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17
A Relevant Analysis of Natural Deduction
 Journal of Logic and Computation
, 1999
"... Linear and other relevant logics have been studied widely in mathematical, philosophical and computational logic. We describe a logical framework, RLF, for defining natural deduction presentations of such logics. RLF consists in a language together, in a manner similar to that of Harper, Honsell and ..."
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Linear and other relevant logics have been studied widely in mathematical, philosophical and computational logic. We describe a logical framework, RLF, for defining natural deduction presentations of such logics. RLF consists in a language together, in a manner similar to that of Harper, Honsell and Plotkin's LF, with a representation mechanism: the language of RLF is the lLcalculus; the representation mechanism is judgementsastypes, developed for relevant logics. The lLcalculus type theory is a firstorder dependent type theory with two kinds of dependent function spaces: a linear one and an intuitionistic one. We study a natural deduction presentation of the type theory and establish the required prooftheoretic metatheory. The RLF framework is a conservative extension of LF. We show that RLF uniformly encodes (fragments of) intuitionistic linear logic, Curry's l I calculus and ML with references. We describe the CurryHowardde Bruijn correspondence of the lLcalculus with a s...
A Natural Deduction Approach to Dynamic Logic
 Proceedings of TYPES'95, LNCS 1158
, 1996
"... . Natural Deduction style presentations of program logics are useful in view of the implementation of such logics in interactive proof development environments, based on type theory, such as LEGO, Coq, etc. In fact, NDstyle systems are the kind of systems which can take best advantage of the possib ..."
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. Natural Deduction style presentations of program logics are useful in view of the implementation of such logics in interactive proof development environments, based on type theory, such as LEGO, Coq, etc. In fact, NDstyle systems are the kind of systems which can take best advantage of the possibility of reasoning "under assumptions" o#ered by proof assistants generated by Logical Frameworks. In this paper we introduce and discuss sound and complete proof systems in Natural Deduction style for representing various "truth" consequence relations of Dynamic Logic. We discuss the design decisions which lead to adequate encodings of these logics in Coq. We derive in Dynamic Logic a set of rules representing a NDstyle system for Hoare Logic.
Higherorder Representation of Substructural Logics
, 2009
"... We present a technique for higherorder representation of substructural logics such as linear or modal logic. We show that such logics can be encoded in the (ordinary) Logical Framework, without any linear or modal extensions. Using this encoding, metatheoretic proofs about such logics can easily be ..."
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We present a technique for higherorder representation of substructural logics such as linear or modal logic. We show that such logics can be encoded in the (ordinary) Logical Framework, without any linear or modal extensions. Using this encoding, metatheoretic proofs about such logics can easily be developed in the Twelf proof assistant.
On the formalization of the modal µcalculus in the Calculus of Inductive Constructions
 Information and Computation
, 2000
"... This paper is part of an ongoing research programme at the Computer Science Department of the University of Udine on proof editors, started in 1992, based on HOAS encodings in dependent typed #calculus for program logics [15, 21, 16]. In this paper, we investigate the applicability of this approach ..."
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This paper is part of an ongoing research programme at the Computer Science Department of the University of Udine on proof editors, started in 1992, based on HOAS encodings in dependent typed #calculus for program logics [15, 21, 16]. In this paper, we investigate the applicability of this approach to the modal calculus. Due to its expressive power, we adopt the Calculus of Inductive Constructions (CIC), implemented in the system Coq. Beside its importance in the theory and verification of processes, the modal calculus is interesting also for its syntactic and proof theoretic peculiarities. These idiosyncrasies are mainly due to a) the negative arity of "" (i.e., the bound variable x ranges over the same syntactic class of x#); b) a contextsensitive grammar due the condition on x#; c) rules with complex side conditions (sequentstyle "proof " rules). These anomalies escape the "standard" representation paradigm of CIC; hence, we need to accommodate special techniques for enforcing these peculiarities. Moreover, since generated editors allow the user to reason "under assumptions", the designer of a proof editor for a given logic is urged to look for a Natural Deduction formulation of the system. Hence, we introduce a new proof system N # K in Natural Deduction style for K. This system should be more natural to use than traditional Hilbertstyle systems; moreover, it takes best advantage of the possibility of manipulating assumptions o#ered by CIC in order to implement the problematic substitution of formul for variables. In fact, substitutions are delayed as much as possible, and are kept in the derivation context by means of assumptions. This mechanism fits perfectly the stack discipline of assumptions of Natural Deduction, and it is neatly formalized in CIC. Bes...
