Introduction to linear logic
نویسنده
چکیده
is manuscript is the lecture notes for the course of the same title I gave at the Summer school on linear logic and geometry of interaction that took place in Torino in August 2013. e aim of this course is to give a broad introduction to linear logic, covering enough ground to present many of the ideas and techniques of the field, while staying at a (hopefully) accessible level for beginners. For this reason, most technical development is carried out in the simple multiplicative fragment, with only hints at generalizations. As prerequisites, some knowledge of classical sequent calculus and some knowledge of the λ-calculus is useful. 1 e proof-program correspondence . . . . . . . . . . . . . . . . . . . 2 1.1 e Curry-Howard isomorphism . . . . . . . . . . . . . . . . . . . 2 1.2 Denotational semantics . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Linearity in logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2 Linear sequent calculus . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.1 Multiplicative linear logic . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Cut elimination and consistency . . . . . . . . . . . . . . . . . . . 8 2.3 One-sided presentation . . . . . . . . . . . . . . . . . . . . . . . . 10 2.4 Full linear logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.5 e notion of fragment . . . . . . . . . . . . . . . . . . . . . . . . 16 3 A bit of semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.1 Provability semantics . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.2 Proof semantics in coherence spaces . . . . . . . . . . . . . . . . . 18 4 A bit of proof theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 4.1 Intuitionistic and classical logics as fragments . . . . . . . . . . . . 19 4.2 Cut elimination and proof equivalence . . . . . . . . . . . . . . . . 21 4.3 Reversibility and focalization . . . . . . . . . . . . . . . . . . . . . 22 5 Proof nets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 5.1 Intuitionistic LL and natural deduction . . . . . . . . . . . . . . . . 23 5.2 Proof structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 5.3 Correctness criteria . . . . . . . . . . . . . . . . . . . . . . . . . . 30 Reference material on the topics discussed here include Girard, Lafont, and Taylor, Proofs and types is a book evolved from lecture notes of a graduate course of the same title. It is a good reference for an introduction to the proof-program correspondence, although it does not cover all topics of the maer. Girard, “Linear Logic” is the historical paper introducing linear logic. It is an unavoidable reference, although not the best way to discover the topic nowadays, since many aspects have been beer understood since then. Girard, “Linear Logic: Its Syntax and Semantics” is an updated andmore accessible presentation, written about ten years later.
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