Ebook: The Semantics and Proof Theory of the Logic of Bunched Implications

This is a monograph about logic. Specifically, it presents the mathe­ matical theory of the logic of bunched implications, BI: I consider Bl's proof theory, model theory and computation theory. However, the mono­ graph is also about informatics in a sense which I explain. Specifically, it is about mathematical models of resources and logics for reasoning about resources. I begin with an introduction which presents my (background) view of logic from the point of view of informatics, paying particular attention to three logical topics which have arisen from the development of logic within informatics: • Resources as a basis for semantics; • Proof-search as a basis for reasoning; and • The theory of representation of object-logics in a meta-logic. The ensuing development represents a logical theory which draws upon the mathematical, philosophical and computational aspects of logic. Part I presents the logical theory of propositional BI, together with a computational interpretation. Part II presents a corresponding devel­ opment for predicate BI. In both parts, I develop proof-, model- and type-theoretic analyses. I also provide semantically-motivated compu­ tational perspectives, so beginning a mathematical theory of resources. I have not included any analysis, beyond conjecture, of properties such as decidability, finite models, games or complexity. I prefer to leave these matters to other occasions, perhaps in broader contexts.

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This monograph provides a thorough account of the model theory, proof theory and computational interpretations of BI, the logic of bunched implications, which freely combines intuitionistic logic and multiplicative intuitionistic linear logic. Starting, on the one hand, from elementary observations about modelling resources and, on the other, from a desire to develop a system of logic within which additive (or extensional) and multiplicative (or intensional) implications co-exist with equal logical status, we give natural deduction, lambda-calculi, sequent calculus, categorical semantics, Kripke models, topological models, logical relations and computational interpretations for both propositional and predicate BI, within which both additive and multiplicative quantifiers also co-exist.

This monograph will be of interest to graduate students and researchers in mathematical logic, philosophical logic, computational logic and theoretical computer science.

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This monograph provides a thorough account of the model theory, proof theory and computational interpretations of BI, the logic of bunched implications, which freely combines intuitionistic logic and multiplicative intuitionistic linear logic. Starting, on the one hand, from elementary observations about modelling resources and, on the other, from a desire to develop a system of logic within which additive (or extensional) and multiplicative (or intensional) implications co-exist with equal logical status, we give natural deduction, lambda-calculi, sequent calculus, categorical semantics, Kripke models, topological models, logical relations and computational interpretations for both propositional and predicate BI, within which both additive and multiplicative quantifiers also co-exist.

This monograph will be of interest to graduate students and researchers in mathematical logic, philosophical logic, computational logic and theoretical computer science.

Content:
Front Matter....Pages i-xlix
Front Matter....Pages 1-1
Introduction to Part I....Pages 3-12
Natural Deduction for Propositional BI....Pages 13-31
Algebraic, Topological, Categorical....Pages 33-49
Kripke Semantics....Pages 51-66
Topological Kripke Semantics....Pages 67-87
Propositional BI as a Sequent Calculus....Pages 89-95
Towards Classical Propositional BI....Pages 97-106
Bunched Logical Relations....Pages 107-119
The Sharing Interpretation, I....Pages 121-144
Front Matter....Pages 145-145
Introduction to Part II....Pages 147-156
The Syntax of Predicate BI....Pages 157-162
Natural Deduction and Sequent Calculus....Pages 163-177
Kripke Semantics for Predicate BI....Pages 179-199
Topological Kripke Semantics for Predicate BI....Pages 201-205
Resource Semantics, Type Theory and Fibred Categories....Pages 207-261
The Sharing Interpretation, II....Pages 263-269
Back Matter....Pages 271-290

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This monograph provides a thorough account of the model theory, proof theory and computational interpretations of BI, the logic of bunched implications, which freely combines intuitionistic logic and multiplicative intuitionistic linear logic. Starting, on the one hand, from elementary observations about modelling resources and, on the other, from a desire to develop a system of logic within which additive (or extensional) and multiplicative (or intensional) implications co-exist with equal logical status, we give natural deduction, lambda-calculi, sequent calculus, categorical semantics, Kripke models, topological models, logical relations and computational interpretations for both propositional and predicate BI, within which both additive and multiplicative quantifiers also co-exist.

This monograph will be of interest to graduate students and researchers in mathematical logic, philosophical logic, computational logic and theoretical computer science.

Content:
Front Matter....Pages i-xlix
Front Matter....Pages 1-1
Introduction to Part I....Pages 3-12
Natural Deduction for Propositional BI....Pages 13-31
Algebraic, Topological, Categorical....Pages 33-49
Kripke Semantics....Pages 51-66
Topological Kripke Semantics....Pages 67-87
Propositional BI as a Sequent Calculus....Pages 89-95
Towards Classical Propositional BI....Pages 97-106
Bunched Logical Relations....Pages 107-119
The Sharing Interpretation, I....Pages 121-144
Front Matter....Pages 145-145
Introduction to Part II....Pages 147-156
The Syntax of Predicate BI....Pages 157-162
Natural Deduction and Sequent Calculus....Pages 163-177
Kripke Semantics for Predicate BI....Pages 179-199
Topological Kripke Semantics for Predicate BI....Pages 201-205
Resource Semantics, Type Theory and Fibred Categories....Pages 207-261
The Sharing Interpretation, II....Pages 263-269
Back Matter....Pages 271-290
....