Results 1  10
of
14
Cutelimination and redundancyelimination by resolution
 Journal of Symbolic Computation
, 2000
"... A new cutelimination method for Gentzen’s LK is defined. First cutelimination is generalized to the problem of redundancyelimination. Then the elimination of redundancy in LKproofs is performed by a resolution method in the following way: A set of clauses C is assigned to an LKproof ψ and it is ..."
Abstract

Cited by 28 (10 self)
 Add to MetaCart
A new cutelimination method for Gentzen’s LK is defined. First cutelimination is generalized to the problem of redundancyelimination. Then the elimination of redundancy in LKproofs is performed by a resolution method in the following way: A set of clauses C is assigned to an LKproof ψ and it is shown that C is always unsatisfiable. A resolution refutation of C then serves as a skeleton of an LKproof ψ ′ with atomic cuts; ψ ′ can be constructed from the resolution proof and ψ by a projection method. In the last step the atomic cuts are eliminated and a cutfree proof is obtained. The complexity of the method is analyzed and it is shown that a nonelementary speedup over Gentzen’s method can be achieved. Finally an application to automated deduction is presented: it is demonstrated how informal proofs (containing pseudocuts) can be transformed into formal ones by the method of redundancyelimination; moreover, the method can even be used to transform incorrect proofs into correct ones. 1.
Proof Transformation by CERES
 MATHEMATICAL KNOWLEDGE MANAGEMENT (MKM) 2006, VOLUME 4108 OF LECTURE NOTES IN ARTIFICIAL INTELLIGENCE
, 2006
"... Cutelimination is the most prominent form of proof transformation in logic. The elimination of cuts in formal proofs corresponds to the removal of intermediate statements (lemmas) in mathematical proofs. The cutelimination method CERES (cutelimination by resolution) works by constructing a set o ..."
Abstract

Cited by 9 (8 self)
 Add to MetaCart
Cutelimination is the most prominent form of proof transformation in logic. The elimination of cuts in formal proofs corresponds to the removal of intermediate statements (lemmas) in mathematical proofs. The cutelimination method CERES (cutelimination by resolution) works by constructing a set of clauses from a proof with cuts. Any resolution refutation of this set then serves as a skeleton of an LKproof with only atomic cuts. In this paper we present an extension of CERES to a calculus LKDe which is stronger than the Gentzen calculus LK (it contains rules for introduction of definitions and equality rules). This extension makes it much easier to formalize mathematical proofs and increases the performance of the cutelimination method. The system CERES already proved efficient in handling very large proofs.
Methods of CutElimination
 PROJECTION, LECTURE
"... This short report presents the main topics of methods of cutelimination which will be presented in the course at the ESSLLI'99. It gives a short introduction addressing the problem of cutelimination in general. Furthermore we give a brief description of several methods and refer to other papers ad ..."
Abstract

Cited by 9 (8 self)
 Add to MetaCart
This short report presents the main topics of methods of cutelimination which will be presented in the course at the ESSLLI'99. It gives a short introduction addressing the problem of cutelimination in general. Furthermore we give a brief description of several methods and refer to other papers added to the course material.
Applying Tree Languages in Proof Theory
 In AdrianHoria Dediu and Carlos MartínVide, editors, Language and Automata Theory and Applications (LATA) 2012, volume 7183 of Lecture Notes in Computer Science
, 2012
"... Abstract. We introduce a new connection between formal language theory and proof theory. One of the most fundamental proof transformations in a class of formal proofs is shown to correspond exactly to the computation of the language of a certain class of tree grammars. Translations in both direction ..."
Abstract

Cited by 7 (6 self)
 Add to MetaCart
Abstract. We introduce a new connection between formal language theory and proof theory. One of the most fundamental proof transformations in a class of formal proofs is shown to correspond exactly to the computation of the language of a certain class of tree grammars. Translations in both directions, from proofs to grammars and from grammars to proofs, are provided. This correspondence allows theoretical as well as practical applications. 1
Asymptotic cyclic expansion and bridge groups of formal proofs
 JOURNAL OF ALGEBRA
, 2001
"... Formal proofs, even simple ones, may hide an unexpected intricate combinatorics. We define a new combinatorial invariant, the bridge group of a proof, which encodes the cyclic structure of proofs in the sequent calculus. We compute the bridge groups of two infinite families of proofs and identify th ..."
Abstract

