# A continuous approach to inductive inference

- 140 Downloads
- 26 Citations

## Abstract

In this paper we describe an interior point mathematical programming approach to inductive inference. We list several versions of this problem and study in detail the formulation based on hidden Boolean logic. We consider the problem of identifying a hidden Boolean function*ℱ*:{0, 1}^{ n } → {0, 1} using outputs obtained by applying a limited number of random inputs to the hidden function. Given this input—output sample, we give a method to synthesize a Boolean function that describes the sample. We pose the Boolean Function Synthesis Problem as a particular type of Satisfiability Problem. The Satisfiability Problem is translated into an integer programming feasibility problem, that is solved with an interior point algorithm for integer programming. A similar integer programming implementation has been used in a previous study to solve randomly generated instances of the Satisfiability Problem. In this paper we introduce a new variant of this algorithm, where the Riemannian metric used for defining the search region is dynamically modified. Computational results on 8-, 16- and 32-input, 1-output functions are presented. Our implementation successfully identified the majority of hidden functions in the experiment.

## Key words

Inductive inference Boolean function synthesis satisfiability artificial intelligence integer programming interior point method Riemannian geometry## Preview

Unable to display preview. Download preview PDF.

## References

- [1]D. Angluin and C.H. Smith, “Inductive inference: Theory and methods,”
*Computing Surveys*15 (1983) 237–265.Google Scholar - [2]E. Boros, P.L. Hammer and J.N. Hooker, “Predicting cause-effect relationships from incomplete discrete observations,” RUTCOR, Rutgers University (Piscataway, NJ, 1991).Google Scholar
- [3]R.K. Brayton, G.D. Hachtel, C.T. McMullen and A.L. Sangiovanni-Vincentelli,
*Logic Minimization Algorithms for VLSI Minimization*(Kluwer Academic Publishers, Dordrecht, 1985).Google Scholar - [4]D.W. Brown, “A state-machine synthesizer-SMS,” in:
*Proceedings of the 18th Design Automation Conference*(1981) pp. 301–304.Google Scholar - [5]Y. Crama, P.L. Hammer and T. Ibaraki, “Cause-effect relationships and partially defined Boolean functions,”
*Annals of Operations Research*16 (1988) 299–325.Google Scholar - [6]M. Davis and H. Putnam, “A computing procedure for quantification theory,”
*Journal of the ACM*7 (1960) 201–215.Google Scholar - [7]J.F. Gimpel, “A method of producing a Boolean function having an arbitrarily prescribed prime implicant table,”
*IEEE Transaction on Computers*14 (1965) 485–488.Google Scholar - [8]S.J. Hong, R.G. Cain and D.L. Ostapko, “MINI: A heuristic approach for logic minimization,”
*IBM Journal of Research and Development*(1974) 443–458.Google Scholar - [9]A.P. Kamath, N.K. Karmarkar, K.G. Ramakrishnan and M.G.C. Resende, “Computational experience with an interior point algorithm on the Satisfiability problem,”
*Annals of Operations Research*25 (1990) 43–58.Google Scholar - [10]N. Karmarkar, “An interior-point approach to NP-complete problems,”
*Contemporary Mathematics*114 (1990) 297–308.Google Scholar - [11]N. Karmarkar, “Riemannian geometry underlying interior-point methods for linear programming,”
*Contemporary Mathematics*114 (1990) 51–75.Google Scholar - [12]N.K. Karmarkar, M.G.C. Resende and K.G. Ramakrishnan, “An interior point algorithm to solve computationally difficult set covering problems,”
*Mathematical Programming*52 (1991) 597–618.Google Scholar - [13]N.K. Karmarkar, M.G.C. Resende and K.G. Ramakrishnan, “An interior-point approach to the maximum independent set problem in dense random graphs,” in:
*Proceedings of the XV Latin American Conference on Informatics*(1989) pp. 241–260.Google Scholar - [14]E.J. McCluskey, “Minimization of Boolean functions,”
*Bell System Technical Journal*351 (1956) 417–1444.Google Scholar - [15]R.E. Miller,
*Switching Theory, Vol. 1: Combinatorial Circuits*(Wiley, New York, 1965).Google Scholar - [16]J.J. Moré and D.C. Sorensen, “Computing a trust region step,”
*SIAM Journal on Scientific and Statistical Computing*4 (1983) 553–572.Google Scholar - [17]E. Morreale, “Recursive operators for prime implicant and irredundant normal form determination,”
*IEEE Transactions on Computers*C-19 (1970) 504.Google Scholar - [18]R. Pai, N. Karmarkar and S.S.S.P. Rao, “A global router based on Karmarkar's interior point method,” CSE, Indian Institute of Technology (Bombay, 1988).Google Scholar
- [19]W.V. Quine, “The problem of simplifying truth functions,”
*American Mathematical Monthly*59 (1952).Google Scholar - [20]W.V. Quine, “A way to simplify truth functions,”
*American Mathematical Monthly*62 (1955).Google Scholar - [21]J.P. Roth, “A calculus and an algorithm for the multiple-output 2-level minimization problem,” IBM Thomas J. Watson Research Center (Yorktown Heights, NY, 1968).Google Scholar
- [22]R. Rudell and A. Sangiovanni-Vincentelli, “Exact minimization of multiple-valued functions for PLA optimization,” in:
*Proceedings of the IEEE International Conference on Computer-Aided Design*(1986) pp. 352–355.Google Scholar - [23]B. Selman, D. Mitchell and H.J. Levesque, “A new method for solving large Satisfiability problems,” AT&T Bell Laboratories (Murray Hill, NJ, 1991).Google Scholar
- [24]J.R. Slagle, C.L. Chang and R.C.T. Lee, “A new algorithm for generating prime implicants,”
*IEEE Transactions on Computers*C-19 (1970) 304.Google Scholar - [25]E. Triantaphyllou, A.L. Soyster and S.R.T. Kumara, “Generating logical expressions from positive and negative examples via a branch-and-bound approach,” Industrial and Management Systems Engineering, Pennsylvania State University (University Park, PA, 1991).Google Scholar
- [26]Y. Ye, “On the interior algorithms for nonconvex quadratic programming,” Integrated Systems Inc. (Santa Clara, CA, 1988).Google Scholar