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Labelled Propositional Modal Logics: Theory and Practice
, 1996
"... We show how labelled deductive systems can be combined with a logical framework to provide a natural deduction implementation of a large and wellknown class of propositional modal logics (including K, D, T , B, S4, S4:2, KD45, S5). Our approach is modular and based on a separation between a base lo ..."
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Cited by 34 (8 self)
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We show how labelled deductive systems can be combined with a logical framework to provide a natural deduction implementation of a large and wellknown class of propositional modal logics (including K, D, T , B, S4, S4:2, KD45, S5). Our approach is modular and based on a separation between a base logic and a labelling algebra, which interact through a fixed interface. While the base logic stays fixed, different modal logics are generated by plugging in appropriate algebras. This leads to a hierarchical structuring of modal logics with inheritance of theorems. Moreover, it allows modular correctness proofs, both with respect to soundness and completeness for semantics, and faithfulness and adequacy of the implementation. We also investigate the tradeoffs in possible labelled presentations: We show that a narrow interface between the base logic and the labelling algebra supports modularity and provides an attractive prooftheory (in comparision to, e.g., semantic embedding) but limits th...
TinkerType: a language for playing with formal systems
, 2003
"... TinkerType is a pragmatic framework for compact and modular description of formal systems (type systems, operational semantics, logics, etc.). A family of related systems is broken down into a set of clauses – individual inference rules – and a set of features controlling the inclusion of clauses in ..."
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Cited by 20 (0 self)
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TinkerType is a pragmatic framework for compact and modular description of formal systems (type systems, operational semantics, logics, etc.). A family of related systems is broken down into a set of clauses – individual inference rules – and a set of features controlling the inclusion of clauses in particular systems. Simple static checks are used to help maintain consistency of the generated systems. We present TinkerType and its implementation and describe its application to two substantial repositories of typed lambdacalculi. The first repository covers a broad range of typing features, including subtyping, polymorphism, type operators and kinding, computational effects, and dependent types. It describes both declarative and algorithmic aspects of the systems, and can be used with our tool, the TinkerType Assembler,to generate calculi either in the form of typeset collections of inference rules or as executable ML typecheckers. The second repository addresses a smaller collection of systems, and provides modularized proofs of basic safety properties.
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...
Metareasoning for multiagent epistemic logics
 In CLIMA V
, 2004
"... Abstract. We present an encoding of a sequent calculus for a multiagent epistemic logic in Athena, an interactive theorem proving system for manysorted firstorder logic. We then use Athena as a metalanguage in order to reason about the multiagent logic an as object language. This facilitates theo ..."
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Cited by 9 (7 self)
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Abstract. We present an encoding of a sequent calculus for a multiagent epistemic logic in Athena, an interactive theorem proving system for manysorted firstorder logic. We then use Athena as a metalanguage in order to reason about the multiagent logic an as object language. This facilitates theorem proving in the multiagent logic in several ways. First, it lets us marshal the highly efficient theorem provers for classical firstorder logic that are integrated with Athena for the purpose of doing proofs in the multiagent logic. Second, unlike modeltheoretic embeddings of modal logics into classical firstorder logic, our proofs are directly convertible into native epistemic logic proofs. Third, because we are able to quantify over propositions and agents, we get much of the generality and power of higherorder logic even though we are in a firstorder setting. Finally, we are able to use Athena’s versatile tactics for proof automation in the multiagent logic. We illustrate by developing a tactic for solving the generalized version of the wise men problem. 1
Interactive Theorem Proving with Temporal Logic
 JOURNAL OF SYMBOLIC COMPUTATION
, 1996
"... In this paper, we present a theorem prover for linear temporal logic. Our goal is to extend the capabilities of existing interactive and automatic systems for verifying temporal properties of software and hardware systems. We focus on increasing the eoeectiveness of user interaction in such systems. ..."
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Cited by 4 (0 self)
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In this paper, we present a theorem prover for linear temporal logic. Our goal is to extend the capabilities of existing interactive and automatic systems for verifying temporal properties of software and hardware systems. We focus on increasing the eoeectiveness of user interaction in such systems. In particular, we extend the techniques of proof by pointing and point and shoot for mousedriven proof construction in ørstorder logic to temporal logic. In addition, we show how to generate text from proofs by extending a previously given translation for ørstorder logic to the temporal operators. Our theorem prover implements an inference system for temporal logic that we have deøned. The inference rules of this system are more intuitive than the rules commonly given for temporal logics and thus they are better suited to our goals. We present this inference system and prove that it is sound and complete with respect to a known system.
