# Proof-theoretical observations of BI and BBI base-logic interactions, and development of phased sequence calculus to define logic combinations

• Ryuta Arisaka

Student thesis: Doctoral Thesis

### Abstract

I study sequent calculus of combined logics in this thesis. Two specific logics are looked at-Logic BI that combines intuitionistic logic and multiplicative intuitionistic linear logic and Logic BBI that combines classical logic and multiplicative linear logic. A proof-theoretical study into logical combinations themsel ves then follows. To consolidate intuition about what this thesis is all about, let us suppose that we know about two different logics, Logic A developed for reasoning about Purpose A and Logic B developed for reasoning about Purpose B. Logic A serves Purpose A very well, but not Purpose B. Logic B serves Purpose B very well but not Purpose A. We wish to fulfill both Purpose A and Purpose B, but presently we can only afford to let one logic guide through our reasoning. What shall we do? One option is to be content with having Logic A with which we handle Purpose A efficiently and Purpose B rather inefficiently. Another option is to choose Logic B instead. But there is yet another option: we combine Logic A and Logic B to derive a new logic Logic C which is still one logic but which serves both Purpose A and Purpose B efficiently. The combined logic is synthetic of the strengths in more basic logics (Logic A and Logic B). As it nicely takes care of our requirements, it may be the best choice among all that have been so far considered. Yet this is not the end of the story. Depending on the manner Logic A and Logic B combine, Logic C may have extensions serving more purposes than just Purpose A and Purpose B. Ensuing is the following problem: we know about Logic A and Logic B, but we may not know about combined logics of the base logics. To understand the combined logics, we need to understand the extensions in which base logics interact each other. Analysis on the interesting parts tends to be non-trivial, however. The mentioned two specific combined logics BI and BBI do not make an exception, for which proof-theoretical development has been particularly slow. It has remained in obscurity how to properly handle base-logic interactions of the combined logics as appearing syntactically. As one objective of this thesis, I provide analysis on the syntactic phenomena of the BI and BBI base-logic interactions within sequent calculus, to augment the knowledge. For BI, I deliver, through appropriate methodologies to reason about the syntactic phenomena of the base-logic interactions, the first BI sequent calculus free of any structural rules. Given its positive consequence to efficient proof searches, this is a significant step forward in further maturity of BI proof theory. Based on the calculus, I prove decidability of a fragment of BI purely syntactically. For BBI which is closely connected to application via separation logic, I develop adequate sequent calculus conventions and consider the implication of the underlying semantics onto syntax. Sound BBI sequent calculi result with a closer syntax-semantics correspondence than previously envisaged. From them, adaptation to separation logic is also considered. To promote the knowledge of combined logics in general within computer science, it is also important that we be able to study logical combinations themselves. Towards this direction of generalisation, I present the concept of phased sequent calculus - sequent calculus which physically separates base logics, and in which a specific manner of logical combination to take place between them can be actually developed and analysed. For a demonstration, the said decidable BI fragment is formulated in phased sequent calculus, and the sense of logical combination in effect is analysed. A decision procedure is presented for the fragment.
Date of Award 12 Aug 2013 English Teesside University Shengchao Qin (Supervisor)

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