A linear logical framework

We present the linear type theory LLF as the formal basis for a conservative extension of the LF logical framework. LLF combines the expressive power of dependent types with linear logic to permit the natural and concise representation of a whole new class of deductive systems, namely those dealing with state. As an example we encode a version of Mini-ML with references including its type system, its operational semantics, and a proof of type preservation. Another example is the encoding of a sequent calculus for classical linear logic and its cut elimination theorem. LLF can also be given an operational interpretation as a logic programming language under which the representations above can be used for type inference, evaluation and cut-elimination.

[1]  Frank Pfenning,et al.  Natural Semantics and Some of Its Meta-Theory in Elf , 1992, ELP.

[2]  Roy Dyckhoff,et al.  Contraction-free sequent calculi for intuitionistic logic , 1992, Journal of Symbolic Logic.

[3]  Dale Miller,et al.  Logic programming in a fragment of intuitionistic linear logic , 1991, [1991] Proceedings Sixth Annual IEEE Symposium on Logic in Computer Science.

[4]  Samson Abramsky,et al.  Computational Interpretations of Linear Logic , 1993, Theor. Comput. Sci..

[5]  Jawahar Chirimar,et al.  Proof theoretic approach to specification languages , 1995 .

[6]  F. Honsell,et al.  A Framework for De ning LogicsRobert Harper , 1987 .

[7]  Anne Sjerp Troelstra.  Strong normalization for typed terms with surjective pairing , 1986, Notre Dame J. Formal Log..

[8]  Alberto Martelli,et al.  Unification in linear time and space: a structured presentation , 1976 .

[9]  C. Pollard,et al.  Center for the Study of Language and Information , 2022 .

[10]  Anne Sjerp Troelstra.  Natural Deduction for Intuitionistic Linear Logic , 1995, Ann. Pure Appl. Log..

[11]  Robin Milner,et al.  Definition of standard ML , 1990 .

[12]  F. Pfenning.  Logic programming in the LF logical framework , 1991 .

[13]  Frank Pfenning,et al.  Elf: A Meta-Language for Deductive Systems (System Descrition) , 1994, CADE.

[14]  Viggo Stoltenberg-hansen,et al.  In: Handbook of Logic in Computer Science , 1995 .

[15]  Alberto Martelli,et al.  An Efficient Unification Algorithm , 1982, TOPL.

[16]  David W. Albrecht,et al.  Curry-Howard Terms for Linear Logic , 1998, Stud Logica.

[17]  Alonzo Church,et al.  A formulation of the simple theory of types , 1940, Journal of Symbolic Logic.

[18]  Frank Pfenning,et al.  A Proof of the Church-Rosser Theorem and its Representation in a Logical Framework , 1992 .

[19]  Dale Miller,et al.  From operational semantics to abstract machines: preliminary results , 1990, LISP and Functional Programming.

[20]  Gesammelte Abhandlungen , 1906, Nature.

[21]  Frank Pfenning,et al.  A Structural Proof of Cut Elimination and Its Representation in a Logical Framework , 1994 .

[22]  Frank Pfenning,et al.  Automated Theorem Proving in a Simple Meta-Logic for LF , 1998, CADE.

[23]  Patrick Lincoln,et al.  Linear logic , 1992, SIGA.

[24]  Alexander Simpson.  Workshop on Types for Proofs and Programs , 1993 .

[25]  J. S. Hodas.  Logic programming in intuitionistic linear logic: theory, design, and implementation , 1995 .

[26]  Michel Parigot,et al.  Lambda-Mu-Calculus: An Algorithmic Interpretation of Classical Natural Deduction , 1992, LPAR.

[27]  Furio Honsell,et al.  A framework for defining logics , 1993, JACM.

[28]  Gopalan Nadathur,et al.  Uniform Proofs as a Foundation for Logic Programming , 1991, Ann. Pure Appl. Log..

[29]  Henk Barendregt,et al.  The Lambda Calculus: Its Syntax and Semantics , 1985 .

[30]  Frank Pfenning,et al.  Linear higher-order pre-unification , 1997, Proceedings of Twelfth Annual IEEE Symposium on Logic in Computer Science.

[31]  Mike Paterson,et al.  Linear unification , 1976, STOC '76.

