Nondeterminism in Algebraic Specifications and Algebraic Programs

Algebraic specification, nondeterminism and term rewriting are three active research areas aiming at concepts for the abstract description of software systems: Algebraic specifications are well-suited for describing data structures and sequential software systems in an abstract way. Term rewriting methods are used in many prototyping systems and form the basis for executing specifi­ cations. Nondeterminism plays a major role in formal language theory; in programming it serves for delaying design decisions in program development and occurs in a "natural" way in formalisations of distributed processes. Heinrich Hussmann presents an elegant extension of equational specification and term rewriting to include nondeterminism. Based on a clean modeltheoretic semantics he considers term rewriting systems without confluence restrictions as a specification language and shows that fundamental properties such as the existence of initial models or the soundness and completeness of narrowing, the basic mechanism for executing equational specifications, can be extended to nondeterministic computations. The work of Heinrich Hussmann is an excellent contribution to Algebraic Programming; it gives a framework that admits a direct approach to program verification, is suitable for describing concurrent and distributed processes, and it can be executed as fast as Prolog.

Contenu

0: Introduction.- 0.1 Preview.- 0.2 Historical Background.- 0.3 Basic Notions.- 1: Nondeterministic Algebraic Specifications.- 1.1 Nondeterministic Algebras.- 1.1.1 A Discussion of Alternative Approaches.- 1.1.2 The Principle of Extensionality.- 1.1.3 The Notion of an Algebra.- 1.2 Inclusion Rules as a Specification Language.- 1.2.1 Axioms and their Semantics.- 1.2.2 The Calculus of Term Rewriting.- 1.2.3 Soundness: A Negative Result.- 1.2.4 Right-Linearity: A Special Case.- 2: Specifications with a Deterministic Basis.- 2.1 Deterministic Basis.- 2.1.1 Soundness and Deterministic Basis.- 2.1.2 Determinacy Predicate.- 2.1.3 Completeness: A Negative Result.- 2.2 Additive Specifications.- 2.2.1 DET-Completeness and DET-Additivity.- 2.2.2 Term Models and Completeness.- 2.3 Junk-Free Models.- 2.3.1 "Junk" in Nondeterministic Models.- 2.3.2 Breadth Induction.- 2.3.4 DET-Generated Models.- 2.3.5 Term-Generated Models.- 2.4 Hierarchical Specifications.- 3: Structure of the Model Classes.- 3.1 Homomorphisms and Extremal Algebras.- 3.2 Initial Models.- 3.3 Initial Models with Deterministic Basis.- 4: Nondeterministic Specifications as a General Framework.- 4.1 Equational Logic.- 4.2 Term Rewriting.- 4.3 Conditional Axioms.- 4.4 Algebraic Programming.- 4.4.1 Constructor-Based Specifications.- 4.4.2. Narrowing without Confluence.- 4.5 Logic Programming.- 4.5.1. Narrowing Simulates Logic Programming.- 4.5.2. Logic Programming Simulates Narrowing.- 5: Implementation and Examples.- 5.1 Term Rewriting.- 5.1.1 Innermost Rewriting.- 5.1.2 Search Strategies.- 5.1.3 Optimizations.- 5.2 Graph Rewriting.- 5.2.1 Representation of Terms by Graphs.- 5.2.2 Rewriting of Term Graphs.- 5.2.3 Soundness and Completeness.- 5.3 Examples.- 5.3.1 Nondeterministic Finite State Automata.- 5.3.2 Petri Nets.- 5.3.3 The Eight Queens Problem.- 5.3.4 The Monkey-Banana Problem.- 5.3.5 Printer Scheduling.- 6: Partial Nondeterministic Specifications.- 6.1 Partial Operations.- 6.1.1 Undefined "Values".- 6.1.2 Partial Multi-Algebras.- 6.2 Partiality and Term Rewriting.- 6.2.1 A Calculus for Partial Specifications.- 6.2.2 Partial DET-Completeness and DET-Additivity.- 6.3 Partial Specifications with Constructor Basis.- 6.4 Structure of the Model Classes.- 6.4.1 Homomorphisms.- 6.4.2 Initial Algebras.- 6.4.3 Terminal Algebras.- 7: Communicating Processes: An Example.- 7.1. Communicating Processes (CP).- 7.2. Semantics of CP.- 7.2.1. Transition Semantics.- 7.2.2. Trace Semantics.- 7.2.3. Refusal Semantics.- 7.3. Improvements and Applications.- 8: Concluding Remarks.- 8.1. Summary and Evaluation.- 8.2. Future Work.- References.- Appendix A: Proofs.- Appendix B: Experiments with RAP.