ABSTRACT
Proof nets for MLL (unit-free Multiplicative Linear Logic) are concise graphical representations of proofs which are canonical in the sense that they abstract away syntactic redundancy such as the order of non-interacting rules. We argue that Girard's extension to MLL1 (first-order MLL) fails to be canonical because of redundant existential witnesses, and present canonical MLL1 proof nets called unification nets without them. For example, while there are infinitely many cut-free Girard nets ∀x Px ⊢ ∃x Px, one per arbitrary witness for ∃x, there is a unique cut-free unification net, with no specified witness.
Cut elimination for unification nets is local and linear time, while Girard's is non-local and exponential time. Since some unification nets are exponentially smaller than corresponding Girard nets and sequent proofs, technical delicacy is required to ensure correctness is polynomial-time (quadratic).
These results transcend MLL1 via a methodological insight: for canonical quantifiers, the standard parallel/sequential dichotomy of proof nets is insufficient; an implicit/explicit witness dichotomy is needed. Current work extends unification nets to additives and uses them to extend combinatorial proofs [Proofs without syntax, Annals of Mathematics, 2006] to classical first-order logic.
- P. B. Andrews. 1981. Theorem Proving via General Matings. J. ACM 28 (1981). Google ScholarDigital Library
- G. Bellin and J. van de Wiele. 1995. Subnets of proof-nets in MLL---. In Advances in Linear Logic. Cambridge University Press, Cambridge, U.K. Google ScholarDigital Library
- W. Bibel. 1981. On Matrices with Connections. J. ACM 28, 4 (1981). Google ScholarDigital Library
- A. Carbone. 2010. A New Mapping Between Combinatorial Proofs and Sequent Calculus Proofs Read out from Logical Flow Graphs. Inf. Comput. 208, 5 (2010). Google ScholarDigital Library
- S. A. Cook and R. A. Reckhow. 1979. The relative efficiency of propositional proof systems. J. Symb. Logic 44 (1979), 36--50. Issue 1.Google ScholarCross Ref
- V. Danos. 1990. La logique linéaire appliquée à l'étude de divers processus de normal. et princip. du lambda calcul. Ph.D. Dissertation. Univ. de Paris.Google Scholar
- V. Danos and L. Regnier. 1989. The Structure of Multiplicatives. Archive for Mathematical Logic 28 (1989), 181--203.Google ScholarCross Ref
- J.-Y. Girard. 1987. Linear Logic. Theor. Comp. Sci. 50 (1987), 1--102. Google ScholarDigital Library
- J.-Y. Girard. 1988. Quantifiers in linear logic. In Temi e prospettive della logica e della filosofia della scienza contemporanee, Vol. 1. CLUEB, Bologna.Google Scholar
- J.-Y. Girard. 1991. Quantifiers in linear logic II. Nuovi problemi della logica e della filosofia della scienza 2 (1991).Google Scholar
- J.-Y. Girard. 1996. Proof-nets: the parallel syntax for proof theory. In Logic and Algebra. Lec. Notes In Pure and Applied Math., Vol. 180.Google Scholar
- J.-Y. Girard. 2011. The Blind Spot: Lectures on Logic. European Math. Soc.Google Scholar
- S. Guerrini. 1999. Correctness of Multiplicative Proof Nets Is Linear. In Proc. Logic in Computer Science '99. Google ScholarDigital Library
- W. Heijltjes. 2010. Classical proof forestry. Annals of Pure and Applied Logic 161, 11 (2010).Google ScholarCross Ref
- J. Herbrand. 1930. Recherches sur la théorie de la démonstration. Ph.D. Dissertation. Sorbonne, Paris.Google Scholar
- D. J. D. Hughes. 2006. Proofs without syntax. Annals of Mathematics 143 (2006), 1065--1076. Issue 3.Google ScholarCross Ref
- D. J. D. Hughes. 2006. Towards Hilbert's 24th Problem: Combinatorial Proof Invariants. In Proc. WOLLiC'06 (Lec. Notes in Comp. Sci), Vol. 165. Google ScholarDigital Library
- D. J. D. Hughes. 2014. First-order Proofs Without Syntax. (2014). Slides for the Berkeley Logic Colloquium, October 17, 2014. http://boole.stanford.edu/~dominic/papers/fopws-blc-2014.Google Scholar
- D. J. D. Hughes and R. J. van Glabbeek. 2003. Proof nets for unit-free multiplicative additive linear logic (Extended abstract). In Proc. Logic in Comp. Sci. '03. Google ScholarDigital Library
- D. J. D. Hughes and R. J. van Glabbeek. 2005. Proof nets for unit-free multiplicative-additive linear logic. ACM Trans. Comput. Log. 6 (2005). Google ScholarDigital Library
- J.-L. Lassez, M. J. Maher, and K. Marriott. 1988. Unification revisited. In Foundations of deductive databases and logic programming. Google ScholarDigital Library
- A. Martelli and U. Montanari. 1976. Unification in linear time and space: a structured presentation. Technical Report B76-16. University of Pisa.Google Scholar
- R. McKinley. 2010. Expansion nets: Proof nets for for propositional classical logic. In LPAR 17 (Lec. Notes in Comp. Sci.), Vol. 6397. Google ScholarDigital Library
- D. A. Miller. 1984. Expansion Tree Proofs and Their Conversion to Natural Deduction Proofs. Lec. Notes in Comp. Sci. 170 (1984). Google ScholarDigital Library
- R. Moot. 2002. Proof nets for lingusitic analysis. Ph.D. Dissertation. Utrecht.Google Scholar
- D. Prawitz. 1970. A proof procedure with matrix reduction. Lecture Notes in Mathematics 125 (1970).Google ScholarCross Ref
- W. V Quine. 1955. A proof procedure for quantification theory. J. Symbolic Logic 20, 2 (1955).Google ScholarCross Ref
- J. A. Robinson. 1965. A Machine-Oriented Logic Based on the Resolution Principle. J. ACM 12, 1 (1965). Google ScholarDigital Library
- L. Straßburger. 2009. Some Observations on the Proof Theory of Second Order Propositional Multiplicative Linear Logic. In TLCA '09. Google ScholarDigital Library
- L. Straßburger. 2017. Combinatorial Flows and Their Normalisation. In Proc. FSCD 2017 (Leibniz Int. Proc. Informal), Vol. 84.Google Scholar
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