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.

Includes supplementary material: sn.pub/extras

Klappentext

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.

Inhalt
I Propositional BI.- 1. Introduction to Part I.- 2. Natural Deduction for Propositional BI.- 3. Algebraic, Topological, Categorical.- 4. Kripke Semantics.- 5. Topological Kripke Semantics.- 6. Propositional BI as a Sequent Calculus.- 7. Towards Classical Propositional BI.- 8. Bunched Logical Relations.- 9. The Sharing Interpretation, I.- II Predicate BI.- 10. Introduction to Part II.- 11. The Syntax of Predicate BI.- 12. Natural Deduction & Sequent Calculus.- 13. Kripke Semantics for Predicate BI.- 14. Topological Kripke Semantics for Predicate BI.- 15. Resource Semantics, Type Theory & Fibred Categories.- 16. The Sharing Interpretation, II.