LFP  A Logical Framework with External Predicates
, 2012
"... The LFP Framework is an extension of the HarperHonsellPlotkin’s Edinburgh Logical Framework LF with external predicates. This is accomplished by defining lock type constructors, which are a sort of ⋄modality constructors, releasing their argument under the condition that a possibly external predi ..."
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The LFP Framework is an extension of the HarperHonsellPlotkin’s Edinburgh Logical Framework LF with external predicates. This is accomplished by defining lock type constructors, which are a sort of ⋄modality constructors, releasing their argument under the condition that a possibly external predicate is satisfied on an appropriate typed judgement. Lock types are defined using the standard pattern of constructive type theory, i.e. via introduction, elimination, and equality rules. Using LFP, one can factor out the complexity of encoding specific features of logical systems which, otherwise, would be awkwardly encoded in LF, e.g. sideconditions in the application of rules in Modal Logics, and substructural rules, as in noncommutative Linear Logic. The idea of LFP is that these conditions need only to be specified, while their verification can be delegated to an external proof engine, in the style of the Poincaré Principle. We investigate and characterize the metatheoretical properties of the calculus underpinning LFP: strong normalization, confluence, and subject reduction. This latter property holds under the assumption that the predicates are wellbehaved, i.e. closed under weakening, permutation, substitution, and reduction in the arguments.
A framework for defining logical frameworks
 University of Udine
, 2006
"... Replace this file with prentcsmacro.sty for your meeting, or with entcsmacro.sty for your meeting. Both can be ..."
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Replace this file with prentcsmacro.sty for your meeting, or with entcsmacro.sty for your meeting. Both can be
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
A Conditional Logical Framework
"... The Conditional Logical Framework LFK is a variant of the HarperHonsellPlotkin’s Edinburgh Logical Framemork LF. It features a generalized form of λabstraction where βreductions fire under the condition that the argument satisfies a logical predicate. The key idea is that the type system memor ..."
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The Conditional Logical Framework LFK is a variant of the HarperHonsellPlotkin’s Edinburgh Logical Framemork LF. It features a generalized form of λabstraction where βreductions fire under the condition that the argument satisfies a logical predicate. The key idea is that the type system memorizes under what conditions and where reductions have yet to fire. Different notions of βreductions corresponding to different predicates can be combined in LFK. The framework LFK subsumes, by simple instantiation, LF (in fact, it is also a subsystem of LF!), as well as a large class of new generalized conditional λcalculi. These are appropriate to deal smoothly with the sideconditions of both Hilbert and Natural Deduction presentations of Modal Logics. We investigate and characterize the metatheoretical properties of the calculus underpinning LFK, such as subject reduction, confluence, strong normalization.
Towards Logical Frameworks in the Heterogeneous Tool Set Hets
"... Abstract. LF is a metalogical framework that has become a standard tool for representing logics and studying their properties. Its focus is proof theoretic, employing the CurryHoward isomorphism: propositions are represented as types, and proofs as terms. Hets is an integration tool for logics, lo ..."
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Abstract. LF is a metalogical framework that has become a standard tool for representing logics and studying their properties. Its focus is proof theoretic, employing the CurryHoward isomorphism: propositions are represented as types, and proofs as terms. Hets is an integration tool for logics, logic translations and provers, with a model theoretic focus, based on the metaframework of institutions, a formalisation of the notion of logical system. In this work, we combine these two worlds. The benefit for LF is that logics represented in LF can be (via Hets) easily connected to various interactive and automated theorem provers, model finders, model checkers, and conservativity checkers thus providing much more efficient proof support than mere proof checking as is done by systems like Twelf. The benefit for Hets is that (via LF) logics become represented formally, and hence trustworthiness of the implementation of logics is increased, and correctness of logic translations can be mechanically verified. Moreover, since logics and logic translations are now represented declaratively, the effort of adding new logics or translations to Hets is greatly reduced. This work is part of a larger effort of building an atlas of logics and translations used in computer science and mathematics. 1
Proofsearch in typetheoretic languages: an introduction
 Theoretical Computer Science
, 2000
"... We introduce the main concepts and problems in the theory of proofsearch in typetheoretic languages and survey some specific, connected topics. We do not claim to cover all of the theoretical and implementation issues in the study of proofsearch in typetheoretic languages; rather, we present som ..."
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Cited by 2 (1 self)
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We introduce the main concepts and problems in the theory of proofsearch in typetheoretic languages and survey some specific, connected topics. We do not claim to cover all of the theoretical and implementation issues in the study of proofsearch in typetheoretic languages; rather, we present some key ideas and problems, starting from wellmotivated points of departure such as a definition of a typetheoretic language or the relationship between languages and proofobjects. The strong connections between different proofsearch methods in logics, type theories and logical frameworks, together with their impact on programming and implementation issues, are central in this context.