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Logic Programming in a Fragment of Intuitionistic Linear Logic
"... When logic programming is based on the proof theory of intuitionistic logic, it is natural to allow implications in goals and in the bodies of clauses. Attempting to prove a goal of the form D ⊃ G from the context (set of formulas) Γ leads to an attempt to prove the goal G in the extended context Γ ..."
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Cited by 303 (40 self)
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When logic programming is based on the proof theory of intuitionistic logic, it is natural to allow implications in goals and in the bodies of clauses. Attempting to prove a goal of the form D ⊃ G from the context (set of formulas) Γ leads to an attempt to prove the goal G in the extended context Γ ∪ {D}. Thus during the bottomup search for a cutfree proof contexts, represented as the lefthand side of intuitionistic sequents, grow as stacks. While such an intuitionistic notion of context provides for elegant specifications of many computations, contexts can be made more expressive and flexible if they are based on linear logic. After presenting two equivalent formulations of a fragment of linear logic, we show that the fragment has a goaldirected interpretation, thereby partially justifying calling it a logic programming language. Logic programs based on the intuitionistic theory of hereditary Harrop formulas can be modularly embedded into this linear logic setting. Programming examples taken from theorem proving, natural language parsing, and data base programming are presented: each example requires a linear, rather than intuitionistic, notion of context to be modeled adequately. An interpreter for this logic programming language must address the problem of splitting contexts; that is, when attempting to prove a multiplicative conjunction (tensor), say G1 ⊗ G2, from the context ∆, the latter must be split into disjoint contexts ∆1 and ∆2 for which G1 follows from ∆1 and G2 follows from ∆2. Since there is an exponential number of such splits, it is important to delay the choice of a split as much as possible. A mechanism for the lazy splitting of contexts is presented based on viewing proof search as a process that takes a context, consumes part of it, and returns the rest (to be consumed elsewhere). In addition, we use collections of Kripke interpretations indexed by a commutative monoid to provide models for this logic programming language and show that logic programs admit a canonical model.
A Logic of Argumentation for Reasoning under Uncertainty.
 Computational Intelligence
, 1995
"... We present the syntax and proof theory of a logic of argumentation, LA. We also outline the development of a category theoretic semantics for LA. LA is the core of a proof theoretic model for reasoning under uncertainty. In this logic, propositions are labelled with a representation of the arguments ..."
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Cited by 107 (3 self)
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We present the syntax and proof theory of a logic of argumentation, LA. We also outline the development of a category theoretic semantics for LA. LA is the core of a proof theoretic model for reasoning under uncertainty. In this logic, propositions are labelled with a representation of the arguments which support their validity. Arguments may then be aggregated to collect more information about the potential validity of the propositions of interest. We make the notion of aggregation primitive to the logic, and then define strength mappings from sets of arguments to one of a number of possible dictionaries. This provides a uniform framework which incorporates a number of numerical and symbolic techniques for assigning subjective confidences to propositions on the basis of their supporting arguments. These aggregation techniques are also described, with examples. Key words: Uncertain reasoning, epistemic probability, argumentation, nonclassical logics, nonmonotonic reasoning 1. Introd...
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 100 (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
Metalogical Frameworks
, 1992
"... In computer science we speak of implementing a logic; this is done in a programming language, such as Lisp, called here the implementation language. We also reason about the logic, as in understanding how to search for proofs; these arguments are expressed in the metalanguage and conducted in the me ..."
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Cited by 57 (15 self)
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In computer science we speak of implementing a logic; this is done in a programming language, such as Lisp, called here the implementation language. We also reason about the logic, as in understanding how to search for proofs; these arguments are expressed in the metalanguage and conducted in the metalogic of the object language being implemented. We also reason about the implementation itself, say to know it is correct; this is done in a programming logic. How do all these logics relate? This paper considers that question and more. We show that by taking the view that the metalogic is primary, these other parts are related in standard ways. The metalogic should be suitably rich so that the object logic can be presented as an abstract data type, and it must be suitably computational (or constructive) so that an instance of that type is an implementation. The data type abstractly encodes all that is relevant for metareasoning, i.e., not only the term constructing functions but also the...
