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17
The ProofTheory and Semantics of Intuitionistic Modal Logic
, 1994
"... Possible world semantics underlies many of the applications of modal logic in computer science and philosophy. The standard theory arises from interpreting the semantic definitions in the ordinary metatheory of informal classical mathematics. If, however, the same semantic definitions are interpret ..."
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Cited by 102 (0 self)
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Possible world semantics underlies many of the applications of modal logic in computer science and philosophy. The standard theory arises from interpreting the semantic definitions in the ordinary metatheory of informal classical mathematics. If, however, the same semantic definitions are interpreted in an intuitionistic metatheory then the induced modal logics no longer satisfy certain intuitionistically invalid principles. This thesis investigates the intuitionistic modal logics that arise in this way. Natural deduction systems for various intuitionistic modal logics are presented. From one point of view, these systems are selfjustifying in that a possible world interpretation of the modalities can be read off directly from the inference rules. A technical justification is given by the faithfulness of translations into intuitionistic firstorder logic. It is also established that, in many cases, the natural deduction systems induce wellknown intuitionistic modal logics, previously given by Hilbertstyle axiomatizations. The main benefit of the natural deduction systems over axiomatizations is their
Proof Search in the Intuitionistic Sequent Calculus
 11th International Conference on Automated Deduction
, 1991
"... The use of Herbrand functions (more popularly known as Skolemization) plays an important role in classical theorem proving and logic programming. We define a notion of Herbrand functions for the full intuitionistic predicate calculus. The definition is based on the view that the prooftheoretic role ..."
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Cited by 42 (1 self)
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The use of Herbrand functions (more popularly known as Skolemization) plays an important role in classical theorem proving and logic programming. We define a notion of Herbrand functions for the full intuitionistic predicate calculus. The definition is based on the view that the prooftheoretic role of Herbrand functions (to replace universal quantifiers), and of unification (to find instances corresponding to existential quantifiers), is to ensure that the eigenvariable conditions on a sequent proof are respected. The propositional impermutabilities that arise in the intuitionistic but not the classical sequent calculus motivate a generalization of the classical notion of Herbrand functions. Proof search using generalized Herbrand functions also provides a framework for generalizing logic programming to subsets of intuitionistic logic that are larger than Horn clauses. The search procedure described here has been implemented and observed to work effectively in practice. The generaliza...
A Resolution Theorem Prover for Intuitionistic Logic
 Proceedings of CADE13
, 1996
"... We use the general scheme of building resolution calculi (also called the inverse method) originating from S.Maslov and G.Mints to design and implement a resolution theorem prover for intuitionistic logic. A number of search strategies is introduced and proved complete. The resolution method is show ..."
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Cited by 42 (4 self)
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We use the general scheme of building resolution calculi (also called the inverse method) originating from S.Maslov and G.Mints to design and implement a resolution theorem prover for intuitionistic logic. A number of search strategies is introduced and proved complete. The resolution method is shown to be a decision procedure for a new syntactically described decidable class of intuitionistic logic. We compare the search strategies suitable for the resolution method with strategies suitable for the tableau method. The performance of our prover is compared with the performance of a tableau prover for intuitionistic logic presented in [17].
Encoding Modal Logics in Logical Frameworks
 Studia Logica
, 1997
"... We present and discuss various formalizations of Modal Logics in Logical Frameworks based on Type Theories. We consider both Hilbert and Natural Deductionstyle proof systems for representing both truth (local) and validity (global) consequence relations for various Modal Logics. We introduce severa ..."
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Cited by 14 (8 self)
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We present and discuss various formalizations of Modal Logics in Logical Frameworks based on Type Theories. We consider both Hilbert and Natural Deductionstyle proof systems for representing both truth (local) and validity (global) consequence relations for various Modal Logics. We introduce several techniques for encoding the structural peculiarities of necessitation rules, in the typed calculus metalanguage of the Logical Frameworks. These formalizations yield readily proofeditors for Modal Logics when implemented in Proof Development Environments, such as Coq or LEGO. Keywords: Hilbert and NaturalDeduction proof systems for Modal Logics, Logical Frameworks, Typed calculus, Proof Assistants. Introduction In this paper we address the issue of designing proof development environments (i.e. "proof editors" or, even better, "proof assistants") for Modal Logics, in the style of [11, 12]. To this end, we explore the possibility of using Logical Frameworks (LF's) based on Type Theory...
Connection Methods in Linear Logic and Proof Nets Construction
 Theoretical Computer Science
, 1999
"... Linear logic (LL) is the logical foundation of some typetheoretic languages and also of environments for specification and theorem proving. In this paper, we analyse the relationships between the proof net notion of LL and the connection notion used for efficient proofsearch in different logics. A ..."
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Cited by 12 (2 self)
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Linear logic (LL) is the logical foundation of some typetheoretic languages and also of environments for specification and theorem proving. In this paper, we analyse the relationships between the proof net notion of LL and the connection notion used for efficient proofsearch in different logics. Aiming at using proof nets as a tool for automated deduction in linear logic, we define a connectionbased characterization of provability in Multiplicative Linear Logic (MLL). We show that an algorithm for proof net construction can be seen as a proofsearch connection method. This central result is illustrated with a specific algorithm that is able to construct, for a provable MLL sequent, a set of connections, a proof net and a sequent proof. From these results we expect to extend to other LL fragments, we analyse what happens with the additive connectives of LL by tackling the additive fragment in a similar way.
