A Propositional Tableaux Based Proof Calculus for Reasoning with Default Rules

  • Valentín Cassano
  • Carlos Gustavo Lopez Pombo
  • Thomas S. E. Maibaum
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9323)


Since introduced by Reiter in his seminal 1980 paper: ‘A Logic for Default Reasoning’, the subject of reasoning with default rules has been extensively dealt with in the literature on nonmonotonic reasoning. Yet, with some notable exceptions, the same cannot be said about its proof theory. Aiming to contribute to the latter, we propose a tableaux based proof calculus for a propositional variant of Reiter’s presentation of reasoning with default rules. Our tableaux based proof calculus is based on a reformulation of the semantics of Reiter’s view of a default theory, i.e., a tuple comprised of a set of sentences and a set of default rules, as a premiss structure. In this premiss structure, sentences stand for definite assumptions, as normally found in the literature, and default rules stand for tentative assumptions, as opposed to rules of inference, as normally found in the literature. On this basis, a default consequence is defined as being such relative to a premiss structure, as is our notion of a default tableaux proof. In addition to its simplicity, as usual in tableaux based proof calculi, our proof calculus allows for the discovery of the non-existence of proofs by providing corresponding counterexamples.


Leaf Node Proof Theory Sequent Calculus Label Tree Default Rule 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Reiter, R.: A logic for default reasoning. Artificial Intelligence 13(1-2), 81–132 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Antoniou, G., Wang, K.: Default logic. In: Gabbay, D.M., Woods, J. (eds.): The Many Valued and Nonmonotonic Turn in Logic. Handbook of the History of Logic, vol. 8, pp. 517–555. North-Holland (2007)Google Scholar
  3. 3.
    Risch, V.: Analytic tableaux for default logics. Journal of Applied Non-Classical Logics 6(1), 71–88 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Amati, G., Aiello, L., Gabbay, D., Pirri, F.: A proof theoretical approach to default reasoning I: Tableaux for default logic. Journal of Logic and Computation 6(2), 205–231 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bonatti, P., Olivetti, N.: A sequent calculus for skeptical default logic. In: Galmiche, D. (ed.) TABLEAUX 1997. LNCS, vol. 1227, pp. 107–121. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  6. 6.
    Smullyan, R.M.: First-Order Logic. Dover (1995)Google Scholar
  7. 7.
    Fitting, M.: Introduction. In: D’Agostino, M., Gabbay, D.M., Hahnle, R., Posegga, J. (eds.) Handbook of Tableau Methods, 1st edn., pp. 1–43. Springer (1999)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Valentín Cassano
    • 1
  • Carlos Gustavo Lopez Pombo
    • 2
    • 3
  • Thomas S. E. Maibaum
    • 1
  1. 1.Department of Computing and SoftwareMcMaster UniversityHamiltonCanada
  2. 2.Departamento de ComputaciónUniversidad Nacional de Buenos AiresBuenos AiresArgentina
  3. 3.Consejo Nacional de Investigaciones Científicas y Tecnológicas (CONICET)Rio de JaneiroArgentina

Personalised recommendations