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The Logic of Bunched Implications
 BULLETIN OF SYMBOLIC LOGIC
, 1999
"... We introduce a logic BI in which a multiplicative (or linear) and an additive (or intuitionistic) implication live sidebyside. The propositional version of BI arises from an analysis of the prooftheoretic relationship between conjunction and implication; it can be viewed as a merging of intuition ..."
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Cited by 190 (38 self)
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We introduce a logic BI in which a multiplicative (or linear) and an additive (or intuitionistic) implication live sidebyside. The propositional version of BI arises from an analysis of the prooftheoretic relationship between conjunction and implication; it can be viewed as a merging of intuitionistic logic and multiplicative intuitionistic linear logic. The naturality of BI can be seen categorically: models of propositional BI's proofs are given by bicartesian doubly closed categories, i.e., categories which freely combine the semantics of propositional intuitionistic logic and propositional multiplicative intuitionistic linear logic. The predicate version of BI includes, in addition to standard additive quantifiers, multiplicative (or intensional) quantifiers # new and # new which arise from observing restrictions on structural rules on the level of terms as well as propositions. We discuss computational interpretations, based on sharing, at both the propositional and predic...
On Bunched Predicate Logic
 Proceedings of the IEEE Symposium on Logic in Computer Science
, 1999
"... We present the logic of bunched implications, BI, in which a multiplicative (or linear) and an additive (or intuitionistic) implication live sidebyside. The propositional version of BI arises from an analysis of the prooftheoretic relationship between conjunction and implication, and may be viewe ..."
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Cited by 29 (17 self)
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We present the logic of bunched implications, BI, in which a multiplicative (or linear) and an additive (or intuitionistic) implication live sidebyside. The propositional version of BI arises from an analysis of the prooftheoretic relationship between conjunction and implication, and may be viewed as a merging of intuitionistic logic and multiplicative, intuitionistic linear logic. The predicate version of BI includes, in addition to usual additive quantifiers, multiplicative (or intensional) quantifiers 8new and 9new , which arise from observing restrictions on structural rules on the level of terms as well as propositions. Moreover, these restrictions naturally allow the distinction between additive predication and multiplicative predication for each propositional connective. We provide a natural deduction system, a sequent calculus, a Kripke semantics and a BHK semantics for BI. We mention computational interpretations, based on locality and sharing, at both the propositiona...
Corrections and Remarks
 University of Edinburgh LFCS Report
, 2000
"... This document contains corrections to errors discovered todate in, and also some remarks upon, both Samin Ishtiaq's thesis, A Relevant Analysis of Natural Deduction [Ish99] and also the JLC [IP98] and CSL [IP99] papers, by Ishtiaq and Pym, that follow from it. 1 Introduction This document cont ..."
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Cited by 5 (4 self)
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This document contains corrections to errors discovered todate in, and also some remarks upon, both Samin Ishtiaq's thesis, A Relevant Analysis of Natural Deduction [Ish99] and also the JLC [IP98] and CSL [IP99] papers, by Ishtiaq and Pym, that follow from it. 1 Introduction This document contains corrections to errors discovered todate in both Samin Ishtiaq's thesis, A Relevant Analysis of Natural Deduction [Ish99] and also the JLC [IP98] and CSL [IP99] papers, by Ishtiaq and Pym, that follow from it. The postscript version of the thesis, available http://www.dcs.qmw.ac.uk/si, is the source of the hardbound copies submitted to the libraries of the University of London (Senate House) and Queen Mary and Westeld College. A copy of this document has been deposited with the Librarians at both of these institutions. 2 Ishtiaq's Ph.D. thesis [Ish99] The corrections are listed by chapter title. 2.1 Introduction On Page 12, Line 10, replace \fragment" by \variant". (Section 2.3 be...
Semantic Labelled Tableaux for Propositional BI
 Journal of Logic and Computation
, 2003
"... In this paper, we study semantic labelled tableaux for the propositional Bunched Implications logic (BI) that freely combines intuitionistic logic (IL) and multiplicative intuitionistic linear logic (MILL). ..."
