One binary horn clause is enough

  • Philippe Devienne
  • Patrick Lebègue
  • Jean-Christophe Routier
  • Jörg Würtz
Conference paper

DOI: 10.1007/3-540-57785-8_128

Part of the Lecture Notes in Computer Science book series (LNCS, volume 775)
Cite this paper as:
Devienne P., Lebègue P., Routier JC., Würtz J. (1994) One binary horn clause is enough. In: Enjalbert P., Mayr E.W., Wagner K.W. (eds) STACS 94. STACS 1994. Lecture Notes in Computer Science, vol 775. Springer, Berlin, Heidelberg

Abstract

This paper proposes an equivalent form of the famous Böhm-Jacopini theorem for declarative languages. C. Böhm and G. Jacopini [1] proved that all programming can be done with at most one single whiledo. That result is cited as a mathematical justification for structured programming. A similar result can be shown for declarative programming. Indeed the simplest class of recursive programs in Horn clause languages can be defined by the following scheme:

where
$$\left\{ {\begin{array}{*{20}c}{\mathcal{A}_1 \leftarrow .} \\{\mathcal{A}_2 \leftarrow } \\{ \leftarrow \mathcal{A}_4 } \\\end{array} } \right.\mathcal{A}_3 . that is \forall x_1 \cdot \cdot \cdot \forall x_m \left[ {\mathcal{A}_1 \wedge \left( {\mathcal{A}_2 \vee \neg \mathcal{A}_3 } \right) \wedge \neg \mathcal{A}_4 } \right]$$
are positive first-order literals. This class is shown here to be as expressive as Turing machines and all simpler classes would be trivial. The proof is based on a remarkable and not enough known codification of any computable function by unpredictable iterations proposed by [5]. Then, we prove effectively by logical transformations that all conjunctive formulas of Horn clauses can be translated into an equivalent conjuctive 4-formula (as above). Some consequences are presented in several contexts (mathematical logic, unification modulo a set of axioms, compilation techniques and other program patterns).

Topics

Logic in Computer Science Theory of Programming Languages 

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Copyright information

© Springer-Verlag 1994

Authors and Affiliations

  • Philippe Devienne
    • 1
  • Patrick Lebègue
    • 1
  • Jean-Christophe Routier
    • 1
  • Jörg Würtz
    • 2
  1. 1.Laboratoire d'Informatique Fondamentale de Lille - CNRS UA 369Université des Sciences et Technologies de Lille Cité ScientifiqueVilleneuve d'Ascq CedexFrance
  2. 2.Deutsches Forschungszentrum für Künstliche Intelligenz - DFKISaarbrücken 11Germany

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