CL 2000: Computational Logic — CL 2000 pp 660-672

# The Theory of Total Unary RPO Is Decidable

• Paliath Narendran
• Michael Rusinowitch
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1861)

## Abstract

The Recursive Path Ordering (rpo) is a syntactic ordering on terms that has been widely used for proving termination of term-rewriting systems [7, 20]. How to combine term-rewriting with ordered resolution and paramodulation is now well-understood and it has been successfully applied in many theorem-proving systems [11, 16, 21]. In this setting an ordering such as rpo is used both to orient rewrite rules and to select maximal literals to perform inferences on. In order to further prune the search space the ordering requirements on conditional inferences are better handled when they are treated as constraints [12, 18]. Typically a non-orientable equation s = t will be split as two constrained rewrite rules: st | s > t and ts | t > s. Such constrained rules are useless when the constraint is unsatisfiable. Therefore it is important for the efficiency of automated reasoning systems to investigate decision procedures for the theory of terms with ordering predicates.

Other types of constraints can be introduced too such as disunification constraints [1]. It is often the case that they can be expressed with ordering constraints (although this might be inefficient).

We prove that the first-order theory of the recursive path ordering is decidable in the case of unary signatures with total precedence. This solves a problem that was mentioned as open in [6]. The result has to be contrasted with the undecidability results of the lexicographic path ordering [6] for the case of symbols with arity ≥ 2 and total precedence and for the case of unary signatures with partial precedence. We recall that lexicographic path ordering (lpo) and the recursive path ordering and many other orderings such as [13, 10] coincide in the unary case.

Among the positive results it is known that the existential theory of total lpo is decidable [3, 17]. The same result holds for the case of total rpo [8, 15].

The proof technique we use for our decidability result might be interesting by itself. It relies on encoding of words as trees and then on building a tree automaton to recognize the rpo relation.

## Keywords

Recursive path ordering first-order theory ground reducibility tree automata ordered rewriting

## References

1. 1.
R. Caferra and N. Peltier. Disinference rules, model building and abduction. Logic at work: Essays dedicated to the memory of Helena Rasiowa(Part 5: Logic in Computer Science, Chap. 20). Physica-Verlag, 1998.Google Scholar
2. 2.
A-C. Caron, J-L. Coquide, and M. Dauchet. Encompassment properties and automata with constraints. In C. Kirchner, editor, Proceedings 5th Conference on Rewriting Techniques and Applications, Montreal (Canada),volume 690 of Lecture Notes in Computer Science, pages 328–342. Springer-Verlag, 1993.Google Scholar
3. 3.
H. Comon. Solving inequations in term algebras. In Proc. 5th IEEE Symposium on Logic in Computer Science (LICS), Philadelphia, June1990.Google Scholar
4. 4.
H. Comon, M. Dauchet, R. Gilleron, F. Jacquemard, D. Lugiez, S. Tison, M. Tommasi. Tree Automata Techniques and Applications. http://www.grappa.univ-lille3.fr/tata/
5. 5.
H. Comon, P. Narendran, R. Nieuwenhuis, and M. Rusinowitch. Decision problems in ordered rewriting. In Proc. 13th IEEE Symp. Logic in Computer Science (LICS’98), Indianapolis, IN, USA, June 1998, pages 276–286, 1998.Google Scholar
6. 6.
H. Comon and R. Treinen. The first-order theory of lexicographic path orderings is undecidable. Theoretical Computer Science 176, April 1997.Google Scholar
7. 7.
N. Dershowitz. Orderings for term-rewriting systems. Theoretical Computer Science 17(3): 279–301. 1982.
8. 8.
J.-P. Jouannaud and M. Okada. Satisfiability of systems of ordinal notations with the subterm ordering is decidable. In 18th International Colloquium on Automata, Languages and Programming (ICALP), volume 510 of Lecture Notes in Computer Science, pages 455–468, Madrid, Spain, July 1991. Springer-Verlag.Google Scholar
9. 9.
D. Kapur, P. Narendran, D. Rosenkrantz and H. Zhang. Sufficient Completeness, Ground-Reducibility and Their Complexity. Acta Informatica 28 (1991) 311–350.
10. 10.
D. Kapur, P. Narendran and G. Sivakumar. A path ordering for proving termination of term rewriting systems. In H. Ehrig (ed.), 10th CAAP, volume 185 of Lecture Notes in Computer Science, pages 173–187, Berlin, March 1985.Google Scholar
11. 11.
D. Kapur and H. Zhang. An Overview of Rewrite Rule Laboratory (RRL), J. of Computer and Mathematics with Applications, 29,2, 1995, 91–114.
12. 12.
C. Kirchner, H. Kirchner, and M. Rusinowitch. Deduction with symbolic constraints. Revue Franaise d’Intelligence Artificielle, 4(3):9–52, 1990. Special issue on automatic deduction.Google Scholar
13. 13.
P. Lescanne. Some properties of the Decomposition Ordering, a simplification ordering to prove termination of rewriting systems. RAIRO, 16(4):331–347, 1982.
14. 14.
U. Martin and E. Scott. The order types of termination orderings on monadic terms, strings and multisets. J. Symbolic Logic 62 (1997) 624–635.
15. 15.
P. Narendran, M. Rusinowitch and R. Verma. RPO constraint solving is in NP. In: Computer Science Logic (CSL 98), Brno, Czech Republic. August 1998. 12p. LNCS 1584, Springer-Verlag, 1999.Google Scholar
16. 16.
W. McCune and R. Padmanabhan. Automated Deduction in Equational Logic and Cubic Curves, Springer-Verlag LNCS 1095 (1996)
17. 17.
R. Nieuwenhuis. Simple LPO constraint solving methods. Inf. Process. Lett. 47(2):65–69, Aug. 1993.Google Scholar
18. 18.
R. Nieuwenhuis and A. Rubio. Theorem proving with ordering constrained clauses. In D. Kapur, editor, Proceedings of 11th Conf. on Automated Deduction, Saratoga Springs, NY, 1992, volume 607 of Lecture Notes in Artificial Intelligence, pages 477–491, June 1992, Springer-Verlag.Google Scholar
19. 19.
D. Plaisted. Semantic confluence tests and completion methods. Information and Control, 65:182–215, 1985.
20. 20.
J. Steinbach. Extensions and comparison of simplification orderings. Proceedings of 3rd RTA, Chapel Hill, NC, volume 355 of Lecture Notes in Computer Science, pages 434–448, 1989, Springer-Verlag.Google Scholar
21. 21.
C. Weidenbach. SPASS: Combining Superposition, Sorts and Splitting in A. Robinson and A. Voronkov, editors, Handbook of Automated Reasoning, Elsevier, 1999. To appear.Google Scholar