Intuitive Semantics for Full Intuitionistic Linear Logic (2014)
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This document provides an overview of linear logic and full intuitionistic linear logic. It introduces linear logic, which treats logical resources as premises that must be used exactly once. Full intuitionistic linear logic is then discussed, which extends intuitionistic linear logic with multiple conclusions. The document explores intuitive semantics for linear logic connectives based on interactions between a storekeeper and buyer in a "FashionMall". It concludes by advocating for game-based semantics and references other works in the field.
![Linear Logic
Full Intuitionistic Linear Logic
Multiplicative Disjunction
Intuitive Semantics
Linear Logic
Linear Logic (LL) introduced by Girard (1987): “[...]linear
logic comes from a proof-theoretic analysis of usual logic.”
LL: the best of both worlds, the dualities of classical logic
plus the constructive content of proofs of intuitionistic logic.
from semantics Coherence Spaces
Sequent calculus system and proof-nets (instead of Natural
Deduction) for classical Linear Logic. to begin with. Lots
since.
Valeria de Paiva VSL 2014](https://image.slidesharecdn.com/vsl2014-210811220236/85/Intuitive-Semantics-for-Full-Intuitionistic-Linear-Logic-2014-2-320.jpg)
Intuitive Semantics for Full Intuitionistic Linear Logic (2014)
- 1. Linear Logic Full Intuitionistic Linear Logic Multiplicative Disjunction Intuitive Semantics Intuitive Semantics for Full Intuitionistic Linear Logic Valeria de Paiva Luiz Carlos Pereira Nuance Communications, CA, US PUC-Rio, Philosophy Dept, RJ, Brazil July, 2014 Valeria de Paiva VSL 2014
- 2. Linear Logic Full Intuitionistic Linear Logic Multiplicative Disjunction Intuitive Semantics Linear Logic Linear Logic (LL) introduced by Girard (1987): “[...]linear logic comes from a proof-theoretic analysis of usual logic.” LL: the best of both worlds, the dualities of classical logic plus the constructive content of proofs of intuitionistic logic. from semantics Coherence Spaces Sequent calculus system and proof-nets (instead of Natural Deduction) for classical Linear Logic. to begin with. Lots since. Valeria de Paiva VSL 2014
- 3. Linear Logic Full Intuitionistic Linear Logic Multiplicative Disjunction Intuitive Semantics Basic Idea In Linear Logic formulas denote resources. Resources are premises, assumptions and conclusions, as they are used in logical proofs. For example: $1 −◦ latte If I have a dollar, I can get a Latte $1 −◦ cappuccino If I have a dollar, I can get a Cappuccino $1 I have a dollar Using my dollar and one of the premisses above, say $1 −◦ latte, gives me a latte, but the dollar is gone. Can conclude either latte or cappuccino with a dollar, but not both Usual logic doesn’t pay attention to uses of premisses, A implies B and A gives me B but I still have A... Valeria de Paiva VSL 2014
- 4. Linear Logic Full Intuitionistic Linear Logic Multiplicative Disjunction Intuitive Semantics Basic Idea Two Assumptions in LL cannot be discarded or duplicated. They must be used exactly once. Other approaches to accounting for logical resources before, e.g relevance logic, Lambek Calculus Great win of Linear Logic: Account for resources when you want to, otherwise fall back on traditional logic, via a modality A → B iff !A −◦ B The exponential or modality ! allows you to duplicate and to destroy resources Valeria de Paiva VSL 2014
- 5. Linear Logic Full Intuitionistic Linear Logic Multiplicative Disjunction Intuitive Semantics Models of Classical Linear Logic Apart from coherence spaces, several other algebraic semantics, e.g. quantales Large body of work on Categorical Semantics, cf. Seely, Bierman, Benton, Barber, Melliès Precursors: Lafont’s Linear Abstract Machine and de Paiva’s Dialectica Categories Some possible world semantics Intuitive semantics, Girard’s “cigarette vending machine”, Lafont’s Menu, Mitchell’s IE models Valeria de Paiva VSL 2014
