Results 1  10
of
15
On the Proof Complexity of Deep Inference
, 2000
"... We obtain two results about the proof complexity of deep inference: 1) deepinference proof systems are as powerful as Frege ones, even when both are extended with the Tseitin extension rule or with the substitution rule; 2) there are analytic deepinference proof systems that exhibit an exponential ..."
Abstract

Cited by 31 (13 self)
 Add to MetaCart
We obtain two results about the proof complexity of deep inference: 1) deepinference proof systems are as powerful as Frege ones, even when both are extended with the Tseitin extension rule or with the substitution rule; 2) there are analytic deepinference proof systems that exhibit an exponential speedup over analytic Gentzen proof systems that they polynomially simulate.
PSPACE bounds for rank 1 modal logics
 IN LICS’06
, 2006
"... For lack of general algorithmic methods that apply to wide classes of logics, establishing a complexity bound for a given modal logic is often a laborious task. The present work is a step towards a general theory of the complexity of modal logics. Our main result is that all rank1 logics enjoy a sh ..."
Abstract

Cited by 25 (15 self)
 Add to MetaCart
For lack of general algorithmic methods that apply to wide classes of logics, establishing a complexity bound for a given modal logic is often a laborious task. The present work is a step towards a general theory of the complexity of modal logics. Our main result is that all rank1 logics enjoy a shallow model property and thus are, under mild assumptions on the format of their axiomatisation, in PSPACE. This leads to a unified derivation of tight PSPACEbounds for a number of logics including K, KD, coalition logic, graded modal logic, majority logic, and probabilistic modal logic. Our generic algorithm moreover finds tableau proofs that witness pleasant prooftheoretic properties including a weak subformula property. This generality is made possible by a coalgebraic semantics, which conveniently abstracts from the details of a given model class and thus allows covering a broad range of logics in a uniform way.
Finality Regained  A Coalgebraic Study of Scottsets and Multisets
 Arch. Math. Logic
, 1999
"... In this paper we study iterated circular multisets in a coalgebraic framework. We will produce two essentially different universes of suchsets. The unisets of the first universe will be shown to be precisely the sets of the Scott universe. The unisets of the second universe will be precisely the ..."
Abstract

Cited by 21 (1 self)
 Add to MetaCart
In this paper we study iterated circular multisets in a coalgebraic framework. We will produce two essentially different universes of suchsets. The unisets of the first universe will be shown to be precisely the sets of the Scott universe. The unisets of the second universe will be precisely the sets of the AFAuniverse. Wewillhave a closer look into the connection of the iterated circular multisets and arbitrary trees. Key words: multiset, nonwellfounded set, Scottuniverse, AFA, coalgebra, modal logic, graded modalities MSC2000 codes: 03B45, 03E65, 03E70, 18A15, 18A22, 18B05, 68Q85 1 Contents 1 Introduction 3 1.1 Multisets on a Given Domain . . . . . . . . . . . . . . . . . . . . 3 1.2 Iterated and Circular Multisets . . . . . . . . . . . . . . . . . . . 6 1.3 Organization of the Paper . . . . . . . . . . . . . . . . . . . . . . 7 2 Prerequisites 8 2.1 Coalgebras and Morphisms . . . . . . . . . . . . . . . . . . . . . 8 2.1.1 A Prototype: Pow . . . . . . . . . . . . . . . ...
Multi Lingual Sequent Calculus and Coherent Spaces
 Fundamenta Informaticae
, 1997
"... We study a Gentzen style sequent calculus where the formulas on the left and right of the turnstile need not necessarily come from the same logical system. Such a sequent can be seen as a consequence between different domains of reasoning. We discuss the ingredients needed to set up the logic ge ..."
Abstract

Cited by 14 (6 self)
 Add to MetaCart
We study a Gentzen style sequent calculus where the formulas on the left and right of the turnstile need not necessarily come from the same logical system. Such a sequent can be seen as a consequence between different domains of reasoning. We discuss the ingredients needed to set up the logic generalized in this fashion.
Proof Theoretic Complexity
 IN PROOF AND SYSTEM RELIABILITY, H. SCHWICHTENBERG AND R. STEINBRÜGGEN, EDS. NATO SCIENCE SERIES
, 2002
"... A weak formal theory of arithmetic is developed, entirely analogous to classical arithmetic but with two separate kinds of variables: induction variables and quantifier variables. The point is that the provably recursive functions are now more feasibly computable than in the classical case, lying be ..."
Abstract