[32]  Mads Tofte,et al.  Type Inference for Polymorphic References , 1990, Inf. Comput..

[33]  M. Gordon,et al.  Introduction to HOL: a theorem proving environment for higher order logic , 1993 .

[34]  Robin Milner,et al.  Edinburgh LCF , 1979, Lecture Notes in Computer Science.

[35]  J. Lambek,et al.  Introduction to higher order categorical logic , 1986 .

[36]  T. Coquand.  An algorithm for testing conversion in type theory , 1991 .

[37]  Jan Friso Groote,et al.  Proceedings of the International Conference on Typed Lambda Calculi and Applications , 1993 .

[38]  Carl A. Gunter,et al.  In handbook of theoretical computer science , 1990 .

[39]  Philip Wadler,et al.  Linear Types can Change the World! , 1990, Programming Concepts and Methods.

[40]  Xavier Leroy,et al.  Polymorphic type inference and assignment , 1991, POPL '91.

[41]  Iliano Cervesato,et al.  Proof-Theoretic Foundation of Compilation in Logic Programming , 1998, IJCSLP.

[42]  P. Martin-Löf.  On the meanings of the logical constants and the justi cations of the logical laws , 1996 .

[43]  Carl A. Gunter,et al.  The machine-assisted proof of programming language properties , 1996 .

[44]  Haskell B. Curry.  Grundlagen der kombinatorischen Logik , 1930 .

[45]  Robert Harper,et al.  A Simplified Account of Polymorphic References , 1994, Inf. Process. Lett..

[46]  Jean-Yves Girard,et al.  On the Unity of Logic , 1993, Ann. Pure Appl. Log..

[47]  Amy P. Felty,et al.  Encoding dependent types in an intuitionistic logic , 1991 .

[48]  J. H. Geuvers.  Logics and type systems , 1993 .

[49]  Nick Benton,et al.  A Term Calculus for Intuitionistic Linear Logic , 1993, TLCA.

[50]  A. Church.  The calculi of lambda-conversion , 1941 .

[51]  M. E. Szabo,et al.  The collected papers of Gerhard Gentzen , 1969 .

[52]  William A. Howard,et al.  The formulae-as-types notion of construction , 1969 .

[53]  Frank Pfenning.  Structural Cut Elimination in Linear Logic. , 1994 .

[54]  David Hilbert.  Neubegründung der Mathematik. Erste Mitteilung , 1922 .

[55]  Patrick J. Hayes,et al.  Computation and Deduction , 1973, MFCS.

[56]  J. A. Robinson,et al.  A Machine-Oriented Logic Based on the Resolution Principle , 1965, JACM.

[57]  Luís Damas,et al.  Type assignment in programming languages , 1984 .

[58]  Dale Miller,et al.  A multiple-conclusion meta-logic , 1994, Proceedings Ninth Annual IEEE Symposium on Logic in Computer Science.

[59]  Andre Scedrov,et al.  A brief guide to linear logic , 1990, Bull. EATCS.

[60]  Andrew Barber,et al.  Dual Intuitionistic Linear Logic , 1996 .

[61]  Dominique Clément,et al.  A simple applicative language: mini-ML , 1986, LFP '86.

[62]  Matthias Felleisen,et al.  A Syntactic Approach to Type Soundness , 1994, Inf. Comput..

[63]  Samson Abramsky,et al.  Handbook of logic in computer science. , 1992 .

[64]  P. Lincoln,et al.  Operational aspects of linear lambda calculus , 1992, [1992] Proceedings of the Seventh Annual IEEE Symposium on Logic in Computer Science.

[65]  Frank Pfenning,et al.  The Practice of Logical Frameworks , 1996, CAAP.

[66]  José Meseguer,et al.  From Petri Nets to Linear Logic through Categories: A Survey , 1991, Int. J. Found. Comput. Sci..

[67]  Jacques Herbrand.  Recherches sur la théorie de la démonstration , 1930 .

[68]  P. Martin-Löf.  Analytic and Synthetic Judgements in Type Theory , 1994 .

[69]  Frank Pfenning,et al.  Structural cut elimination , 1995, Proceedings of Tenth Annual IEEE Symposium on Logic in Computer Science.

[70]  David J. Pym,et al.  A Relevant Analysis of Natural Deduction , 1998, J. Log. Comput..