Completeness theorems for automata
These notes present completeness results for varieties of products, state mappings and auxiliary variable constructions, for a (Mealy) automata-theoretic model of computation that generalizes the I/O automaton model of Lynch and Tuttle [Lyn88, LT87]. Conditions are examined under which these tools suffice to demonstrate that one specification implements another. The major theorem is a restatement of a completeness theorem due to Abadi and Lamport [AL88], translated from their (Moore) state machine model. The multivalued possibilities mappings of Lynch and Tuttle are used in place of the single-valued refinement mappings of Abadi and Lamport. A new kind of state mapping, prophecy mappings, is defined. Prophecy mappings are the time-reversal of possibilities mappings. This definition admits greater modularity in the proofs of Abadi and Lamport's results. Additional results explore properties of products of automata, developing more fully ideas implicit in Abadi and Lamport's work.
Key wordsSpecification implementation completeness automata state mappings products
Unable to display preview. Download preview PDF.
- [AL88]M. Abadi and L Lamport. The existence of refinement mappings. In Proceedings of the Third Annual Symposium on Logic in Computer Science, pages 165–175, July 1988. Edinburgh, Scotland. Also available as a Digital Systems Research Center technical report, 130 Lytton Avenue, Palo Alto, CA 94301.Google Scholar
- [CM88]K.M. Chandy and J. Misra. Parallel Program Design: A Foundation. Addison-Wesley, 1988.Google Scholar
- [Knu73]D. E. Knuth. Fundamental Algorithms. Volume 1 of The Art of Computer Programming. Addison-Wesley, 1973. Reading, Massachusetts, second edition.Google Scholar
- [LT87]N. Lynch and M. Tuttle. Hierarchical correctness proofs for distributed algorithms. In Proceedings of 6th ACM Symposium on Principles of Distributed Computation, pages 137–151, August 1987. Expanded version available as Technical Report MIT/LCS/TR-387, Laboratory for Computer Science, Massachusetts Institute Technology, Cambridge, MA., April 1987.Google Scholar
- [Lyn88]N. Lynch. I/O automata: A model for discrete event systems. Technical Memo MIT/LCS/TM-351, Massachusetts Institute Technology, Laboratory for Computer Science, March 1988. Also, in 22nd Annual Conference on Information Science and Systems, Princeton University, Princeton, N.J., March 1988.Google Scholar
- [Lyn89]N. Lynch. Multivalued possibilities mappings. In Lecture Notes in Computer Science, 1989. This volume.Google Scholar
- [WLL88]J. Welch, L Lamport, and N. Lynch. A lattice-structured proof of a minimum spanning tree algorithm. In Proceedings of the Seventh Annual Symposium on Principles of Distributed Computation, August 1988. Vancouver, BC.Google Scholar