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31
Strong Normalisation of CutElimination in Classical Logic
, 2000
"... In this paper we present a strongly normalising cutelimination procedure for classical logic. This procedure adapts Gentzen's standard cutreductions, but is less restrictive than previous strongly normalising cutelimination procedures. In comparison, for example, with works by Dragalin and D ..."
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Cited by 36 (4 self)
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In this paper we present a strongly normalising cutelimination procedure for classical logic. This procedure adapts Gentzen's standard cutreductions, but is less restrictive than previous strongly normalising cutelimination procedures. In comparison, for example, with works by Dragalin and Danos et al., our procedure requires no special annotations on formulae and allows cutrules to pass over other cutrules. In order to adapt the notion of symmetric reducibility candidates for proving the strong normalisation property, we introduce a novel term assignment for sequent proofs of classical logic and formalise cutreductions as term rewriting rules.
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 ..."
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Cited by 31 (13 self)
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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.
Tableaubased decision procedures for hybrid logic
 Journal of Logic and Computation
, 2005
"... Abstract. Hybrid logics are a principled generalization of both modal logics and description logics. It is wellknown that various hybrid logics without binders are decidable, but decision procedures are usually not based on tableau systems, a kind of formal proof procedure that lends itself towards ..."
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Cited by 21 (4 self)
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Abstract. Hybrid logics are a principled generalization of both modal logics and description logics. It is wellknown that various hybrid logics without binders are decidable, but decision procedures are usually not based on tableau systems, a kind of formal proof procedure that lends itself towards computer implementation. In this paper we give four different tableaubased decision procedures for a very expressive hybrid logic including the universal modality; three of the procedures are based on different tableau systems, and one procedure is based on a Gentzen system. The decision procedures make use of socalled loopchecks which is a technique standardly used in connection with tableau systems for other logics, namely prefixed tableau systems for transitive modal logics, as well as prefixed tableau systems for certain description logics. The loopchecks used in our four decision procedures are similar, but the four proof systems on which the procedures are based constitute a spectrum of different systems: prefixed and internalized systems, tableau and Gentzen systems.
Making Knowledge Explicit: How Hard It Is
, 2005
"... Artemov’s logic of proofs LP is a complete calculus of propositions and proofs, which is now becoming a foundation for the evidencebased approach to reasoning about knowledge. Additional atoms in LP have form t: F, read as “t is a proof of F ” (or, more generally, as “t is an evidence for F”) for a ..."
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Cited by 13 (1 self)
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Artemov’s logic of proofs LP is a complete calculus of propositions and proofs, which is now becoming a foundation for the evidencebased approach to reasoning about knowledge. Additional atoms in LP have form t: F, read as “t is a proof of F ” (or, more generally, as “t is an evidence for F”) for an appropriate system of terms t called proof polynomials. In this paper, we answer two wellknown questions in this area. One of the main features of LP is its ability to realize modalities in any S4derivation by proof polynomials thus revealing a statement about explicit evidences encoded in that derivation. We show that the original Artemov’s algorithm of building such realizations can produce proof polynomials of exponential length in the size of the initial S4derivation. We modify the realization algorithm to produce proof polynomials of at most quadratic length. We also found a modal formula, any realization of which necessarily requires selfreferential constants of type c: A(c). This demonstrates that the evidencebased reasoning encoded by the modal logic S4 is inherently selfreferential.
Lectures on the curryhoward isomorphism
, 1998
"... The CurryHoward isomorphism states an amazing correspondence between systems of formal logic as encountered in proof theory and computational calculi as found in type theory. For instance, minimal propositional logic corresponds to simply typed λcalculus, firstorder logic corresponds to dependent ..."
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Cited by 8 (0 self)
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The CurryHoward isomorphism states an amazing correspondence between systems of formal logic as encountered in proof theory and computational calculi as found in type theory. For instance, minimal propositional logic corresponds to simply typed λcalculus, firstorder logic corresponds to dependent types, secondorder logic corresponds to polymorphic types, etc. The isomorphism has many aspects, even at the syntactic level: formulas correspond to types, proofs correspond to terms, provability corresponds to inhabitation, proof normalization corresponds to term reduction, etc. But there is much more to the isomorphism than this. For instance, it is an old idea—due to Brouwer, Kolmogorov, and Heyting, and later formalized by Kleene’s realizability interpretation—that a constructive proof of an implication is a procedure that transforms proofs of the antecedent into proofs of the succedent; the CurryHoward isomorphism gives syntactic representations of such procedures. These notes give an introduction to parts of proof theory and related
A Solver for QBFs in Negation Normal Form
"... Various problems in artificial intelligence can be solved by translating them into a quantified boolean formula (QBF) and evaluating the resulting encoding. In this approach, a QBF solver is used as a black box in a rapid implementation of a more general reasoning system. Most of the current solvers ..."
