A system for reasoning within and about algebraic specifications

  • Jaoek Leszczylowski
  • Martin Wirsing
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 137)


A machine-implemented system to support the reasoning about algebraic specifications is presented. The PAT-system is an attempt to ease design, analysis, and implementation of partial abstract data types. More precisely, the PAT-system

  • allows to write "axiomatic abstract types", i.e. parameterised hierarchical algebraic specifications, as well as "domain types", i.e. abstract types defined by domain equations,

  • introduces automatically the semantic conventions for the theory of partial abstract types including axioms for strictness (of partial functions) and for the validity of data type induction,

  • facilitates proofs of derived properties: PAT provides (semiautomatic) strategies for using derived rules of inference as well as for simplifying conditional equations. It allows to use a metalanguage — Edinburgh-ML /LCF 79/ — for generating and performing proofs interactively;

  • makes first attempts for analysing algebraic specifications by checking sufficient conditions for the existence of initial algebras (of hierachical specifications and/or parameterised specifications — for nonhierarchical and nonparameterised specifications initial algebras always exist) and for the existence of weakly terminal algebras,

  • supports development and proof of correctness of implementations; in particular it is possible to verify implementations of "axiomatic types" by "domain types".

The PAT-system is implemented in Edinburgh-ML being part of an interactive verification system, Edinburgh-LCF, which seems particularly appropriate to support proofs with algebraic specifications.

The system PAT tries to use as much as possible the facilities and properties of the underlying Edinburgh-LCF system. Only necessary changes — such as "smash" product instead of "cartesian" product — and extensions — such as the introduction of certain induction schemata — are made.

A short characterisation of LCF and a description of the PAT-system is given. The PAT-system is shortly compared with other systems manipulating algebraic specifications. Finally, as an example for the reasoning with and about algebraic specifications priority queues over linearly ordered data types are described as a parameterised abstract type, PQ.


Priority Queue Domain Type Structural Induction Domain Equation Abstract Type 


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Copyright information

© Springer-Verlag Berlin Heidelberg 1982

Authors and Affiliations

  • Jaoek Leszczylowski
    • 1
  • Martin Wirsing
    • 2
  1. 1.Institute of Computer SciencePolish Academy of SciencesWarszawa, PKiNPoland
  2. 2.Institut für InformatikTechnische Universität MünchenMünchen 2FRG

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