- 6. Linear Logic Full Intuitionistic Linear Logic Multiplicative Disjunction Intuitive Semantics Duality vs. constructivity? Lafont and de Paiva ⇒ categorical models for Intuitionistic Linear Logic (ILL). ILL is obtained by restricting sequents of Classical Linear Logic to have a single conclusion. Hence no multiplicative disjunction ` nor linear possibiility ? How well does this correspond to a semantic restriction? Hyland and de Paiva introduced Full Intuitionistic Linear Logic (APAL, 1993) showing show that the constructive character of logical system is not given by syntactic size-restrictions on sequent calculus. Valeria de Paiva VSL 2014
- 7. Linear Logic Full Intuitionistic Linear Logic Multiplicative Disjunction Intuitive Semantics Full Intuitionistic Linear Logic A multiple conclusion version of multiplicative Linear Logic where connectives ⊗, ` and −◦ are independent of each other, as in IL. Valeria de Paiva VSL 2014
- 8. Linear Logic Full Intuitionistic Linear Logic Multiplicative Disjunction Intuitive Semantics Full Intuitionistic Linear Logic System obtained from categorical semantics, didn’t satisfy cut-elimination, as originally formulated. Journal version of system (Hyland and de Paiva, APAL 1993) relies on Curry-Howard correspondence, as the necessary restriction is on the associated lambda-term. (x is not a variable in ∆). Braüner and de Paiva’s improvement, condition on the dependencies between propositional atoms, plus calculus for modalities. de Paiva and Pereira similar solution for (full) Intuitionistic Logic, (2005) Valeria de Paiva VSL 2014
- 9. Linear Logic Full Intuitionistic Linear Logic Multiplicative Disjunction Intuitive Semantics What is the meaning of `? Games are perhaps the most interesting notion of semantics for Linear Logic. Dialectica categories can be considered a superpower game semantics. In a Dialectica space, the players (prover and denier) face-off in one single move. Both have to show their full strategies and the pay-off function simply computes whether player won (the relation is satisfied), the denier won (the negation of the relation is satisfied) or neither of the players won. (Formalism not so straightforward, if you do not like games...) Valeria de Paiva VSL 2014
- 10. Linear Logic Full Intuitionistic Linear Logic Multiplicative Disjunction Intuitive Semantics Can this ` be more intuitive? Pereira’s suggestion: Understand the connectives of FILL as actions that a user/buyer engages in, when interacting with the stock keeper of a big store, FashionMall semantics. Propositions A are considered basic resources that the store has (or not) in stock. The five independent propositional connectives (multiplicative conjunction and disjunction, additive conjunction and disjunction and linear implication) are interactions between the buyer and the store keeper described next. Negation (as usual in constructive logic) is taken as implication into falsum A −◦ ⊥ Valeria de Paiva VSL 2014
- 11. Linear Logic Full Intuitionistic Linear Logic Multiplicative Disjunction Intuitive Semantics FashionMall Interactions: If the user/buyer requests a multiplicative conjunction A ⊗ B the store keeper has to have both A and B and it has to provide both to the buyer, who takes both home. an additive conjunction A&B the store keeper has to have both A and B, but the buyer only takes one home. The buyer is allowed to choose which one. an additive disjunction A ⊕ B the store keeper only has to have one of either A or B, the buyer takes that one home. The buyer is not allowed to choose which one, the choice is the store keeper’s. a linear implication A −◦ B we have a higher-order rule: If the buying of the antecedent A is satisfied, then the buying of the consequent B will be satisfied too. a multiplicative disjunction A ` B the store keeper has to have both A and B, but the buyer only takes one home. The buyer is allowed to choose which one and can even come back and swap it. Valeria de Paiva VSL 2014
