ABSTRACT
This paper concerns both the complexity aspects of PA and the pragmatics of improving algorithms for dealing with restricted subcases of PA for uses such as program verification. We improve the Cooper-Presburger decision procedure and show that the improved version permits us to obtain time and space upper bounds for PA classes restricted to a bounded number of alternations of quantifiers. The improvement is one exponent less than the upper bounds for the decision problem for full PA. The pragmatists not interested in complexity bounds can read the results as substantiation of the intuitive feeling that the improvement to the Cooper-Presburger algorithm is a real, rather than ineffectual, improvement. (It can be easily shown that the bounds obtained here are not achievable using the Cooper-Presburger procedure).
- 1.Borosh I. and Treybig, L. B. Bounds on positive integral solutions of linear diophantine equations. Proc. AMS 55, March, 1976.Google ScholarCross Ref
- 2.Cooper, D. C. Theorem-proving in arithmetic without multiplication. Machine Intell. 7, J. Wiley, 1972.Google Scholar
- 3.Ferrante, J. and Rackoff, C. A decision procedure for the first order theory of real addition with order. SIAM J. Comp., March, 1975.Google Scholar
- 4.Fischer, M. and Rabin, M. O. Super-exponential complexity of Presburger arithmetic. Project MAC. Tech. Memo 43, MIT, Cambridge, 1974. Google ScholarDigital Library
- 5.Oppen, D. C. Elementary bounds for Presburger arithmetic. |5th SIGACT,# May, 1973. Google ScholarDigital Library
- 6.Presburger, M. Uber die Vollstandigkeit eines gewissen Systems der Arithmetic ganzer Zahlen, in welchem die Addition als einzige Operation hervortritt. Compte-Rendus dei Congres des Math. des pays slavs, Warsaw, 1929.Google Scholar
- 7.Shostak, R. An efficient decision algorithm for arithmetic with function symbols. Talk at Workshop on Auto. Deduction, Aug. 1977.Google Scholar
- 8.Suzuki, N. and Jefferson, D. Verification decidability of Presburger array programs. CMU Comp. Sci. Report,, June, 1977.Google Scholar
- Presburger arithmetic with bounded quantifier alternation
Recommendations
Bounds on the automata size for Presburger arithmetic
Automata provide a decision procedure for Presburger arithmetic. However, until now only crude lower and upper bounds were known on the sizes of the automata produced by the automata-based approach for Presburger arithmetic. In this article, we give an ...
On the Automata Size for Presburger Arithmetic
LICS '04: Proceedings of the 19th Annual IEEE Symposium on Logic in Computer ScienceAutomata provide an effective mechanization of decision procedures for Presburger arithmetic. However, only crude lower and upper bounds are known on the sizes of the automata produced by this approach. In this paper, we prove that the number of states ...
Alternation and Bounded Concurrency Are Reverse Equivalent
Numerous models of concurrency have been considered in the framework of automata. Among the more interesting concurrency models are classical nondeterminism and pure concurrency, the two facets of alternation, and the bounded concurrency model. Bounded ...
Comments