Cited by 7 (2 self)
 Add to MetaCart
A weak formal theory of arithmetic is developed, entirely analogous to classical arithmetic but with two separate kinds of variables: induction variables and quantifier variables. The point is that the provably recursive functions are now more feasibly computable than in the classical case, lying between Grzegorczyk's E² and E³, and their computational complexity can be characterized in terms of the logical complexity of their termination proofs. Previous results of Leivant are reworked and extended in this new setting, with quite di#erent proof theoretic methods.
Implementation of Proof Search in the Imperative Programming Language Pizza
 In Int. Conference TABLEAUX'98, LNAI 1397
, 1998
"... . Automated proof search can be easily implemented in logic programming languages. We demonstrate the technique of success continuations, which provides an equally simple method for encoding proof search in imperative programming languages. This technique is exemplified by developing an interpreter ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
. Automated proof search can be easily implemented in logic programming languages. We demonstrate the technique of success continuations, which provides an equally simple method for encoding proof search in imperative programming languages. This technique is exemplified by developing an interpreter for the calculus G4ip in the language Pizza. Keywords: Success Continuations, G4ip, Pizza 1 Introduction A sequentstyle formulation of a logic calculus is a convenient startingpoint for automating proof search because the corresponding inference rules are `local' operations on proofs. A sequent can be proved by applying inference rules until one reaches axioms, or can make no further progress in which case one must backtrack or even abandon the search. This proving method is a simple depthfirst strategy; it is preferred over a less efficient breadthfirst strategy. However, this method requires the mechanism of choice points in order to facilitate the backtracking. Logic programming lan...
An abstract machine based on linear logic and explicit substitutions
, 1997
"... a mis hermanas, Patricia y Paula, y a mi sobrino y ahijado, Nicol'as. Acknowledgements First of all, I would like to express my gratitude to my supervisor, Eike Ritter, for his wisdom, insight, uncountably many discussions, and invaluable friendship. I am indebted to my tutor, Valeria de Paiva, who ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
a mis hermanas, Patricia y Paula, y a mi sobrino y ahijado, Nicol'as. Acknowledgements First of all, I would like to express my gratitude to my supervisor, Eike Ritter, for his wisdom, insight, uncountably many discussions, and invaluable friendship. I am indebted to my tutor, Valeria de Paiva, who also believed in me from the very beginning, encouraged me to work in this area, showed me the beauty of logic, and, above all, honoured me with her friendship. This thesis would not exist if it were not for their constant support. Thanks to my old friends, Cecilia C. Crespo, Santiago M. Peric'as, and, especially, Mat'ias Giovannini, for being always a wonderful critic of my work. Many thanks to Mathias Kegelmann for showing me the thrill of theorem proving; and to my former supervisor, Achim Jung, for introducing me to semantics.
Elementary arithmetic
 Annals of Pure and Applied Logic
, 2001
"... There is a quite natural way in which the safe/normal variable discipline of BellantoniCook recursion (1992) can be imposed on arithmetical theories like PA: quantify over safes and induct on normals. This weakens the theory severely, so that the provably recursive functions become more realistical ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
There is a quite natural way in which the safe/normal variable discipline of BellantoniCook recursion (1992) can be imposed on arithmetical theories like PA: quantify over safes and induct on normals. This weakens the theory severely, so that the provably recursive functions become more realistically computable (slow growing rather than fast growing). Earlier results of Leivant (1995) are reworked and extended in this new context, giving prooftheoretic characterizations (according to the levels of induction used) of complexity classes between Grzegorczyk’s E 2 and E 3. This is a contribution to the search for “natural ” theories, without explicitlyimposed bounds on quantifiers as in Buss [3], whose provably recursive functions form “more feasible ” complexity classes (than for example the primitive recursive functions). We develop a quite different, alternative treatment of Leivant’s results in [6], where ramified inductions over N are cleverly used to obtain prooftheoretic characterizations of PTIME and Grzegorczyk’s classes E2 and E3; and we further extend the characterization
PSPACE bounds for rank 1 modal logics
 In LICS’06
, 2006
"... For lack of general algorithmic methods that apply to wide classes of logics, establishing a complexity bound for a given modal logic is often a laborious task. The present work is a step towards a general theory of the complexity of modal logics. Our main result is that all rank1 logics enjoy a sh ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
For lack of general algorithmic methods that apply to wide classes of logics, establishing a complexity bound for a given modal logic is often a laborious task. The present work is a step towards a general theory of the complexity of modal logics. Our main result is that all rank1 logics enjoy a shallow model property and thus are, under mild assumptions on the format of their axiomatisation, in PSPACE. This leads to a unified derivation of tight PSPACEbounds for a number of logics including K, KD, coalition logic, graded modal logic, majority logic, and probabilistic modal logic. Our generic algorithm moreover finds tableau proofs that witness pleasant prooftheoretic properties including a weak subformula property. This generality is made possible by a coalgebraic semantics, which conveniently abstracts from the details of a given model class and thus allows covering a broad range of logics in a uniform way.
A classical sequent calculus free of structural rules
, 2005
"... Gentzen’s system LK, classical sequent calculus, has explicit structural rules for contraction and weakening. They can be absorbed (in a rightsided formulation) by replacing the axiom P,¬P by Γ,P,¬P for any context Γ, and replacing the original disjuction rule with Γ,A,B implies Γ, A ∨ B. This pape ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
Gentzen’s system LK, classical sequent calculus, has explicit structural rules for contraction and weakening. They can be absorbed (in a rightsided formulation) by replacing the axiom P,¬P by Γ,P,¬P for any context Γ, and replacing the original disjuction rule with Γ,A,B implies Γ, A ∨ B. This paper presents a classical sequent calculus which is also free of contraction and weakening, but more symmetrically: both contraction and weakening are absorbed into conjunction, leaving the axiom rule untouched. It uses a hybrid conjunction rule, combining the standard contextsharing and contextsplitting rules: Γ,∆,A and Γ,Σ,B implies Γ,∆,Σ, A ∧ B. We call this system Hybrid Logic. Hybrid conjunction is critical for the liberation from structural rules: relaxing it to the pair of standard conjunction rules breaks completeness. We prove a minimality theorem for hybrid logic: any sequent calculus (within a standard class of rightsided calculi) is complete iff it contains hybrid logic. Thus one can view hybrid logic as a “complete core ” of Gentzen’s LK. 1