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Cited by 6 (1 self)
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Various problems in artificial intelligence can be solved by translating them into a quantified boolean formula (QBF) and evaluating the resulting encoding. In this approach, a QBF solver is used as a black box in a rapid implementation of a more general reasoning system. Most of the current solvers for QBFs require formulas in prenex conjunctive normal form as input, which makes a further translation necessary, since the encodings are usually not in a specific normal form. This additional step increases the number of variables in the formula or disrupts the formula’s structure. Moreover, the most important part of this transformation, prenexing, is not deterministic. In this paper, we focus on an alternative way to process QBFs without these drawbacks and describe a solver, qpro, which is able to handle arbitrary formulas. To this end, we extend algorithms for QBFs to the nonnormal form case and compare qpro with the leading normal form provers on several problems from the area of artificial intelligence. We prove properties of the algorithms generalized to nonclausal form by using a novel approach based on a sequentstyle formulation of the calculus. 1.
Revisiting cutelimination: One difficult proof is really a proof
 RTA 2008
, 2008
"... Powerful proof techniques, such as logical relation arguments, have been developed for establishing the strong normalisation property of termrewriting systems. The first author used such a logical relation argument to establish strong normalising for a cutelimination procedure in classical logic. ..."
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Cited by 5 (3 self)
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Powerful proof techniques, such as logical relation arguments, have been developed for establishing the strong normalisation property of termrewriting systems. The first author used such a logical relation argument to establish strong normalising for a cutelimination procedure in classical logic. He presented a rather complicated, but informal, proof establishing this property. The difficulties in this proof arise from a quite subtle substitution operation. We have formalised this proof in the theorem prover Isabelle/HOL using the Nominal Datatype Package, closely following the first authors PhD. In the process, we identified and resolved a gap in one central lemma and a number of smaller problems in others. We also needed to make one informal definition rigorous. We thus show that the original proof is indeed a proof and that present automated proving technology is adequate for formalising such difficult proofs.
A systematic presentation of quantified modal logics
 University of Edinburgh
, 2002
"... this paper is an attempt at providing a systematic presentation of Quantified Modal Logics (with constant domains and rigid designators). We present a set of modular, uniform, normalizing, sound and complete labelled sequent calculi for all QMLs whose frame properties can be expressed as a finite se ..."
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Cited by 5 (3 self)
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this paper is an attempt at providing a systematic presentation of Quantified Modal Logics (with constant domains and rigid designators). We present a set of modular, uniform, normalizing, sound and complete labelled sequent calculi for all QMLs whose frame properties can be expressed as a finite set of firstorder sentences with equality. We first present CQK, a calculus for the logic QK, and then we extend it to any such logic QL. Each calculus, called CQL, is modular (obtained by adding rules to CQK), uniform (each added rule is clearly related to a property of the frame), normalizing (frame reasoning only happens at the top of the proof tree) and Kripkesound and complete for QL. We improve on the existing literature on the subject (mainly, [21]) by extending the class of logics for which such a presentation is given, and by giving a new proof of soundness and completeness.
Ordinals and Interactive Programs
, 2000
"... The work reported in this thesis arises from the old idea, going back to the origins of constructive logic, that a proof is fundamentally a kind of program. If proofs can be ..."
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Cited by 5 (2 self)
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The work reported in this thesis arises from the old idea, going back to the origins of constructive logic, that a proof is fundamentally a kind of program. If proofs can be
Resolution In The Calculus Of Structures
, 2003
"... nctive normal form, is obtained in SKSg as 2 i#  [([aa],R,T),(a,R),(a,T),U] s;s  [(a,R),(a,T),(a,R),(a,T),U] c#;c#  . Proving the structure Q by h resolution steps means finding a proof t . . ..."
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Cited by 5 (2 self)
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nctive normal form, is obtained in SKSg as 2 i#  [([aa],R,T),(a,R),(a,T),U] s;s  [(a,R),(a,T),(a,R),(a,T),U] c#;c#  . Proving the structure Q by h resolution steps means finding a proof t . . . [ t,P] r#  h1 . . . r#  , Where P is in disjunctive normal form. A refutation by resolution is simply obtained by topdown flipping the derivation above. This means that both styles of resolution are directly supported by the calculus of structures. By flipping the derivation above one introduces cuts (in correspondence to i# rules). Please notice that these cuts are finitary, since the atoms introduced by them are present in the conclusion as well (thanks to Kai Brnnler for this observation, see the paper [BG] about finitary cuts). This is not so important anyway, because in a refutation one builds the derivation topdown, so the presence of cuts, finitary or otherwise,