Efficiently Deciding Intuitionistic Propositional Logic via Translation into Classical Logic
, 1996
"... We present a technique that efficiently translates propositional intuitionistic formulas into propositional classical formulas. This technique allows the use of arbitrary classical theorem provers for deciding the intuitionistic validity of a given propositional formula. The translation is based on ..."
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Cited by 6 (3 self)
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We present a technique that efficiently translates propositional intuitionistic formulas into propositional classical formulas. This technique allows the use of arbitrary classical theorem provers for deciding the intuitionistic validity of a given propositional formula. The translation is based on the constructive description of a finite countermodel for any intuitionistic nontheorem. This enables us to replace universal quantification over all accessible worlds by a conjunction over the constructed finite set of these worlds within the encoding of a refuting Kripkeframe. This way, no additional theory handling by the theorem prover is required.
Automated Theorem Proving in Multiple Valued Logics
"... algebras 4 2.2 Syntax and semantics 5 2.3 Examples of multiplevalued logics 11 2.3.1 The logics of Kleene and of Rosser and Turquette 12 2.3.2 Lukasiewicz Logics 13 2.3.3 Post logics 14 3 The Logical Basis: Signed Analytic Tableaux 15 ..."
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Cited by 6 (0 self)
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algebras 4 2.2 Syntax and semantics 5 2.3 Examples of multiplevalued logics 11 2.3.1 The logics of Kleene and of Rosser and Turquette 12 2.3.2 Lukasiewicz Logics 13 2.3.3 Post logics 14 3 The Logical Basis: Signed Analytic Tableaux 15
Relations Between Propositional Normal Modal Logics: An Overview
 Journal of Logic and Computation
, 1995
"... The modal logic literature is notorious for multiple axiomatisations of the same logic and for conflicting overloading of axiom names. Many of the interesting interderivability results are still scattered over the often hard to obtain classics. We catalogue the most interesting axioms, their numerou ..."
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Cited by 5 (3 self)
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The modal logic literature is notorious for multiple axiomatisations of the same logic and for conflicting overloading of axiom names. Many of the interesting interderivability results are still scattered over the often hard to obtain classics. We catalogue the most interesting axioms, their numerous variants, and explore the relationships between them in terms of interderivability as both axiom (schema) and as simple formulae. In doing so we introduce the Logics Workbench (LWB, see http://lwbwww.unibe.ch:8080/LWBinfo.html), a versatile tool for proving theorems in numerous propositional (nonclassical) logics. As a side effect we fulfill a call from the modal theorem proving community for a database of known theorems. 1 Introduction The modal logic literature is quite diverse but most beginners turn to the introductory works by Hughes and Cresswell [HC68, HC84], Bull and Segerberg [BS86], Lemmon and Scott [LS77], or Chellas [Che80]. The classic by Segerberg [Seg71] is indispensable...
AnnotationBased Deduction in Temporal Logic
 Journal of the Association for Computing Machinery
, 1994
"... . This paper presents a deductive system for predicate temporal logic with induction. Representing temporal operators by firstorder expressions enables temporal deduction to use the already developed techniques of firstorder deduction. But when translating from temporal logic to firstorder logic ..."
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Cited by 4 (0 self)
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. This paper presents a deductive system for predicate temporal logic with induction. Representing temporal operators by firstorder expressions enables temporal deduction to use the already developed techniques of firstorder deduction. But when translating from temporal logic to firstorder logic is done indiscriminately, the ensuing quantifications and comparisons of time expressions encumber formulas, hindering deduction. So in the deductive system presented here, translation occurs more carefully, via reification rules. These rules paraphrase selected temporal formulas as nontemporal firstorder formulas with time annotations. This time reification process suppresses quantifications (the process is analogous to quantifier skolemization) and uses addition instead of complicated combinations of comparisons. Some ordering conditions on arithmetic expressions can arise, but such are handled automatically by a specialpurpose unification algorithm plus a decision procedure for Presburg...
On the modal logic K plus theories
 Proc. CSL'95
, 1996
"... . K + T is the propositional modal logic K with the elements of the nite set T as additional axioms. We develop a sequent calculus that is suited for proof search in K + T and discuss methods to improve the eciency. An implementation of the resulting decision procedure is part of the Logics Workben ..."
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Cited by 3 (1 self)
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. K + T is the propositional modal logic K with the elements of the nite set T as additional axioms. We develop a sequent calculus that is suited for proof search in K + T and discuss methods to improve the eciency. An implementation of the resulting decision procedure is part of the Logics Workbench LWB. Then we show that { in contrast to K, KT, S4 { there are theories T and formulas A where a countermodel must have a superpolynomial diameter in the size of T plus A. In the last part we construct an embedding of S4 in K + T . 1 Introduction A Hilbertstyle calculus for (propositional) K + T is obtained from the usual Hilbertstyle calculus for the modal logic K by adding the formulas of T as additional axioms. There is also a natural characterization of K + T from the point of view of Kripke structures: We restrict ourselves to Kripke structures whose worlds satisfy T . One application of K+T is the proof of the equivalence of propositional normal modal logics ([5]). Another app...