Skip to main content
SpringerLink
Log in

−-match: An inference rule for incrementally elaborating set instantiations

  • Published:
Journal of Automated Reasoning Aims and scope Submit manuscript

Abstract

In this paper we describe a new inference rule, called −-match, which is used for finding set instantiations within an automated reasoning program. We have implemented −-match within a theorem prover called & and have used the system to prove some non-trivial theorems in mathematics, including Cantor's theorem and the correctness of transfinite induction. While not complete, we believe that −-match is a generally useful inference rule for finding set instantiations. One of the major contributions of the −-match rule is the ability to instantiate a term as an incompletely specified set abstraction, and then subsequently elaborate the identity of this set by considering other subgoals in the proof. This elaboration happens as a consequence of the deduction rules of the prover, and requires no additional machinery in the prover.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bailin, S. C., ‘A λ unifiability test for set theory’,Journal of Automated Reasoning 4(3) 269–286 (September 1988).

    Google Scholar 

  2. Bledsoe, W. W. and Feng, G., ‘Completeness ofSet-var’,Technical Report ATP104, University of Texas at Austin, Department of Computer Sciences, Austin, Texas (December 1990).

    Google Scholar 

  3. Bledsoe, W. W. and Feng, G., ‘Soundness and incompleteness results forSet-var’,Technical Report ATP105, University of Texas at Austin, Department of Computer Sciences, Austin, Texas (1991). Draft only.

    Google Scholar 

  4. Bledsoe, W. W., ‘A maximal method for set variables in automatic theorem proving’, inMachine Intelligence 9, pp. 53–100 Ellis Harwood, Ltd., Chichester (1979).

    Google Scholar 

  5. Bledsoe, W. W., ‘TheUt interactive prover’,Memo ATP-17B, Mathematics Department, University of Texas (April 1983).

  6. Barker-Plummer, D., ‘The gazer theorem prover: Description and users' guide’,Technical Report CS-1989-15, Department of Computer Science, Duke University, Durham, North Carolina (1989).

    Google Scholar 

  7. Barker-Plummer, D. and Bailin, S. C., ‘Graphical theorem proving: An approach to reasoning with the help of diagrams’, inProceedings European Conference on Artificial Intelligence (ECAI-92), Vienna, Austria, 3–7 August, pp. 55–59. John Wiley and Sons (August 1992).

  8. Barker-Plummer, D. and Bailin, S. C., ‘Proofs and pictures: Proving the diamond lemma with theGrover theorem proving system’, inWorking Notes of the AAAI Symposium on Reasoning with Diagrammatic Representations, March 25–27 1992, Stanford (March 1992).

  9. Barker-Plummer, D., Bailin, S. C. and Merrill, A. S., ‘The & theorem proving system’, in D. Kapur (Ed.),CADE-11, Lecture Notes in Artificial Intelligence, Vol. 607, pp. 716–720.11th International Conference on Automated Deduction (1992).

  10. Brown, F. M., ‘Towards the automation of set theory and its logic’,Artificial Intelligence 10 281–316 (1978).

    Google Scholar 

  11. Bledsoe, W. W. and Tyson, M., ‘TheUt interactive prover’,Memo ATP-17, Mathematics Department, University of Texas (May 1975).

  12. Cvetkovic, D. and Pevac, I., ‘Man-machine theorem proving in graph theory’,Artificial Intelligence 35 1–23 (May 1988).

    Google Scholar 

  13. Gentzen, G., ‘Untersuchen über das logische schliessen’ [Investigations into logical deduction], in M. E. Szabo (Ed.),The Collected Papers of Gerhard Gentzen, Ch. 3, pp. 68–131. North-Holland (1969). First appeared in 1935 as an article inMathematische Zeitschrift 39. This work was accepted as Gentzen's inaugural dissertation at the University of Göttingen.

  14. Nevins, A. J., ‘A human oriented logic for automatic theorem proving’,ACM 4 606–621 (1974).

    Google Scholar 

  15. Nevins, A. J., ‘A relaxation approach to splitting in an automatic theorem prover’,Artificial Intelligence 6 25–39 (1975).

    Google Scholar 

  16. Robinson, J. A., ‘A machine-oriented logic based on the resolution principle’,J. ACM 12 23–41 (1965).

    Article  Google Scholar 

  17. Russell, B., Letter to Frege in J. Van Heijenoort (Ed.),From Frege to Gödel: A Sourcebook in Mathematical Logic 1879–1931, Ch. 7, pp. 124–125. Harvard University Press, Cambridge, Massachusetts (1967). This letter is dated 16 June 1902.

    Google Scholar 

  18. Sterling, L. and Shapiro, E.,The Art of Prolog, MIT Press (1986).

  19. Wos, L., ‘The problem of finding an inference rule for set theory’,Journal of Automated Reasoning 5(1), 93–95 (1989).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. CTA Incorporated, 6116 Executive Blvd., Suite 800, 20852, Rockville, Maryland, USA

    Sidney C. Bailin

  2. Computer Science Program, Swarthmore College, 19081, Swarthmore, Pennsylvania, USA

    Dave Barker-Plummer

Authors
  1. Sidney C. Bailin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dave Barker-Plummer
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bailin, S.C., Barker-Plummer, D. −-match: An inference rule for incrementally elaborating set instantiations. J Autom Reasoning 11, 391–428 (1993). https://doi.org/10.1007/BF00881874

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00881874

Key words

Search

Navigation