Logical systems for structured specifications

We study proof systems for reasoning about logical consequences and refinement of structured specifications, based on similar systems proposed earlier in the literature (Inform. and Comput. 76 (1988) 165; in: F.L. Bauer, W. Brauer, H. Schwichtenberg (Eds.), Logic and Algebra of Specification, NATO ASI Series F: Computer and Systems Sciences, vol. 94, Springer, Berlin, 1991, p. 411). Following Goguen and Burstall, the notion of an underlying logical system over which we build specifications is formalized as an institution and extended to a more general notion, called (d, J)-institution. We show that under simple assumptions (essentially: amalgamation and interpolation) the proposed proof systems are sound and complete. The completeness proofs are inspired by proofs due to Cengarle (Ph.D. Thesis, Institut fur Informatik, Ludwig-Maximilians-Universitat Muenchen, 1994) for specifications in first-order logic and the logical systems for reasoning about them. We then propose a methodology for reusing proof systems built over institutions rich enough to satisfy the properties required for the completeness results for specifications built over poorer institutions where these properties need not hold.

[1]  Andrzej Tarlecki.  Moving Between Logical Systems , 1995, COMPASS/ADT.

[2]  Antonino Salibra,et al.  Interpolation and Compactness in Categories of Pre-Institutions , 1996, Math. Struct. Comput. Sci..

[3]  Arnon Avron,et al.  Simple Consequence Relations , 1988, Inf. Comput..

[4]  Jan A. Bergstra,et al.  Module algebra , 1990, JACM.

[5]  Andrzej Tarlecki,et al.  Towards Heterogeneous Specifications , 1998, FroCoS.

[6]  Jorge Farres-Casals.  Verification in ASL and related specification languages , 1992 .

[7]  Donald Sannella,et al.  Toward formal development of programs from algebraic specifications: Implementations revisited , 1987, TAPSOFT, Vol.1.

[8]  Peter Burmeister,et al.  Partial algebras—survey of a unifying approach towards a two-valued model theory for partial algebras , 1982 .

[9]  Tomasz Borzyszkowski.  Moving Specification Structures Between Logical Systems , 1998, WADT.

[10]  John Slaney.  A general logic , 1990 .

[11]  P. Burmeister.  A Model Theoretic Oriented Approach to Partial Algebras , 1986 .

[12]  Egidio Astesiano,et al.  Multiparadigm Specification Languages: A First Attempt at Foundations , 1993, Semantics of Specification Languages.

[13]  R. J. vanGlabbeek,et al.  An interpolation theorem in equational logic , 1988 .

[14]  Rod M. Burstall,et al.  Structured Theories in LCF , 1983, CAAP.

[15]  Till Mossakowski.  Representations, hierarchies and graphs of institutions , 1996 .

[16]  Donald Sannella,et al.  Extended ML: Past, Present, and Future , 1990, ADT.

[17]  José Meseguer,et al.  May I Borrow Your Logic? (Transporting Logical Structures Along Maps) , 1997, Theor. Comput. Sci..

[18]  Donald Sannella,et al.  Specifications in an Arbitrary Institution , 1988, Inf. Comput..

[19]  Michel Bidoit,et al.  Observational Logic , 1998, AMAST.

[20]  T. Borzyskowski.  Correctness of a logical system for structured specifications , 1997, WADT.

[21]  Joseph A. Goguen,et al.  Institutions: abstract model theory for specification and programming , 1992, JACM.

[22]  I. S. Gradshtein,et al.  THE ELEMENTS OF MATHEMATICAL LOGIC , 1963 .

[23]  T. Borzyszkowski.  Completeness of a logical system for structured specifications , 1998 .

[24]  Egidio Astesiano,et al.  Algebraic Foundations of Systems Specification , 1999, IFIP State-of-the-Art Reports.

[25]  Joseph A. Goguen,et al.  The Semantics of CLEAR, A Specification Language , 1979, Abstract Software Specifications.

[26]  Robert Harper,et al.  Structured Theory Presentations and Logic Representations , 1994, Ann. Pure Appl. Log..

[27]  Peter D. Mosses,et al.  CoFI: The Common Framework Initiative for Algebraic Specification and Development , 2001, Current Trends in Theoretical Computer Science.

[28]  Martin Wirsing,et al.  Structured Specifications: Syntax, Semantics and Proof Calculus , 1993 .

[29]  T. Broadbent,et al.  Module Algebra , 1970, Nature.

[30]  Chen C. Chang,et al.  Model Theory: Third Edition (Dover Books On Mathematics) By C.C. Chang;H. Jerome Keisler;Mathematics , 1966 .

[31]  G. Kreisel,et al.  Elements of Mathematical Logic: Model Theory , 1971 .

[32]  Peter D. Mosses.  CoFI: The Common Framework Initiative for Algebraic Specification , 1996, Bull. EATCS.

[33]  Andrzej Tarlecki,et al.  Bits and Pieces of the Theory of Institutions , 1985, ADT.