Focusing and Polarization in Linear, Intuitionistic, and Classical Logics
, 2009
"... A focused proof system provides a normal form to cutfree proofs in which the application of invertible and noninvertible inference rules is structured. Within linear logic, the focused proof system of Andreoli provides an elegant and comprehensive normal form for cutfree proofs. Within intuitioni ..."
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Cited by 44 (19 self)
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A focused proof system provides a normal form to cutfree proofs in which the application of invertible and noninvertible inference rules is structured. Within linear logic, the focused proof system of Andreoli provides an elegant and comprehensive normal form for cutfree proofs. Within intuitionistic and classical logics, there are various different proof systems in the literature that exhibit focusing behavior. These focused proof systems have been applied to both the proof search and the proof normalization approaches to computation. We present a new, focused proof system for intuitionistic logic, called LJF, and show how other intuitionistic proof systems can be mapped into the new system by inserting logical connectives that prematurely stop focusing. We also use LJF to design a focused proof system LKF for classical logic. Our approach to the design and analysis of these systems is based on the completeness of focusing in linear logic and on the notion of polarity that appears in Girard’s LC and LU proof systems.
CutElimination and a PermutationFree Sequent Calculus for Intuitionistic Logic
, 1998
"... We describe a sequent calculus, based on work of Herbelin, of which the cutfree derivations are in 11 correspondence with the normal natural deduction proofs of intuitionistic logic. We present a simple proof of Herbelin's strong cutelimination theorem for the calculus, using the recursive path ..."
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Cited by 41 (6 self)
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We describe a sequent calculus, based on work of Herbelin, of which the cutfree derivations are in 11 correspondence with the normal natural deduction proofs of intuitionistic logic. We present a simple proof of Herbelin's strong cutelimination theorem for the calculus, using the recursive path ordering theorem of Dershowitz.
Focusing the inverse method for linear logic
 Proceedings of CSL 2005
, 2005
"... 1.1 Quantification and the subformula property.................. 3 1.2 Ground forward sequent calculus......................... 5 1.3 Lifting to free variables............................... 10 ..."
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Cited by 37 (11 self)
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1.1 Quantification and the subformula property.................. 3 1.2 Ground forward sequent calculus......................... 5 1.3 Lifting to free variables............................... 10
Modality in Dialogue: Planning, Pragmatics and Computation
, 1998
"... Natural language generation (NLG) is first and foremost a reasoning task. In this reasoning, a system plans a communicative act that will signal key facts about the domain to the hearer. In generating action descriptions, this reasoning draws on characterizations both of the causal properties of the ..."
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Cited by 36 (9 self)
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Natural language generation (NLG) is first and foremost a reasoning task. In this reasoning, a system plans a communicative act that will signal key facts about the domain to the hearer. In generating action descriptions, this reasoning draws on characterizations both of the causal properties of the domain and the states of knowledge of the participants in the conversation. This dissertation shows how such characterizations can be specified declaratively and accessed efficiently in NLG. The heart of this dissertation is a study of logical statements about knowledge and action in modal logic. By investigating the prooftheory of modal logic from a logic programming point of view, I show how many kinds of modal statements can be seen as straightforward instructions for computationally manageable search, just as Prolog clauses can. These modal statements provide sufficient expressive resources for an NLG system to represent the effects of actions in the world or to model an addressee whose knowledge in some respects exceeds and in other respects falls short of its own. To illustrate the use of such statements, I describe how the SPUD sentence planner exploits a modal knowledge base to
A formulaeastypes interpretation of subtractive logic
 Journal of Logic and Computation
, 2004
"... We present a formulaeastypes interpretation of Subtractive Logic (i.e. biintuitionistic logic). This presentation is twofold: we first define a very natural restriction of the λµcalculus which is closed under reduction and whose type system is a constructive restriction of the Classical Natural ..."
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Cited by 23 (1 self)
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We present a formulaeastypes interpretation of Subtractive Logic (i.e. biintuitionistic logic). This presentation is twofold: we first define a very natural restriction of the λµcalculus which is closed under reduction and whose type system is a constructive restriction of the Classical Natural Deduction. Then we extend this deduction system conservatively to Subtractive Logic. From a computational standpoint, the resulting calculus provides a type system for firstclass coroutines (a restricted form of firstclass continuations). Keywords: CurryHoward isomorphism, Subtractive Logic, control operators, coroutines. 1