## On the complexity of proof deskolemization

Venue: | J. Symbolic Logic |

Citations: | 5 - 4 self |

### BibTeX

@ARTICLE{Baaz_onthe,

author = {Matthias Baaz and Stefan Hetzl and Daniel Weller},

title = {On the complexity of proof deskolemization},

journal = {J. Symbolic Logic},

year = {}

}

### OpenURL

### Abstract

Abstract. We consider the following problem: Given a proof of the Skolemization of a formula F, what is the length of the shortest proof of F? For the restriction of this question to cut-free proofs we prove corresponding exponential upper and lower bounds. §1. Introduction. The Skolemization of formulas is a standard technique in logic. It consists of replacing existential quantifiers by new function symbols whose arguments reflect the dependencies of the quantifier. The Skolemization of a formula is satisfiability-equivalent to the original formula. This transformation has a number of applications, it is for example crucial for automated theorem

### Citations

560 |
Untersuchungen über das logische Schliessen
- Gentzen
- 1935
(Show Context)
Citation Context ...roposition 1. Let F be a prenex formula and π an LK-proof of ⊢ sk(F ). Then there exists an LK-proof ψ of ⊢ F such that |ψ| ≤ |π| ∗ (qocc(F ) + 1). Proof. First, we apply Gentzen’s midsequent theorem =-=[6]-=- to π to obtain an LK-proof ϕ of ⊢ sk(F ) such that ϕ contains a sequent S such that above S, only propositional inferences are applied and below S, only weak quantifier inferences are applied. Furthe... |

35 |
bounds on Herbrand’s theorem
- Statman, Lower
- 1979
(Show Context)
Citation Context ..., and 3. for all cut-free proofs πn of Sn, |πn| ≥ s(n − d) (i. e. non-elementary). Proof. Consider the sequence of sequents Tn Statman uses to show the non-elementary complexity of cut-elimination in =-=[10]-=-. Statman constructs short proofs with cut πn of Tn. Consider the end-sequent T ′ n of πn ˇ T where T is the set of eigenvariables of πn. We take sk(T ′ n) for Mn. For Sn we take a certain “bad prenex... |

31 |
A compact representation of proofs
- Miller
- 1987
(Show Context)
Citation Context ...upper bound. A central technique for the upper bound will be to collect instances of a formula that appear in a proof. To that aim we will use a variant of expansion trees, introduced by D. Miller in =-=[9]-=- in the setting of higher-order logic. In fact, what we are going to define as expansion below corresponds most closely to the Skolem expansion trees of [9]. In order to simplify the notation we do no... |

28 | Cut-elimination and redundancy-elimination by resolution
- Baaz, Leitsch
(Show Context)
Citation Context ...from π by replacing every cut by an →l-inference followed by ∀linferences to bind all occurrences of terms from T (this is a slightly more general version of proof transformations that also appear in =-=[2, 4]-=-). If the endsequent of π is Γ ⊢ ∆, then the endsequent of ˇπ T is Σ, Γ ⊢ ∆ where Σ = ∀x1 . . . ∀xm (A1 → A1), . . . , ∀x1 . . . ∀xm (Ak → Ak)12 MATTHIAS BAAZ, STEFAN HETZL, AND DANIEL WELLER and A1{... |

23 |
On skolemization and proof complexity
- Baaz
- 1994
(Show Context)
Citation Context ... Then the length of π, denoted by |π|, is the number of sequent occurrences in π. Having set up our calculus, we will now introduce Skolemization (for Skolemization in the context of proofs, see also =-=[2, 3]-=-). We postulate a countably infinite set of Skolem symbols SK = {fn | n ∈ N} and define an operator for structural Skolemization as follows. ∃lON THE COMPLEXITY OF PROOF DESKOLEMIZATION 3 Definition ... |

13 |
Eliminating definitions and Skolem functions in first-order logic
- Avigad
- 2001
(Show Context)
Citation Context ... WELLER the language. The question of the degree of this impact has been formulated by P. Pudlák (in a slightly different form) as problem 22 in [5]. A partial solution has been given by J. Avigad in =-=[1]-=-: theories that allow the encoding of finite functions have polynomial deskolemization. In this paper we consider a different type of restriction: instead of restricting the theories we restrict the p... |

11 |
P.H.: The liberalized δ-rule in free variable semantic tableaux
- Hähnle, Schmitt
- 1994
(Show Context)
Citation Context ...mbols. This optimized version sm-Skolemization of structural Skolemization allows minimization of quantifier scope. This is analogous to the δ + rule for free variable semantic tableaux introduced in =-=[7]-=-. Definition 12. We define a rewrite relation →sm on formulas that “pushes quantifiers down”: (∀x)¬F →sm ¬(∃x)F, (∀x)(F ∨ G) →sm (∀x)F ∨ G, (∀x)(G ∨ F ) →sm G ∨ (∀x)FON THE COMPLEXITY OF PROOF DESKOL... |

6 |
ome open problems in arithmetic, proof theory and computational complexity
- CLOTE, KRAJfEK
- 1993
(Show Context)
Citation Context ...0000/$00.00 12 MATTHIAS BAAZ, STEFAN HETZL, AND DANIEL WELLER the language. The question of the degree of this impact has been formulated by P. Pudlák (in a slightly different form) as problem 22 in =-=[5]-=-. A partial solution has been given by J. Avigad in [1]: theories that allow the encoding of finite functions have polynomial deskolemization. In this paper we consider a different type of restriction... |

1 |
normal forms and proof complexity
- Cut
- 1999
(Show Context)
Citation Context ... Then the length of π, denoted by |π|, is the number of sequent occurrences in π. Having set up our calculus, we will now introduce Skolemization (for Skolemization in the context of proofs, see also =-=[2, 3]-=-). We postulate a countably infinite set of Skolem symbols SK = {fn | n ∈ N} and define an operator for structural Skolemization as follows. ∃lON THE COMPLEXITY OF PROOF DESKOLEMIZATION 3 Definition ... |

1 |
Basic proof theory, second ed., Cambridge Tracts in Theoretical Computer Science, Cambridge
- Troelstra, Schwichtenberg
- 2000
(Show Context)
Citation Context ... sides of implications), and weak otherwise. A dual definition is made for ∃. The number of strong quantifiers in a formula F is denoted by qocc(F ). We use a variant of the sequent calculus G3c from =-=[11]-=-, with the difference that we add the appropriate axiom for ⊤ and that we work in a purely cut-free setting. Definition 1 (Sequent calculus). Sequents are pairs of multisets of formulas, written Γ ⊢ ∆... |