Cited by 5 (1 self)
 Add to MetaCart
Formal proofs, even simple ones, may hide an unexpected intricate combinatorics. We define a new combinatorial invariant, the bridge group of a proof, which encodes the cyclic structure of proofs in the sequent calculus. We compute the bridge groups of two infinite families of proofs and identify them with the Baumslag–Solitar and Gersten groups. We observe that the distortion of cyclic subgroups in these groups equals the asymptotic growth of the procedure of elimination of lemmas from the proofs.
Herbrand sequent extraction
 IN INTELLIGENT COMPUTER MATHEMATICS
, 2008
"... Computer generated proofs of interesting mathematical theorems are usually too large and full of trivial structural information, and hence hard to understand for humans. Techniques to extract specific essential information from these proofs are needed. In this paper we describe an algorithm to extr ..."
Abstract

Cited by 5 (3 self)
 Add to MetaCart
Computer generated proofs of interesting mathematical theorems are usually too large and full of trivial structural information, and hence hard to understand for humans. Techniques to extract specific essential information from these proofs are needed. In this paper we describe an algorithm to extract Herbrand sequents from proofs written in Gentzen’s sequent calculus LK for classical firstorder logic. The extracted Herbrand sequent summarizes the creative information of the formal proof, which lies in the instantiations chosen for the quantifiers, and can be used to facilitate its analysis by humans. Furthermore, we also demonstrate the usage of the algorithm in the analysis of a proof of the equivalence of two different definitions for the mathematical concept of lattice, obtained with the proof transformation system CERES.
Cut elimination for a class of propositional based logics
, 2005
"... Sufficient conditions for propositional based logics to enjoy cut elimination are established. These conditions are satisfied by a wide class of logics encompassing among others classical and intuitionistic logic, modal logic S4, and classical and intuitionistic linear logic and some of their fragme ..."
Abstract

Cited by 4 (2 self)
 Add to MetaCart
Sufficient conditions for propositional based logics to enjoy cut elimination are established. These conditions are satisfied by a wide class of logics encompassing among others classical and intuitionistic logic, modal logic S4, and classical and intuitionistic linear logic and some of their fragments. The class of logics is characterized by the type of rules and provisos used in their sequent calculi. The conditions can be checked in finite time and define relations between the rules and the provisos so that the calculus can enjoy cut elimination. A general proof of cut elimination is presented for any calculus satisfying those conditions.
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.
Transforming and Analyzing Proofs in the CERESsystem ∗
"... Cutelimination is the most prominent form of proof transformation in logic. The elimination of cuts in formal proofs corresponds to the removal of intermediate statements (lemmas) in mathematical proofs. Cutelimination can be applied to mine real mathematical proofs, i.e. for extracting explicit a ..."
Abstract
 Add to MetaCart
Cutelimination is the most prominent form of proof transformation in logic. The elimination of cuts in formal proofs corresponds to the removal of intermediate statements (lemmas) in mathematical proofs. Cutelimination can be applied to mine real mathematical proofs, i.e. for extracting explicit and algorithmic information. The system CERES (cutelimination by resolution) is based on automated deduction and was successfully applied to the analysis of nontrivial mathematical proofs. In this paper we focus on the inputoutput environment of CERES, and show how users can interact with the system and extract new mathematical knowledge. 1
Cut Elimination In Situ Comments appreciated.
, 2012
"... We present methods for removing toplevel cuts from a sequent calculus or Taitstyle proof without significantly increasing the space used for storing the proof. For propositional logic, this requires converting a proof from treelike to daglike form, but it most doubles the number of lines in the ..."
Abstract
 Add to MetaCart
We present methods for removing toplevel cuts from a sequent calculus or Taitstyle proof without significantly increasing the space used for storing the proof. For propositional logic, this requires converting a proof from treelike to daglike form, but it most doubles the number of lines in the proof. For firstorder logic, the proof size can grow exponentially, but the proof has a succinct description and is polynomialtimeuniform. We usedirect, globalconstructionsthat give polynomial time methods for removing all toplevel cuts from proofs. Byexploitingprenexrepresentations,this extendsto removingall cuts, with final proof size bounded superexponentially in the alternation of quantifiers in cut formulas. 1