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Proof Search in Lax Logic
, 2000
"... This paper describes two new sequent calculi for Lax Logic. One calculus is for proof enumeration for quanti ed Lax Logic, the other calculus is for theorem proving in propositional Lax Logic ..."
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This paper describes two new sequent calculi for Lax Logic. One calculus is for proof enumeration for quanti ed Lax Logic, the other calculus is for theorem proving in propositional Lax Logic
A monadic formalization of ML5
 In Prepreceedings of Workshop on Logical Frameworks and Metalanguages: Theory and Practice
, 2010
"... ML5 is a programming language for spatially distributed computing, based on a CurryHoward correspondence with the modal logic S5. However, the ML5 programming language differs from the logic in several ways. In this paper, we give a semantic embedding of ML5 into the dependently typed programming l ..."
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ML5 is a programming language for spatially distributed computing, based on a CurryHoward correspondence with the modal logic S5. However, the ML5 programming language differs from the logic in several ways. In this paper, we give a semantic embedding of ML5 into the dependently typed programming language Agda, which both explains these discrepancies between ML5 and S5 and suggests some simplifications and generalizations of the language. Our embedding translates ML5 into a slightly different logic: intuitionistic S5 extended with a lax modality that encapsulates effectful computations in a monad. Rather than formalizing lax S5 as a proof theory, we embed it as a universe within the the dependently typed host language, with the universe elimination given by implementing the modal logic’s Kripke semantics. 1
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
Monads and eects
 Lecture Notes in Computer Science
, 2002
"... Abstract. A tension in language design has been between simple semantics on the one hand, and rich possibilities for sideeects, exception handling and so on on the other. The introduction of monads has made a large step towards reconciling these alternatives. First proposed by Moggi as a way of st ..."
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Abstract. A tension in language design has been between simple semantics on the one hand, and rich possibilities for sideeects, exception handling and so on on the other. The introduction of monads has made a large step towards reconciling these alternatives. First proposed by Moggi as a way of structuring semantic descriptions, they were adopted by Wadler to structure Haskell programs, and now oer a general technique for delimiting the scope of eects, thus reconciling referential transparency and imperative operations within one programming language. Monads have been used to solve longstanding problems such as adding pointers and assignment, interlanguage working, and exception handling to Haskell, without compromising its purely functional semantics. The course will introduce monads, eects and related notions, and exemplify their applications in programming (Haskell) and in compilation (MLj). The course will present typed metalanguages for monads and related categorical notions, and describe how they can be further rened by introducing eects.
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.
On the Logical Content of Computational Type Theory: A Solution to Curry's Problem
 In Types for Proofs and Programs
, 2002
"... In this paper we relate the lax modality O to Intuitionistic Propositional Logic (IPL) and give a complete characterisation of inhabitation in Computational Type Theory (CTT) as a logic of constraint contexts. This solves a problem open since the 1940's, when Curry was the first to suggest a fo ..."
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In this paper we relate the lax modality O to Intuitionistic Propositional Logic (IPL) and give a complete characterisation of inhabitation in Computational Type Theory (CTT) as a logic of constraint contexts. This solves a problem open since the 1940's, when Curry was the first to suggest a formal syntactic interpretation of O in terms of contexts.
On Lukasiewicz's fourvalued modal logic
, 2000
"... . # Lukasiewicz's fourvalued modal logic is surveyed and analyzed, together with # Lukasiewicz's motivations to develop it. A faithful interpretation of it into classical (nonmodal) twovalued logic is presented, and some consequences are drawn concerning its classification and its algeb ..."
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. # Lukasiewicz's fourvalued modal logic is surveyed and analyzed, together with # Lukasiewicz's motivations to develop it. A faithful interpretation of it into classical (nonmodal) twovalued logic is presented, and some consequences are drawn concerning its classification and its algebraic behaviour. Some counterintuitive aspects of this logic are discussed under the light of the presented results, # Lukasiewicz's own texts, and related literature. 1 Introduction The Polish philosopher and logician Jan # Lukasiewicz (Lwow, 1878  Dublin, 1956) is one of the fathers of modern manyvalued logic, and some of the systems he introduced are presently a topic of deep investigation. In particular his infinitelyvalued logic belongs to the core systems of mathematical fuzzy logic as a logic of comparative truth, see [5, 15, 14, 16]. However, it must be stressed here that # Lukasiewicz's logical work bears also a special relationship to modal logic. Actually, modal notions were part of #...
Under consideration for publication in Math. Struct. in Comp. Science Proof Search in Lax Logic
, 2000
"... Copyright & reuse City University London has developed City Research Online so that its users may access the research outputs of City University London's staff. Copyright © and Moral Rights for this paper are retained by the individual author(s) and / or other copyright holders. All materia ..."
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Copyright & reuse City University London has developed City Research Online so that its users may access the research outputs of City University London's staff. Copyright © and Moral Rights for this paper are retained by the individual author(s) and / or other copyright holders. All material in City Research Online is checked for eligibility for copyright before being made available in the live archive. URLs from City Research Online may be freely distributed and linked to from other web pages. Versions of research The version in City Research Online may differ from the final published version. Users are advised to check the Permanent City Research Online URL above for the status of the paper. Enquiries If you have any enquiries about any aspect of City Research Online, or if you wish to make contact with the author(s) of this paper, please email the team at publications@city.ac.uk.