Advertisement

Best-Path Theorem Proving: Compiling Derivations

  • Martin FrickéEmail author
Chapter
Part of the Studies in History and Philosophy of Science book series (AUST, volume 28)

Abstract

When computers answer our questions in mathematics and logic they need also to be able to supply justification and explanatory insight. Typical theorem provers do not do this. The paper focuses on tableau theorem provers for First Order Predicate Calculus. The paper introduces a general construction and a technique for converting the tableau data structures of these to human friendly linear proofs using any familiar rule set and ‘laws of thought’. The construction uses a type of tableau in which only leaf nodes are extended. To produce insightful proofs, improvements need to be made to the intermediate output. Dependency analysis and refinement, ie compilation of proofs, can produce benefits. To go further, the paper makes other suggestions including a perhaps surprising one: the notion of best proof or insightful proof is an empirical matter. All possible theorems, or all possible proofs, distribute evenly, in some sense or other, among the possible uses of inference steps. However, with the proofs of interest to humans this uniformity of distribution does not hold. Humans favor certain inferences over others, which are structurally very similar. The author’s research has taken many sample questions and proofs from logic texts, scholastic tests, and similar sources, and analyzed the best proofs for them (‘best’ here usually meaning shortest). This empirical research gives rise to some suggestions on heuristic. The general point is: humans are attuned to certain forms inference, empirical research can tell us what those are, and that empirical research can educate as to how tableau theorem provers, and their symbiotic linear counterparts, should run. In sum, tableau theorem provers, coupled with transformations to linear proofs and empirically sourced heuristic, can provide transparent and accessible theorem proving.

Keywords

Leaf Node Theorem Prover Natural Deduction Standard Tableau Skolem 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.

References

  1. Bergmann, M., J. Moor, and J. Nelson. 1998. The logic book, 4th ed. New York: McGraw-Hill.Google Scholar
  2. Beth, E.W. 1969. Semantic entailment and formal derivability. In The philosophy of mathematics, ed. J. Hintikka, 9–41. London: Oxford University Press.Google Scholar
  3. Black, P.E. 2005. British Museum technique. Dictionary of algorithms and data structures. Retrieved July 2009, from http://www.itl.nist.gov/div897/sqg/dads/HTML/britishMuseum.html.
  4. Copeland, B.J., and D.R. Murdoch. 1991. The Arthur Prior memorial conference: Christchurch 1989. The Journal of Symbolic Logic 56(1): 372–382.CrossRefGoogle Scholar
  5. D’Agostino, M., et al. (eds.). 1999. Handbook of tableau methods. Dordrecht: Kluwer Academic Publishers.Google Scholar
  6. Fitting, M. 1996. First-order logic and automated theorem proving, 2nd ed. Berlin: Springer.Google Scholar
  7. Fitting, M. 1998. Introduction. In Handbook of tableau methods, ed. M. D’Agostino et al. Dordrecht: Kluwer Academic Publishers.Google Scholar
  8. Frické, M. 1989a. Derivation planner. Dunedin: Unisoft.Google Scholar
  9. Frické, M. 1989b. Deriver plus. Ventura: Kinko’s Academic Courseware Exchange.Google Scholar
  10. Gallier, J.H. 1986. Logic for computer science: Foundations of automatic theorem proving. New York: Harper Row.Google Scholar
  11. Gentzen, G. 1935. Investigations into logical deduction. In The collected papers of Gerhard Gentzen, ed. M.E. Szabo. Amsterdam: North-Holland.Google Scholar
  12. Hähnle, R. 2001. Tableaux and related methods. In Handbook of automated reasoning, ed. J.A. Robinson and A. Voronkov. Cambridge: MIT Press.Google Scholar
  13. Howson, C. 1997. Logic with trees: An introduction to symbolic logic. London: Routledge.Google Scholar
  14. Jeffrey, R.C. 1967. Formal logic: Its scope and limits. New York: McGraw Hill.Google Scholar
  15. Letz, R. 1999. First-order tableau methods. In Handbook of tableau methods, ed. M. D’Agostino et al. Dordrecht: Kluwer Academic Publishers.Google Scholar
  16. Lindstrom, P. 1969. On extensions of elementary logic. Theoria 35: 1–11.CrossRefGoogle Scholar
  17. Manzano, M. 1996. Extensions of first-order logic. Cambridge: Cambridge University Press.Google Scholar
  18. Mill, J.S. 1869. II. Of the liberty of thought and discussion. In On liberty, ed. J.S. Mill. London: Longman, Roberts & Green.Google Scholar
  19. Quine, W.V. 1970. Philosophy of logic, 2nd ed. Oxford: Oxford University Press.Google Scholar
  20. Robinson, J.A., and A. Voronkov (eds.). 2001. Handbook of automated reasoning. Cambridge: MIT Press.Google Scholar
  21. Sieg, W., and J. Byrnes. 1998. Normal natural deduction proofs (in classical logic). Studia Logica 60: 67–106.CrossRefGoogle Scholar
  22. Smullyan, R. 1968. First order logic. Berlin: Springer.Google Scholar
  23. Wolfram, S. 2002. A new kind of science. Champaign: Wolfram Media, Inc.Google Scholar
  24. Zalta, E.N. 2009. Achieving Leibniz’s goal of a computational metaphysics. The 2009 North American Conference on Computing and Philosophy. Bloomington.Google Scholar
  25. Zalta, E.N., B. Fitelson, and P. Oppenheimer. 2011. Computational metaphysics. Retrieved 18 November 2011, from http://mally.stanford.edu/cm/.

Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.School of Information Resources and Library ScienceUniversity of ArizonaTucsonUSA

Personalised recommendations