This Article 
 Bibliographic References 
 Add to: 
Quasi-Acyclic Propositional Horn Knowledge Bases: Optimal Compression
October 1995 (vol. 7 no. 5)
pp. 751-762

Abstract—Horn knowledge bases are widely used in many applications. This paper is concerned with the optimal compression of propositional Horn production rule bases - one of the most important knowledge bases used in practice. The problem of knowledge compression is interpreted as a problem of Boolean function minimization. It was proved in [16] that the minimization of Horn functions, i.e., Boolean functions associated with Horn knowledge bases, is NP-complete.

This paper deals with the minimization of quasi-acyclic Horn functions, the class of which properly includes the two practically significant classes of quadratic and of acyclic functions. A procedure is developed for recognizing in quadratic time the quasi-acyclicity of a function given by a Horn CNF, and a graph-based algorithm is proposed for the quadratic time minimization of quasi-acyclic Horn functions.

[1] D. Angluin,Learning Propositional Horn Sentences with Hints, technical report Yale/DCS/RR-590, Dept. of Computer Science,Yale Univ., 1987.
[2] D. Angluin,M. Frazier,, and L. Pitt,“Learning conjunctions of Horn clauses,” Machine Learning, vol. 9, pp. 147-164, 1992.
[3] B. Apsvall,M.F. Plass,, and R.E Tarjan,“A linear-time algorithm for testing the truth of certain quantified Boolean formulas,” Information Processing Letters, vol. 8, pp. 121-123, 1979.
[4] O. Cepek,Restricted Consensus Method and Quadratic Implicates of Pure HornFunctions, RUTCOR Research Report, RRR 31-94, Rutgers Univ., New Brunswick, N.J., Sept. 1994.
[5] E. Boros,Y. Crama,, and P.L. Hammer,“Polynomial-time inference of all valid implications for Horn and related formulae,” Annals of Mathematics and Artificial Intelligence, vol. 1, pp. 21-32, 1990.
[6] V. Chandru,C.R. Coullard,P.L. Hammer,M. Montanez,, and X. Sun,“On renamable Horn and generalized Horn functions,” Annals of Mathematics and Artificial Intelligence, vol. 1, pp. 33-47, 1990.
[7] E. Charles and O. Dubois, "MELODIA: Logical Methods for Checking Knowledge Bases," Validation, Verification and Test of Knowledge Based Systems, M. Ayel and J.-P. Laurent, eds. Chichester, U.K.: John Wiley, 1991.
[8] R.M. Colomb and C.Y.C. Chung,“Very fast decision table execution of propositional expert systems,” Proc. Eighth Nat’l Conf. Artificial Intelligence (AAAI 90), pp. 671-676, 1990.
[9] R. Dechter and J. Pearl,“Structure identification in relational data,” Artificial Intelligence, vol. 58, pp. 237-270, 1992.
[10] W.F. Dowling and J.H. Gallier,“Linear time algorithms for testing the satisfiability of propositional Horn formulae,” J. Logic Programming, vol. 3, pp. 267-284, 1984.
[11] M.R. Garey and D.S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness.New York: W.H. Freeman, 1979.
[12] A. Ginsberg,“Knowledge-base reduction: A new approach to checking knowledge bases for inconsistency&redundancy,” Proc. Seventh Nat’l Conf. Artificial Intelligence (AAAI 88), pp. 585-589, 1988.
[13] F. Glover and H.J. Greenberg,“Logical testing for rule-base management,” Annals of Operations Research, vol. 12, pp. 199-215, 1988.
[14] P.L. Hammer and A. Kogan,“Horn functions and their DNFs,” Information Processing Letters, vol. 44, pp. 23-29, 1992.
[15] P.L. Hammer and A. Kogan,Horn Function Minimization and Knowledge Compression in Production RuleBases, RUTCOR Research Report RRR 8-92, Rutgers Univ., New Brunswick, N.J., Mar. 1992.
[16] P.L. Hammer and A. Kogan,“Optimal compression of propositional Horn knowledge bases: Complexity and approximation,” Artificial Intelligence, vol. 64, pp. 131-145, 1993.
[17] P.L Hammer and B. Simeone,“Quadratic functions of binary variables,” in Combinatorial Optimization, B. Simeone, ed. Lecture Notes in Mathematics 1403.Berlin: Springer-Verlag, pp. 1-56, 1989.
[18] J.P. Ignizio, Introduction to Expert Systems: The Development and Implementation of Rule-Based Expert Systems. McGraw-Hill, 1991.
[19] H.R. Lewis,“Renaming a set of clauses as a Horn set,” J. ACM, vol. 25, pp. 134-135, 1978.
[20] M. Minoux,“LTUR: A simplified linear-time unit resolution algorithm for Horn formulae and computer implementation,” Information Processing Letters, vol. 29, pp. 1-12, 1988.
[21] M. Minoux,“The Unique Horn-satisfiability problem and quadratic Boolean equations,” Annals of Mathematics and Artificial Intelligence, vol. 6, pp. 253-266, 1992.
[22] T.A. Nguyen, W.A. Perkins, T.J. Laffey, and D. Pecora, “Knowledge Base Verification,” AI Magazine, pp. 69–75, Summer 1987.
[23] W. Quine,“A way to simplify truth functions,” Am. Math. Monthly, vol. 62, pp. 627-631, 1955.
[24] B. Selman and H. Kautz,“Knowledge compilation using Horn approximations,” Proc. Ninth Nat’l Conf. Artificial Intelligence (AAAI 91), pp. 904-909, 1991.
[25] R.E. Tarjan,“Depth first search and linear graph algorithms,” SIAM J. Computing, vol. 1, no. 2, pp. 146-160, 1972.
[26] J. Tepandi,“Comparison of exper1708t system verification criteria: Redundancy,” Validation, Verification, and Test of Knowledge-based Systems, M. Ayel and J.-P. Laurent, eds. Wiley∧Sons, 1991, pp. 49-62.

Index Terms:
Expert systems, propositional knowledge bases, Horn clauses, logical equivalence, Boolean functions, logic minimization, knowledge compression, acyclic rule bases.
Peter L. Hammer, Alexander Kogan, "Quasi-Acyclic Propositional Horn Knowledge Bases: Optimal Compression," IEEE Transactions on Knowledge and Data Engineering, vol. 7, no. 5, pp. 751-762, Oct. 1995, doi:10.1109/69.469822
Usage of this product signifies your acceptance of the Terms of Use.