A Topography Of Labelled Modal Logics
"... . Labelled Deductive Systems provide a general method for representing logics in a modular and transparent way. A Labelled Deductive System consists of two parts, a base logic and a labelling algebra, which interact through a fixed interface. The labelling algebra can be viewed as an independent par ..."
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Cited by 3 (2 self)
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. Labelled Deductive Systems provide a general method for representing logics in a modular and transparent way. A Labelled Deductive System consists of two parts, a base logic and a labelling algebra, which interact through a fixed interface. The labelling algebra can be viewed as an independent parameter: the base logic stays fixed for a given class of related logics from which we can generate the one we want by plugging in the appropriate algebra. Our work identifies an important property of the structured presentation of logics, their combination, and extension. Namely, there is tension between modularity and extensibility: a narrow interface between the base logic and labelling algebra can limit the degree to which we can make use of extensions to the labelling algebra. We illustrate this in the case of modal logics and apply simple results from proof theory to give examples. 1. Introduction The idea of a Labelled Deductive System (LDS) has been proposed as a general technique for...
Logic of Predicates With Explicit Substitutions
 Mathematical Foundations of Computer Science 1996, 21st Symposium
, 1996
"... This paper aims at supporting the same idea. Our justification of the claim is, however, quite different from the one offered by Girard. The latter, cf. [9], translates every sequent of the usual propositional logic (classical, or intuitionistic) into a sequent of commutative linear logic. Then one ..."
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Cited by 3 (3 self)
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This paper aims at supporting the same idea. Our justification of the claim is, however, quite different from the one offered by Girard. The latter, cf. [9], translates every sequent of the usual propositional logic (classical, or intuitionistic) into a sequent of commutative linear logic. Then one shows that a sequent can be proved classically, resp., intuitionistically, iff its translation can be proved linearly. By contrast, our embedding only works on the level of predicate logic. We show that every theory of classical logic of predicates with equality lives as a theory within a noncommutative intuitionistic substructural logic: the logic of predicates with equality and explicit substitution. Also, our explanation does not require to call upon so called exponentials  the modalities introduced by Girard just to facilitate his embedding. Our construction is also different from other proposals to move substitutions from the level of metatheory to the theory of logic, cf. [16]. They add substitutions as modal constructions. Here, substitutions are considered new atomic formulae.
Proof Tactics for a Theory of State Machines in a Graphical Environment
 In Proc. 14th Intenational Conference on Automated Deduction (CADE14), Lecture Notes in Artificial Intelligence
, 1997
"... . The state machine paradigm is a popular and convenient means for expressing designs of critical systems. State machines can be readily represented by transition graphs, thus enhancing human understanding of even quite complex problems. In the case of state machines, tracing a path through the ..."
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Cited by 2 (2 self)
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. The state machine paradigm is a popular and convenient means for expressing designs of critical systems. State machines can be readily represented by transition graphs, thus enhancing human understanding of even quite complex problems. In the case of state machines, tracing a path through the transition graph can represent a critical sequence in the execution of a machine. State machine notations are also amenable to formal treatment. A highlevel of assurance can be gained by a combination of both these aspects: a machinechecked, formal proof together with a higherlevel argument that can be understood by humans. This paper describes proof tactics that support reasoning about state machines at the level of diagrams and paths, and the construction of a corresponding formal proof. A tool, called Veracity [3], has been developed, which links these powerful proof tactics to a graphical userinterface. The proof tactics are implemented in Isabelle, and the paper discusses s...
Logic of Predicates Versus Linear Logic
 ICS PAS Reports, Vol 795
, 1995
"... This paper aims at supporting the same idea. Our justification of the claim is, however, quite different from the one envisaged by Girard. The latter, cf. [11], is prooftheoretic in nature. Firstly, every sequent of classical, resp., intuitionistic, logic is translated into a sequent of commutative ..."
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Cited by 2 (2 self)
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This paper aims at supporting the same idea. Our justification of the claim is, however, quite different from the one envisaged by Girard. The latter, cf. [11], is prooftheoretic in nature. Firstly, every sequent of classical, resp., intuitionistic, logic is translated into a sequent of commutative linear logic with exponentials. Then one shows that the former can be proved classically, resp., intuitionistically, iff its translation can be proved linearly. Here it is shown that every theory of classical logic of predicates with equality lives in a sufficiently rich theory built over a noncommutiative intuitionistic substructural logic: the logic of predicates with explicit substitution. This perspective does not require to call upon