Results 1 
3 of
3
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 X... ..."
Abstract

Cited by 27 (0 self)
 Add to MetaCart
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 X...
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 ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
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.