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Cited by 4 (0 self)
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In this paper, we study semantic labelled tableaux for the propositional Bunched Implications logic (BI) that freely combines intuitionistic logic (IL) and multiplicative intuitionistic linear logic (MILL).
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.
Corrections and Remarks
 University of Edinburgh LFCS Report
, 1993
"... This document contains corrections to errors discovered todate in, and also some remarks upon, both Samin Ishtiaq's thesis, A Relevant Analysis of Natural Deduction [Ish99] and also the JLC [IP98] and CSL [IP99] papers, by Ishtiaq and Pym, that follow from it. 1 Introduction This document cont ..."
Abstract
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This document contains corrections to errors discovered todate in, and also some remarks upon, both Samin Ishtiaq's thesis, A Relevant Analysis of Natural Deduction [Ish99] and also the JLC [IP98] and CSL [IP99] papers, by Ishtiaq and Pym, that follow from it. 1 Introduction This document contains corrections to errors discovered todate in both Samin Ishtiaq's thesis, A Relevant Analysis of Natural Deduction [Ish99] and also the JLC [IP98] and CSL [IP99] papers, by Ishtiaq and Pym, that follow from it. The postscript version of the thesis, available http://www.dcs.qmw.ac.uk/si, is the source of the hardbound copies submitted to the libraries of the University of London (Senate House) and Queen Mary and Westeld College. A copy of this document has been deposited with the Librarians at both of these institutions. 2 Ishtiaq's Ph.D. thesis [Ish99] The corrections are listed by chapter title. 2.1 Introduction On Page 12, Line 10, replace \fragment" by \variant". (Section 2.3 be...
Notes Towards a Semantics for Proofsearch
"... Algorithmic proofsearch is an essential enabling technology throughout informatics. Proofsearch is the prooftheoretic realization of the formulation of logic not as a theory of deduction but rather as a theory of reduction. Whilst deductive logics typically have a welldeveloped semantics of proo ..."
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Algorithmic proofsearch is an essential enabling technology throughout informatics. Proofsearch is the prooftheoretic realization of the formulation of logic not as a theory of deduction but rather as a theory of reduction. Whilst deductive logics typically have a welldeveloped semantics of proofs, reductive logics are typically wellunderstood only operationally. Each deductive system can, typically, be read as a corresponding reductive system. We discuss some of the problems which must be addressed in order to provide a semantics of proofsearches of comparable value to the corresponding semantics of proofs. Just as the semantics of proofs is intimately related to the model theory of the underlying logic, so too should be the semantics of proofsearches. We discuss how to solve the problem of providing a semantics for proofsearches which adequately models both operational and logical aspects of the reductive system. 1
Semantic Labelled Tableaux for Propositional BI (BI without ...)
 Journal of Logic and Computation
, 2003
"... In this paper, we propose a semantic tableau method for the propositional logic of Bunched Implications BI that is a logic merging intuitionistic logic (IL) and multiplicative intuitionistic linear logic (MILL). With its resourcebased sharing interpretation, BI is the basis of new logical foundat ..."
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In this paper, we propose a semantic tableau method for the propositional logic of Bunched Implications BI that is a logic merging intuitionistic logic (IL) and multiplicative intuitionistic linear logic (MILL). With its resourcebased sharing interpretation, BI is the basis of new logical foundations for logic programming or for reasoning about mutable data structures. The characterization of provability is based here on the construction of a labelled tableau with a related socalled dependency graph that is a central semantical structure. Thus, we dene a procedure for tableau construction that terminates and that, from a given BI formula A, yields either a tableau proof of A or a nite countermodel of A in terms of the Kripke resource semantics. The restrictions of the method to IL and MILL provide new tableau methods for both logics including an easy countermodel generation. Moreover, we analyze some algorithmic aspects of labelled tableau construction by providing one calculus with free variables and another with liberalized rules. 1