Noncommutative logic

Abstract: 1. Introduction 2. Sequent calculus for linear logic 3. Some elementary syntactic results 4. The calculus of two implications: a digression 5. Embeddings and approximations 6. Natural deduction systems for linear logic 7. Hilbert-type systems 8. Algebraic semantics 9. Combinatorial linear logic 10. Girard domains 11. Coherence in symmetric monoidal categories 12. The storage operator as a coffee comonoid 13. Evaluation in typed calculi 14. Computation by lazy evaluation in CCC's 15. Computation by lazy evaluation in SMC's and ILC's 16. The categorical and linear machine 17. Proofnets for the multiplicative fragment 18. The algorithm of cut elimination for proof nets 19. Multiplicative operators 20. The undecidability of linear logic 21. Cut elimination and strong normalization References Index.

Abstract: This article introduces a logical system, called BV, which extends multiplicative linear logic by a noncommutative self-dual logical operator. This extension is particularly challenging for the sequent calculus, and so far, it is not achieved therein. It becomes very natural in a new formalism, called the calculus of structures, which is the main contribution of this work. Structures are formulas subject to certain equational laws typical of sequents. The calculus of structures is obtained by generalizing the sequent calculus in such a way that a new top-down symmetry of derivations is observed, and it employs inference rules that rewrite inside structures at any depth. These properties, in addition to allowing the design of BV, yield a modular proof of cut elimination.

Abstract: Linear logic: its syntax and semantics J. Y. Girard Part I. Categories and Semantics: 1. Bilinear logic in algebra and linguistics J. Lambek 2. A category arising in linear logic, complexity theory and set theory A. Blass 3. Hypercoherences: a strongly stable model of linear logic T. Erhard Part II. Complexity and Expressivity: 4. Deciding provability of linear logic formulas P. D. Lincoln 5. The direct simulation of Minsky machines in linear logic M. I. Kanovich 6. Stochastic interaction and linear logic P. D. Lincoln, J. Mitchell and A. Scedrov 7. Inheritance with exceptions C. Fouquere and J. Vauzeilles Part III. Proof Theory: 8. On the fine structure of the exponential rule S. Martini and A. Masini 9. Sequent calculi for second order logic V. Danos, J. B. Joinet and H. Schellinx Part IV. Proff Nets: 10. From proof nets to interaction nets Y. Lafont 11. Empires and kingdoms in MLL G. Bellin and J. Van De Wiele 12. Noncommutative proof nets V. M. Abrusci 13. Volume of multiplicative formulas and provability F. Metayer Part V. Geometry of Interaction: 14. Proof nets and Hilbert space V. Danos and L. Regnier 15. Geometry of interacion III: accomodating the additives J. Y. Girard.

Abstract: The linear logic introduced in [3] by J.-Y. Girard keeps one of the so-called structural rules of the sequent calculus: the exchange rule. In a one-sided sequent calculus this rule can be formulated asThe exchange rule allows one to disregard the order of the assumptions and the order of the conclusions of a proof, and this means, when the proof corresponds to a logically correct program, to disregard the order in which the inputs and the outputs occur in a program.In the linear logic introduced in [3], the exchange rule allows one to prove the commutativity of the multiplicative connectives, conjunction (⊗) and disjunction (⅋). Due to the presence of the exchange rule in linear logic, in the phase semantics for linear logic one starts with a commutative monoid. So, the usual linear logic may be called commutative linear logic.The aim of the investigations underlying this paper was to see, first, what happens when we remove the exchange rule from the sequent calculus for the linear propositional logic at all, and then, how to recover the strength of the exchange rule by means of exponential connectives (in the same way as by means of the exponential connectives ! and ? we recover the strength of the weakening and contraction rules).