but theequivalent quantified formula∃x1 . . . ∃xnA(x1, . . . , xn)has size in the order of n + |A|.10
  11. 11. The quantified propositional calculus is firstdiscussed by B. Russell in “The Theory ofImplication”, American Journal ofMathematics in 1906. Also, the Polishmathematician Le´sniewski treated it in his“Protothetic”.Martin Dowd investigated the connectionbetween quantified propositional proofsystems and the complexity class PSPACE inhis PhD thesis in 1979.A modest source of information is also a 5page section in Jan Kraj´ıˇcek’s book“Bounded Arithmetic, Propositional Logic,and Complexity Theory”, 1995.Still, little is known about quantifiedpropositional proof systems.11
  12. 12. G is PK with ∃ and ∀I follow the results and notation from JanKraj´ıˇcek’s “Bounded Arithmetic,Propositional Logic, and Complexity Theory”.The quantified propositional proof system Gextends LK by allowing quantifiedpropositional formulas in sequents, and byadding the following quantifier rules:A(B), Γ → ∆∀xA(x), Γ → ∆Γ → ∆, A(p)Γ → ∆, ∀xA(x)A(p), Γ → ∆∃xA(x), Γ → ∆Γ → ∆, A(B)Γ → ∆, ∃xA(x)where B is any formula, and with therestriction that the atom p does not occur inthe lower sequents of ∀-right and ∃-left.12
  13. 13. Let G1 be the restriction of G where we allowonly strict Σq1 formulas, that is, all formulasmust begin with a block of existentialquantifiers (possibly empty) followed by aquantifier free formula. In other words, allformulas are either quantifier free, or of theform∃x1 . . . ∃xnA(x1, . . . , xn)where A(x1, . . . , xn) is quantifier free.For example, the sequent∃x∃y((¬x ∨ p) ∧ (¬y ∨ q)) → ∃x(x ∨ ¬p ∨ q), p ∧ qhas only Σq1 formulas.13
  14. 14. WitnessingGiven a sequent S, we want to have a way ofcomputing the values of the existentialvariables on the right in terms of the freevariables, and the quantified variables on theleft. For example, consider the sequent∃yA(x, y) → ∃yA(x, y) (∗)where x, y are the only variables in thissequent. In this case it is very easy. Thequantified variable y on the right can bewitnessed by the following function:f(x, y) = ythat is, the sequentA(x, y) → A(x, f(x, y))is valid if (∗) was valid.14
  15. 15. Suppose that π is a G1 proof. We show thatthe existential quantifiers can be witnessed byBoolean programs. We do this by inductionon the number of sequents in π.For example, consider the contraction rule:C(¯p) → D(¯p), ∃xB(x), ∃xB(x)C(¯p) → D(¯p), ∃xB(x)By Induction Hypothesis, we know that thereare Boolean programs g, h such that thesequentC(¯p) → D(¯p), B(g(¯p)), B(h(¯p))is valid. Now, define f as follows:f(¯p) =g(¯p) if B(g(¯p))h(¯p) otherwiseThe bottom sequent can thus be convertedto the valid sequentC(¯p) → D(¯p), B(f(¯p)),15
  16. 16. Main IdeaIf π is a G1 proof of a propositional tautology,and π is tree-like, then the existentialquantifiers can be feasibly computed byevaluating Boolean circuits. This was alreadyshown in Jan Kraj´ıˇcek’s book “BoundedArithmetic, Propositional Logic, andComplexity Theory”.If, on the other hand, π is not tree-like, thenit seems that evaluating the existentialquantifiers requires a lot of computationalpower, i.e. Boolean programs whoseevaluation is a PSPACE complete problem.This is shown in our paper “BooleanPrograms and Quantified Propositional ProofSystems”.16
  17. 17. References“Boolean Programs and QuantifiedPropositional Proof Systems”, by S.Cook andM. Soltys, in this paper we introduce thenotion of Boolean programs, and wecharacterize PSPACE in terms of families ofBoolean programs. We then show how to useBoolean programs to witness quantifiers inG1.“The Complexity of Propositional Proofs”, byAlasdair Urquhart, in The Bulletin ofSymbolic Logic, December 1995, acomprehensive introduction to the complexityof propositional proofs.“Bounded Arithmetic, Propositional Logic,and Complexity Theory”, by Jan Kraj´ıˇcek. Ashort introduction to quantified propositionalproof systems.17