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Almost duplicationfree tableau calculi for propositional Lax logics
 In TABLEAUX'96
, 1996
"... In this paper we provide tableau calculi for the intuitionistic modal logics PLL and PLL 1 , where the calculus for PLL 1 is duplicationfree while among the rules for PLL there is just one rule that allows duplication of formulas. These logics have been investigated by Fairtlough and Mendler in re ..."
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In this paper we provide tableau calculi for the intuitionistic modal logics PLL and PLL 1 , where the calculus for PLL 1 is duplicationfree while among the rules for PLL there is just one rule that allows duplication of formulas. These logics have been investigated by Fairtlough and Mendler in relation to the problem of Formal Hardware Verification. In order to develop these calculi we extend to the modal case some ideas presented by Miglioli, Moscato and Ornaghi for intuitionistic logic. Namely, we enlarge the language containing the usual sings T and F with the new sign F c . PLL and PLL 1 logics are characterized by a Kripkesemantics which is a "weak" version of the semantics for ordinary intuitionistic modal logics. In this paper we establish the soundness and completeness theorems for these calculi.
A Permutationfree Calculus for Lax Logic
, 1998
"... this paper the same `permutationfree' techniques used to develop MJ are applied to Lax Logic, giving a `permutationfree' calculus for Lax Logic. As our starting point we take the above cited papers of Fairtlough & Mendler and of Benton, Bierman & de Paiva. 2 Natural Deduction ..."
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this paper the same `permutationfree' techniques used to develop MJ are applied to Lax Logic, giving a `permutationfree' calculus for Lax Logic. As our starting point we take the above cited papers of Fairtlough & Mendler and of Benton, Bierman & de Paiva. 2 Natural Deduction
Provably Correct Hardware Compilation using Timing Diagrams
, 1997
"... In this article we present a framework within which hardware implementations are proven correct from specifications given in an OCCAMlike language called Handel by the use of a robust set of mathematical transformational laws. The semantical basis for Handel and its hardware implementations are ..."
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In this article we present a framework within which hardware implementations are proven correct from specifications given in an OCCAMlike language called Handel by the use of a robust set of mathematical transformational laws. The semantical basis for Handel and its hardware implementations are simple functions of time which are called timing diagrams. This basis allows to denote the abstract properties of the Handel programs and hence the implementations in a modal logic, called Duration Calculus. The semantical treatment by one model for all three levels including the abstract properties, Handel and the level of gates, is one of the outstanding features of our approach. The delicate mathematical model which is used is able to cope with the complex form of parallelism used in Handel and with the detailed treatment of the relation between parallelism and timing. An immediate benefit of this approach is that Handel is a language already in use by hardware designers for spe...
Cover semantics for quantified lax logic
 Journal of Logic and Computation
"... Lax modalities occur in intuitionistic logics concerned with hardware verification, the computational lambda calculus, and access control in secure systems. They also encapsulate the logic of LawvereTierneyGrothendieck topologies on topoi. This paper provides a complete semantics for quantified la ..."
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Lax modalities occur in intuitionistic logics concerned with hardware verification, the computational lambda calculus, and access control in secure systems. They also encapsulate the logic of LawvereTierneyGrothendieck topologies on topoi. This paper provides a complete semantics for quantified lax logic by combining the BethKripkeJoyal cover semantics for firstorder intuitionistic logic with the classical relational semantics for a “diamond ” modality. The main technique used is the lifting of a multiplicative closure operator (nucleus) from a Heyting algebra to its MacNeille completion, and the representation of an arbitrary locale as the lattice of “propositions ” of a suitable cover system. In addition, the theory is worked out for certain constructive versions of the classical logics K and S4. An alternative completeness proof is given for (nonmodal) firstorder intuitionistic logic itself with respect to the cover semantics, using a simple and explicit Henkinstyle construction of a characteristic model whose points are principal theories rather than prime saturated ones. The paper provides further evidence that there is more to intuitionistic modal logic than the generalisation of properties of boxes and diamonds from Boolean modal logic.
Abstraction and Constraints: Two Sides of the Same Coin
, 1997
"... ion and Constraints: Two Sides of the Same Coin M.Walton November 1997 University of Sheffield Department of Computer Science Technical Report CS9718 Abstraction and Constraints: Two Sides of the Same Coin M. Walton PhD Supervisor: M.V.H. Fairtlough Abstract This report presents a highlev ..."
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ion and Constraints: Two Sides of the Same Coin M.Walton November 1997 University of Sheffield Department of Computer Science Technical Report CS9718 Abstraction and Constraints: Two Sides of the Same Coin M. Walton PhD Supervisor: M.V.H. Fairtlough Abstract This report presents a highlevel survey of some approaches to abstraction and the wide range of fields in which their applications may be found. Whilst there is much to be found in the literature about either abstraction or constraints, very little is to be found about them both, or the relationship between them. We examine the role of a novel intuitionistic modal logic (called Lax Logic) in capturing the dual notions of abstraction and constraints, and a particular notion of refinement. As a specific application of Lax Logic, we look at its use as an abstraction framework for the paradigm of Constraint Logic Programming. This provides a novel declarative and operational semantics for CLP which offers a clean separation...
Ternary Simulation: Refinement of Binary Functions
"... ©Copyright in this paper belongs to the author(s) Published in collaboration with the ..."
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©Copyright in this paper belongs to the author(s) Published in collaboration with the
Under consideration for publication in Math. Struct. in Comp. Science A Judgmental Reconstruction of Modal Logic
, 2000
"... We reconsider the foundations of modal logic, following MartinLöf’s methodology of distinguishing judgments from propositions. We give constructive meaning explanations for necessity and possibility which yields a simple and uniform system of natural deduction for intuitionistic modal logic which d ..."
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We reconsider the foundations of modal logic, following MartinLöf’s methodology of distinguishing judgments from propositions. We give constructive meaning explanations for necessity and possibility which yields a simple and uniform system of natural deduction for intuitionistic modal logic which does not exhibit anomalies found in other proposals. We also give a new presentation of lax logic and find that the lax modality is already expressible using possibility and necessity. Through a computational interpretation of proofs in modal logic we further obtain a new formulation of Moggi’s monadic metalanguage.