Halting problem of one binary Horn clause is undecidable

  • Philippe Devienne
  • Patrick Lebègue
  • Jean-Christophe Routier
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 665)


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).


Finite Number Turing Machine Logic Programming Function Symbol Piecewise Linear Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [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
  2. [BHW92]
    Bibel W., Hölldobler S., Würtz J. “Cycle Unification”. CADE pp. 94–108. June 1992.Google Scholar
  3. [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
  4. [Con72]
    Conway J.H. “Unpredictable Iterations”. Proc. 1972 Number Theory Conference. University of Colorado, pp 49–52. 1972.Google Scholar
  5. [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
  6. [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
  7. [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
  8. [DLD90]
    Devienne P., Lebègue P., Dauchet M. “Weighted Systems of Equations”. Informatika 91, Grenoble, Special issue of TCS. 1991.Google Scholar
  9. [DLR92]
    Devienne P., Lebègue P., Routier J.C. “Cycle Unification is Undecidable”. LIFL Technical Report n∘IT 241, Lille. 1992.Google Scholar
  10. [GM87]
    Gaiman, Mairson “Undecidable optimisation problems for database logic programs”. Symposium on Logic in Computer Science, New-York, pp. 106–115. 1987.Google Scholar
  11. [Lag85]
    Lagarias J.C. “The 3x+1 problem and its generalizations”. Amer. Math Monthly 92, pp. 3–23. 1985.Google Scholar
  12. [Min67]
    Minsky M. “Computation: Finite and Infinite Machines”. Prentice-Hall. 1967.Google Scholar
  13. [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
  14. [Rog87]
    Rogers H. “Theory of Recursive Functions and Effective Computability”. The MIT Press. 1987.Google Scholar
  15. [SS88]
    Schmidt-Schauss M. “Implication of clauses is undecidable”. Journal of Theoretical Computer Science, n∘59, pp. 287–296. 1988.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1993

Authors and Affiliations

  • Philippe Devienne
    • 1
  • Patrick Lebègue
    • 1
  • Jean-Christophe Routier
    • 1
  1. 1.Laboratoire d'Informatique Fondamentale de LilleCNRS U.A. 369 Université des Sciences et Technologies de LilleVilleneuve d'Ascq CedexFrance

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