On-Line Algorithms for Polynomially Solvable Satisfiability Problems

Abstract Given a propositional Horn formula, we show how to maintain on-line information about its satisfiability during the insertion of new clauses. A data structure is presented which answers each satisfiability question in O (1) time and inserts a new clause of length q in O ( q ) amortized time. This significantly outperforms previously known solutions of the same problem. This result is extended also to a particular class of non-Horn formulae already considered in the literature, for which the space bound is improved. Other operations are considered, such as testing whether a given hypothesis is consistent with a satisfying interpretation of the given formula and determining a truth assignment which satisfies a given formula. The on-line time and space complexity of these operations is also analyzed.

[1]  Jeffrey D. Ullman,et al.  Principles of Database Systems , 1980 .

[2]  R. Tarjan.  Amortized Computational Complexity , 1985 .

[3]  Giorgio Ausiello,et al.  Minimal Representation of Directed Hypergraphs , 1986, SIAM J. Comput..

[4]  Shuji Doshita,et al.  The Satisfiability Problem for a Class Consisting of Horn Sentences and Some Non-Horn Sentences in Proportional Logic , 1983, Inf. Control..

[5]  C. A. Petri.  Communication with automata , 1966 .

[6]  Jean H. Gallier,et al.  Linear-Time Algorithms for Testing the Satisfiability of Propositional Horn Formulae , 1984, J. Log. Program..

[7]  Giorgio Ausiello,et al.  Graph Algorithms for Functional Dependency Manipulation , 1983, JACM.

[8]  Neil D. Jones,et al.  Complete problems for deterministic polynomial time , 1974, STOC '74.

[9]  Shimon Even,et al.  An On-Line Edge-Deletion Problem , 1981, JACM.

[10]  Greg N. Frederickson,et al.  Data Structures for On-Line Updating of Minimum Spanning Trees, with Applications , 1985, SIAM J. Comput..

[11]  Robert E. Tarjan,et al.  A data structure for dynamic trees , 1981, STOC '81.

[12]  Alon Itai,et al.  Unification as a Complexity Measure for Logic Programming , 1987, J. Log. Program..

[13]  Alfred V. Aho,et al.  The Design and Analysis of Computer Algorithms , 1974 .

[14]  Vikraman Arvind,et al.  An O(n²) Algorithm for the Satisfiability Problem of a Subset of Propositional Sentences in CNF That Includes All Horn Sentences , 1987, Inf. Process. Lett..

[15]  Kenneth McAloon,et al.  Stratification and Knowledge Base Management , 1987, J. Symb. Comput..

[16]  Neil D. Jones,et al.  Complete problems for deterministic polynomial time , 1974, Symposium on the Theory of Computing.

[17]  Giuseppe F. Italiano,et al.  Amortized Efficiency of a Path Retrieval Data Structure , 1986, Theor. Comput. Sci..