Halting problem of one binary Horn clause is undecidable
This paper proposes a codification of the halting problem of any Turing machine in the form of only one right-linear binary Horn clause as follows: p(t) ← p(tt). where t (resp. tt) is any (resp. linear) term. Recursivity is well-known to be a crucial and fundamental concept in programming theory. This result proves that in Horn clause languages there is no hope to control it without additional hypotheses even for the simplest recursive schemes.
Some direct consequences are presented here. For instance, there exists an explicitly constructible right-linear binary Horn clause for which no decision algorithm, given a goal, always decides in a finite number of steps whether or not the resolution using this clause is finite. The halting problem of derivations w.r.t. one binary Horn clause had been shown decidable if the goal is ground [SS88] or if the goal is linear [Dev88, Dev90, DLD90]. The undecidability in the non-linear case is an unexpected extension.
The proof of the main result is based on the unpredictable iterations of periodically linear functions defined by J.H. Conway within number theory. Let us note that these new undecidability results are proved w.r.t. any type of resolution (bottom-up or top-down, depth-first or breadth-first, unification with or without occur-check).
Unable to display preview. Download preview PDF.
- [ABK89]Apt K.R., Bol R.N., Klop J.W. “On the safe termination of PROLOG programs”. ICLP'89, Lisbon, pp. 353–368. 1989.Google Scholar
- [BHW92]Bibel W., Hölldobler S., Würtz J. “Cycle Unification”. CADE pp. 94–108. June 1992.Google Scholar
- [BJ66]Böhm C, Jacopini G. “Flow diagrams, Turing machines and languages with only two formation rules”. Communications of the Association for Computing Machinery, Vol.9, pp. 366–371. 1966.Google Scholar
- [Con72]Conway J.H. “Unpredictable Iterations”. Proc. 1972 Number Theory Conference. University of Colorado, pp 49–52. 1972.Google Scholar
- [Dau92]Dauchet M. “Simulation of Turing Machines by a regular rewrite rule”. Journal of Theoretical Computer Science. n∘103. pp. 409–420 1992.Google Scholar
- [Dev88]Devienne P. “Weighted graphs — tool for studying the halting problem and time complexity in term rewriting systems and logic programming (extended abstract)”. Fifth Generation Computer Systems 88, Tokyo, Japan. 1988.Google Scholar
- [Dev90]Devienne P. “Weighted graphs — tool for studying the halting problem and time complexity in term rewriting systems and logic programming”. Journal of Theoretical Computer Science, n∘75, pp. 157–215, 1990.Google Scholar
- [DLD90]Devienne P., Lebègue P., Dauchet M. “Weighted Systems of Equations”. Informatika 91, Grenoble, Special issue of TCS. 1991.Google Scholar
- [DLR92]Devienne P., Lebègue P., Routier J.C. “Cycle Unification is Undecidable”. LIFL Technical Report n∘IT 241, Lille. 1992.Google Scholar
- [GM87]Gaiman, Mairson “Undecidable optimisation problems for database logic programs”. Symposium on Logic in Computer Science, New-York, pp. 106–115. 1987.Google Scholar
- [Lag85]Lagarias J.C. “The 3x+1 problem and its generalizations”. Amer. Math Monthly 92, pp. 3–23. 1985.Google Scholar
- [Min67]Minsky M. “Computation: Finite and Infinite Machines”. Prentice-Hall. 1967.Google Scholar
- [PDL91]Parrain A., Devienne P., Lebègue P. “Prolog programs transformations and Meta-Interpreters”. Logic program synthesis and transformation, Springer-Verlag, LOPSTR'91, Manchester. 1991.Google Scholar
- [Rog87]Rogers H. “Theory of Recursive Functions and Effective Computability”. The MIT Press. 1987.Google Scholar
- [SS88]Schmidt-Schauss M. “Implication of clauses is undecidable”. Journal of Theoretical Computer Science, n∘59, pp. 287–296. 1988.Google Scholar