Automated Mathematical Induction pp 39-78 | Cite as

# New Uses of Linear Arithmetic in Automated Theorem Proving by Induction

## Abstract

Zhang, Kapur, and Krishnamoorthy introduced a cover set method for designing induction schemes for automating proofs by induction from specifications expressed as equations and conditional equations. This method has been implemented in the theorem prover *Rewrite Rule Laboratory* (*RRL*) and a proof management system *Tecton* built on top of *RRL*, and it has been used to prove many nontrivial theorems and reason about sequential as well as parallel programs. The cover set method is based on the assumption that a function symbol is defined by using a finite set of terminating (conditional or unconditional) rewrite rules. The termination ordering employed in orienting the rules is used to perform proofs by well-founded induction. The left sides of the rules are used to design different cases of an induction scheme, and recursive calls to the function made in the right side can be used to design appropriate instantiations for generating induction hypotheses. A weakness of this method is that it relies on syntactic unification for generating an induction scheme for a conjecture. This paper goes a step further by proposing semantic analysis for generating an induction scheme for a conjecture from a cover set. We discuss the use of a decision procedure for Presburger arithmetic (quantifier-free theory of numbers with the addition operation and relational predicates >, <, ≠, =, ≥, ≤) for performing semantic analysis about numbers. The decision procedure is used to generate appropriate induction schemes for a conjecture by using cover sets of function taking numbers as arguments. This extension of the cover set method automates proofs of many theorems that otherwise require human guidance and hints. The effectiveness of the method is demonstrated by using some examples that commonly arise in reasoning about specifications and programs. It is also shown how semantic analysis using a Presburger arithmetic decision procedure can be used for checking the completeness of a cover set of a function defined by using operations such as + and − on numbers. With this check, many function definitions used in a proof of the prime factorization theorem stating that every number can be factored uniquely into prime factors, which had to be checked manually, can now be checked automatically in *RRL*. The use of the decision procedure for guiding generalization for generating conjectures and merging induction schemes is also illustrated.

## Key words

Induction automated theorem proving heuristics linear arithmetic Presburger arithmetic Generalization semantic unification## Preview

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## References

- 1.Boyer, R. S. and Moore, J S.: A
*Computational Logic*, ACM Monographs in Computer Science, 1979.Google Scholar - 2.Boyer, R. S. and Moore, J S.:
*A Computational Logic Handbook*, Academic Press, New York, 1988.zbMATHGoogle Scholar - 3.Boyer, R. S. and Moore, J S.: Integrating decision procedures into heuristic theorem provers: A case study of linear arithmetic,
*Machine Intelligence***11**(1988), 83–157.MathSciNetGoogle Scholar - 4.Dershowitz, N.: Termination of rewriting,
*J. Symbolic Computation***3**(1987), 69–116.MathSciNetCrossRefzbMATHGoogle Scholar - 5.Jouannaud, J.-P. and Kounalis, E.: Automatic proofs by induction in theories without constructors,
*Information and Computation***82**(1989), 1–33.MathSciNetCrossRefzbMATHGoogle Scholar - 6.Kapur, D.: An automated tool for analyzing completeness of equational specifications, in
*Proc. Int. Symp. Software Testing and Analysis (ISSTA)*, Seattle, August 1994, pp. 28–43.Google Scholar - 7.Kapur, D., Musser, D. R., and Nie, X.: An overview of the Tecton proof system, in
*Theoret. Computer Science Journal*, special issue on*Formal Methods in Databases and Software Engineering*(ed. V. Alagar), Vol. 133, October 1994, pp. 307–339.Google Scholar - 8.Kapur, D., Narendran, P., Rosenkrantz, D., and Zhang, H.: Sufficient-completeness, quasi-reducibility and their complexity,
*Acta Infortnatica***28**(1991), 311–350.MathSciNetCrossRefzbMATHGoogle Scholar - 9.Kapur, D. and Nie, X.: Reasoning about numbers in Tecton, in
*Proc. 8th Int. Symp. Methodologies for Intelligent Systems (ISMIS’94)*, Charlotte, NC, October 1994, pp. 57–70.Google Scholar - 10.Kapur, D. and Zhang, H.: An overview of Rewrite Rule Laboratory (RRL),
*J. Computer Math. Appt*.**29**(2) (1995), 91–114.MathSciNetCrossRefGoogle Scholar - 11.Walther, C.: Combining induction axioms by machine,
*Proc. 12th Int. Joint Conf. Artificial Intelligence*, Chambery, France, 1993.Google Scholar - 12.Zhang, H.: Reduction, superposition and induction: Automated reasoning in an equational logic, Ph.D. Thesis, Department of Computer Science, Rensselaer PolytechnicInstitute, Troy, NY, 1988.Google Scholar
- 13.Zhang, H., Kapur, D., and Krishnamoorthy, M. S.: A mechanizable induction principle for equational specifications, in
*Proc. 9th Int. Conf. Automated Deduction (CADE-9)*, Argonne, IL, LNCS 310, Springer-Verlag, 1988, pp. 250–265.Google Scholar