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Structured proof procedures

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Abstract

In this paper, we address the problem of enriching an interactive theorem prover with complex proof procedures. We show that the approach of building complex proof procedures out of deciders for (decidable) quantifier-free theories has many advantages: (i) deciders for quantifier-free theories provide powerful, high level functionalities which greatly simplify the activity of designing and implementing complex and higher level proof procedures; (ii) this approach is of wide applicability since most of the proof procedures are composed by steps of propositional reasoning intermixed with steps carrying out higher level strategical functionalities; (iii) decidability and efficiency are retained on important (decidable) subclasses, while they are often sacrificed by uniform proof strategies for the sake of generality; and finally (iv), from a software engineering perspective, the modularity of the procedures guarantees that any modification in the implementation can be accomplished locally. As a case study, we present and discuss a proof procedure for the existential fragment of first-order logic (prenex existential formulas without function symbols) built on top of a propositional decider.

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References

  1. A. Armando and E. Giunchiglia, Embedding complex decision procedures inside an interactive theorem prover, Ann. Math. and AI 8(1993)475–502.

    Google Scholar 

  2. M. Abadi and Z. Manna, Nonclausal deduction in first-order temporal logic, J. ACM 37(1990)279–317.

    Google Scholar 

  3. P.B. Andrews, Theorem proving via general matings, J. ACM 28(1981)193–214.

    Google Scholar 

  4. M. Buro and H.K. Büning, Report on a SAT competition, Technical Report No. 110, FB-17, Mathematik/Informatik Universität Paderborn (1992).

  5. W. Bibel,Automated Theorem Proving (Vieweg, Braunschweig, 1982).

    Google Scholar 

  6. R.S. Boyer and J.S. More, Integrating decision procedures into heuristic theorem provers: A case study of linear arithmetic, Machine Intelligence 11(1988)83–124.

    Google Scholar 

  7. B. Dreben and W.D. Goldfarb,The Decision Problem — Solvable Classes of Quantificational Formulas (Addison-Wesley, 1979).

  8. M. Davis, G. Longemann and D. Loveland, A machine program for theorem proving, J. ACM 5(7) (1962).

  9. M. Davis and H. Putnam, A computing procedure for quantification theory, J. ACM 7(1960)201–215.

    Google Scholar 

  10. P.C. Gilmore, A proof method for quantification theory: Its justification and realization, IBM J. Res. Develop. 4(1960)28–35.

    Google Scholar 

  11. F. Giunchiglia, The GETFOL Manual — GETFOL Version 1, Technical Report 9204-01, DIST, University of Genova, Genoa, Italy (1992). Forthcoming IRST Technical Report.

    Google Scholar 

  12. F. Harche, J.N. Hooker and G.L. Thompson, A computational study of satisfiability algorithms for propositional logic, Working Paper 1991-27 (1991).

  13. R.G. Jeroslow, Computation-oriented reduction of predicate to propositional logic, Decision Support Syst. 4(1988)183–197.

    Google Scholar 

  14. W.H. Joyner, Resolution strategies as decision procedures, J. ACM 23(1976)398–417.

    Google Scholar 

  15. S.J. Lee and D.A. Plaisted, Eliminating duplication with the hyper-linking strategy, Technical Report TR90-032, Department of Computer Science, University of North Carolina (1990).

  16. N.V. Murray and E. Rosenthal, Dissolution: Making paths vanish, J. ACM, to appear.

  17. G. Nelson and D.C. Oppen, Simplification by cooperating decision procedures, Technical Report STAN-CS-78-652, Stanford Computer Science Department (1978).

  18. F.J. Pelletier, Seventy-five problems for testing automatic theorem provers, J. Autom. Reasoning 2(1986)191–216. See also Errata Corrige in J. Autom. Reasoning 4(1988)235–236.

    Google Scholar 

  19. D. Prawitz, An improved proof procedure, Theoria 26(1960)102–139.

    Google Scholar 

  20. D. Prawitz,Natural Deduction — A Proof Theoretical Study (Almquist-Wiksell, Stockholm, 1965).

    Google Scholar 

  21. R. Reiter, Deductive questions-answering on relational databases, in:Logic and Data Bases, eds. H. Gallaire and J. Minker (Plenum Press, New York, 1978).

    Google Scholar 

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

    Google Scholar 

  23. M. Stickel, Automated deduction by theory resolution, J. Autom. Reasoning 4(1985)333–356.

    Google Scholar 

  24. G. Winskel,The Formal Semantics of Programming Languages (MIT Press, 1993).

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Author notes

  1. Enrico Giunchiglia

    Present address: Computer Science Department, University of Texas at Austin, Austin, USA

Authors and Affiliations

  1. Mechanized Reasoning Group, Dipartimento di Informatica, Sistemistica e Telematica, Università di Genova, Genova, Italy

    Enrico Giunchiglia, Alessandro Armando & Paolo Pecchiari

Authors
  1. Enrico Giunchiglia
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  2. Alessandro Armando
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  3. Paolo Pecchiari
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Giunchiglia, E., Armando, A. & Pecchiari, P. Structured proof procedures. Ann Math Artif Intell 15, 1–18 (1995). https://doi.org/10.1007/BF01535839

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