On equivalence and canonical forms in the LF type theory

Decidability of definitional equality and conversion of terms into canonical form play a central role in the meta-theory of a type-theoretic logical framework. Most studies of definitional equality are based on a confluent, strongly normalizing notion of reduction. Coquand has considered a different approach, directly proving the correctness of a practical equivalance algorithm based on the shape of terms. Neither approach appears to scale well to richer languages with, for example, unit types or subtyping, and neither provides a notion of canonical form suitable for proving adequacy of encodings.In this article, we present a new, type-directed equivalence algorithm for the LF type theory that overcomes the weaknesses of previous approaches. The algorithm is practical, scales to richer languages, and yields a new notion of canonical form sufficient for adequate encodings of logical systems. The algorithm is proved complete by a Kripke-style logical relations argument similar to that suggested by Coquand. Crucially, both the algorithm itself and the logical relations rely only on the shapes of types, ignoring dependencies on terms.

[1]  Conal Elliott,et al.  Higher-order Unification with Dependent Function Types , 1989, RTA.

[2]  Frank Pfenning,et al.  System Description: Twelf - A Meta-Logical Framework for Deductive Systems , 1999, CADE.

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

[4]  Adriana B. Compagnoni,et al.  Anti-Symmetry of Higher-Order Subtyping , 1999, CSL.

[5]  Herman Geuvers,et al.  Some logical and syntactical observations concerning the first-order dependent type system λP , 1999, Mathematical Structures in Computer Science.

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

[7]  Roberto Virga,et al.  Higher-order rewriting with dependent types (lambda calculus) , 1999 .

[8]  Herman Geuvers,et al.  The Church-Rosser property for beta eta -reduction in typed lambda -calculi , 1992, [1992] Proceedings of the Seventh Annual IEEE Symposium on Logic in Computer Science.

[9]  H. Geuvers.  The Church-Rosser property for βη-reduction in typed λ-calculi , 1992, LICS 1992.

[10]  Frank Pfenning,et al.  A linear logical framework , 1996, Proceedings 11th Annual IEEE Symposium on Logic in Computer Science.

[11]  Amy P. Felty,et al.  Encoding a Dependent-Type Lambda-Calculus in a Logic Programming Language , 1990, CADE.

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

[13]  Frank Pfenning,et al.  Intensionality, extensionality, and proof irrelevance in modal type theory , 2001, Proceedings 16th Annual IEEE Symposium on Logic in Computer Science.

[14]  Frank Pfenning,et al.  Natural Deduction for Intuitionistic Non-communicative Linear Logic , 1999, TLCA.

[15]  Neil Ghani,et al.  The virtues of eta-expansion , 1995, Journal of Functional Programming.

[16]  HarperRobert,et al.  On equivalence and canonical forms in the LF type theory , 2005 .

[17]  Healfdene Goguen.  Soundness of the Logical Framework for Its Typed Operational Semantics , 1999, TLCA.

[18]  Robert Harper,et al.  Abstract 1 Introduction , 2022 .

[19]  Karl Crary,et al.  A Simplified Account of the Metatheory of Linear LF , 2002, Electron. Notes Theor. Comput. Sci..

[20]  Healfdene Goguen.  Typed Operational Semantics , 1995, TLCA.

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

[22]  Richard Statman,et al.  Higher-Order Rewriting with Dependent Types , 1999 .

[23]  Healfdene Goguen.  A typed operational semantics for type theory , 1994 .

[24]  Bengt Nordström,et al.  Programming in Martin-Lo¨f's type theory: an introduction , 1990 .

[25]  Jeff Polakow,et al.  Ordered linear logic and applications , 2001 .

[26]  Neil Ghani.  Eta-Expansions in Dependent Type Theory - The Calculus of Constructions , 1997, TLCA.

[27]  T. Streicher.  Semantics of Type Theory , 1991, Progress in Theoretical Computer Science.