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The Undecidability of kProvability
 Annals of Pure and Applied Logic
, 1989
"... The kprovability problem is, given a first order formula φ and an integer k, to determine if φ has a proof consisting of k or fewer lines (i.e., formulas or sequents). This paper shows that the kprovability problem for the sequent calculus is undecidable. Indeed, for every r.e. srt ..."
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The kprovability problem is, given a first order formula &phi; and an integer k, to determine if &phi; has a proof consisting of k or fewer lines (i.e., formulas or sequents). This paper shows that the kprovability problem for the sequent calculus is undecidable. Indeed, for every r.e. srt X...
Logical Approaches to the Complexity of Search Problems: Proof Complexity, Quantified Propositional Calculus, and Bounded Arithmetic
, 2005
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Nearlyexponential size lower bounds for symbolic quantifier elimination algorithms and OBDDbased proofs of unsatisfiability
 Electronic Colloquium on Computational Complexity
, 2007
"... We demonstrate a family of propositional formulas in conjunctive normal form so that a formula of size N requires size 2 Ω ( 7 √ N/logN) to refute using the treelike OBDD refutation system of Atserias, Kolaitis and Vardi [3] with respect to all variable orderings. All known symbolic quantifier elim ..."
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We demonstrate a family of propositional formulas in conjunctive normal form so that a formula of size N requires size 2 Ω ( 7 √ N/logN) to refute using the treelike OBDD refutation system of Atserias, Kolaitis and Vardi [3] with respect to all variable orderings. All known symbolic quantifier elimination algorithms for satisfiability generate treelike proofs when run on unsatisfiable CNFs, so this lower bound applies to the runtimes of these algorithms. Furthermore, the lower bound generalizes earlier results on OBDDbased proofs of unsatisfiability in that it applies for all variable orderings, it applies when the clauses are processed according to an arbitrary schedule, and it applies when variables are eliminated via quantification. 1
Term Induction
, 2001
"... In this thesis we study a formal first order system T (tind) in the standard language L, of Gentzen’s LK, see [Tak87]. T (tind) extends LK by the following valid firstorder inference rule (A is quantifierfree). Γ, A(a), Λ → ∆, A(s(a)), Θ Γ, A(0), Λ → ∆, A(s n (0)), Θ (tind) This rule is called ter ..."
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In this thesis we study a formal first order system T (tind) in the standard language L, of Gentzen’s LK, see [Tak87]. T (tind) extends LK by the following valid firstorder inference rule (A is quantifierfree). Γ, A(a), Λ → ∆, A(s(a)), Θ Γ, A(0), Λ → ∆, A(s n (0)), Θ (tind) This rule is called term induction, it derives a restricted term built from successor s and the constant 0. We call such terms numerals. To characterise the difference between T (tind) and pure logic, we employ proof theoretic methods. Firstly we establish a variant of Herbrand’s Theorem for T (tind). Let ∃¯xF (¯x) be a Σ1 formula; provable by Π. Then there exists a disjunction � N i1 · · · � N il M1(s i1 (0),..., s il(0)) ∨ · · · ∨ Mm(s i1 (0),..., s il(0)), denoted by H that is valid for some N ∈ IN, furthermore the Mi are instances of F (ā). In T (tind) it is not possible to bound the length of Herbrand disjunctions in terms of proof length and logical complexity of the endformula as usual. The main result is that we can bound the length of the {s, 0}matrix of the above disjunctions in this way.
Academy of Sciences, Prague
"... Abstract We study diagonalization in the context of implicit proofs of [10]. We prove that at least one of the following three conjectures is true: ffl There is a function f: f0; 1g \Lambda! f0; 1g computable in E that has circuit complexity 2 \Omega (n) ffl N P 6 = coN P. ffl There is no poptimal ..."
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Abstract We study diagonalization in the context of implicit proofs of [10]. We prove that at least one of the following three conjectures is true: ffl There is a function f: f0; 1g \Lambda! f0; 1g computable in E that has circuit complexity 2 \Omega (n) ffl N P 6 = coN P. ffl There is no poptimal propositional proof system. We note that a variant of the statement (either N P 6 = coN P or N E &quot; coN E contains a function 2 \Omega (n) hard on average) seems to have a bearing on the existence of good proof complexity generators. In particular, we prove that if a minor variant of a recent conjecture of Razborov [17, Conjecture 2] is true (stating conditional lower bounds for the Extended Frege proof system EF) then actually unconditional lower bounds would follow for EF. The only method for demonstrating unprovability of a \Pi 0
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"... This paper consists of two parts both of which attempt to provide tautologies which might be hard for a propositional proof system P. Finding hard tautologies is of interest as a possible approach to the NP versus co − NP problem. The first part of the paper continues the study of the τ(g)b(x) tauto ..."
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This paper consists of two parts both of which attempt to provide tautologies which might be hard for a propositional proof system P. Finding hard tautologies is of interest as a possible approach to the NP versus co − NP problem. The first part of the paper continues the study of the τ(g)b(x) tautologies begun in Krajíček [1]. These tautologies express that b ∈ {0, 1} m is not the output of g(x) for a function g: {0, 1} n − → {0, 1} m with m> n computed by a polynomial sized circuit family. In the earlier Krajíček paper, τ(g)b(x) tautologies, where g computes a certain pseudorandom number generator, were put forward as candidate hard tautologies for propositional proof systems like extended Frege. Another choice of g suggested in this earlier paper was the function tt which takes as input a circuit C of size at most 2 k/2 with k inputs and outputs the truth table for C. The first part of the present paper gives two example tautologies, which if they had short proofs, would imply the τ(g)b(x) tautologies have short proofs. The second part of the present paper gives a family of tautologies which would be hard
Proof Complexity and the KneserLovász Theorem (I)
, 2014
"... We investigate the proof complexity of a class of propositional formulas expressing a combinatorial principle known as the KneserLovász Theorem. This is a family of propositional tautologies, indexed by an nonnegative integer parameter k that generalizes the Pigeonhole Principle (obtained for k = ..."
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We investigate the proof complexity of a class of propositional formulas expressing a combinatorial principle known as the KneserLovász Theorem. This is a family of propositional tautologies, indexed by an nonnegative integer parameter k that generalizes the Pigeonhole Principle (obtained for k = 1). We show, for all fixed k, 2Ω(n) lower bounds on resolution complexity and exponential lower bounds for bounded depth Frege proofs. These results hold even for the more restricted class of formulas encoding Schrijver’s strenghtening of the KneserLovász Theorem. On the other hand for the cases k = 2, 3 (for which combinatorial proofs of the KneserLovász Theorem are known) we give polynomial size Frege (k = 2), respectively extended Frege (k = 3) proofs. The paper concludes with a brief announcement of the results (presented in subsequent work) on the complexity of the general case of the KneserLovász theorem. 1