- 12. Linear Logic Full Intuitionistic Linear Logic Multiplicative Disjunction Intuitive Semantics Take Home Introduced you to FILL (Full Intuitionistic Linear Logic). Hope to have convinced you that it came from categorical semantics, but can be understood in terms of game-like interactions We believe we have soundness of the system, but a bit at a loss on how to prove completeness Valeria de Paiva VSL 2014
- 13. Linear Logic Full Intuitionistic Linear Logic Multiplicative Disjunction Intuitive Semantics Thanks! Valeria de Paiva VSL 2014
- 14. Linear Logic Full Intuitionistic Linear Logic Multiplicative Disjunction Intuitive Semantics References Linear logic, J-Y Girard, Theoretical computer science 50.1 (1987): 1-101. Full intuitionistic linear logic, M Hyland, V de Paiva - Annals of Pure and Applied Logic, 1993 A short note on intuitionistic propositional logic with multiple conclusions, V De Paiva, LC Pereira Manuscrito, Rev. Int. Fil., Campinas 28 (2), 317-329 Lorenzen Games for Full Intuitionistic Linear Logic, Valeria de Paiva, 14th Congress of Logic, Methodology and Philosophy of Science, Nancy, July, 19-26, 2011 The carcinogenic example, WPR Mitchell, Logic Journal of IGPL 5 (6), 795-810 http://math.stackexchange.com/questions/50340/what-is-the- intuition-behind-the-par-operator-in-linear-logic Valeria de Paiva VSL 2014
- 15. Linear Logic Full Intuitionistic Linear Logic Multiplicative Disjunction Intuitive Semantics 2014, XVI, 279 p. 24 illus. Printed book Hardcover ▶ 129,99 € | £117.00 | $179.00 ▶ *139,09 € (D) | 142,99 € (A) | CHF 173.50 eBook Available from your library or ▶ springer.com/shop MyCopy Printed eBook for just ▶ € | $ 24.99 ▶ springer.com/mycopy Order online at springer.com ▶ or for the Americas call (toll free) 1-800-SPRINGER ▶ or email us at: orders- ny@springer.com. ▶ For outside the Americas call +49 (0) 6221-345-4301 ▶ or email us at: orders-hd-individuals@springer.com. The first € price and the £ and $ price are net prices, subject to local VAT. Prices indicated with * include VAT for books; the €(D) includes 7% for Germany, the €(A) includes 10% for Austria. Prices indicated with ** include VAT for electronic products; 19% for Germany, 20% for Austria. All prices exclusive of carriage charges. Prices and other details are subject to change without notice. All errors and omissions excepted. L.C. Pereira, E.H. Haeusler, V. de Paiva (Eds.) Advances in Natural Deduction A Celebration of Dag Prawitz's Work Series: Trends in Logic, Vol. 39 ▶ First comprehensive collection to cover the diverse elements of natural deduction, and a celebration of the ground-breaking work of Dag Prawitz ▶ Surveys the full range of novel research directions ▶ Collects original papers covering the work of celebrated figures in the field of natural deduction This collection of papers celebrating the contributions of Swedish logician Dag Prawitz to Proof Theory, has been assembled from those presented at the Natural Deduction conference organized in Rio de Janeiro to honour his seminal research. Dag Prawitz’s work forms the basis of intuitionistic type theory and his inversion principle constitutes the foundation of most modern accounts of proof-theoretic semantics in Logic, Linguistics and Theoretical Computer Science. The range of contributions includes material on the extension of natural deduction with higher-order rules, as opposed to higher-order connectives, and a paper discussing the application of natural deduction rules to dealing with equality in predicate calculus. The volume continues with a key chapter summarizing work on the extension of the Curry-Howard isomorphism (itself a by-product of the work on natural deduction), via methods of category theory that have been successfully applied to linear logic, as well as many other contributions from highly regarded authorities. With an illustrious group of contributors addressing a wealth of topics and applications, this volume is a valuable addition to the libraries of academics in the multiple disciplines whose development has been given added scope by the methodologies supplied by natural deduction. The volume is representative of the rich and varied directions that Prawitz work has inspired in the area of natural deduction. Valeria de Paiva VSL 2014