Chin-Liang Chang

A New Algorithm for Generating Prime Implicants

This paper describes an algorithm which will generate all the prime implicants of a Boolean function. The algorithm is different from those previously given in the literature, and in many cases it is more efficient. It is proved that the algorithm will find all the prime implicants. The algorithm may possibly generate some nonprime implicants. However, using frequency orderings on literals, the experiments with the algorithm show that it usually generates very few ( possibly none) nonprime implicants. Furthermore, the algorithm may be used to find the minimal sums of a Boolean function. The algorithm is implemented by a computer program in the LISP language.

Some properties of fuzzy logic

In this paper, the fuzzy set ZZadeh (1965)] is viewed as a multivalued logic with a continuum of truth values in the interval Z0, 1]. The concepts of inconsistency, validity, prime implicant and prime implicate are extended to fuzzy logic and various properties of these notions in the context of fuzzy logic are established. It is proved that a formula is valid (inconsistent) in fuzzy logic iff it is valid (inconsistent) in two-valued logic. An algorithm that generates fuzzy prime implicants (implicates) is introduced. A proof of the completeness of this algorithm is also given.

A Heuristic Relaxation Method for Nonlinear Mapping in Cluster Analysis

A relaxation method mapping high-dimensional sample points to low-dimensional sample points is discussed. This method tries to preserve the local interdistance of sample points. Some special heuristics have been introduced to handle the difficulty arising from a large amount of data. Experimental results with handwritten character data and Iris data show that the method runs fast, converges rapidly, and requires a small amount of memory space.

The Unit Proof and the Input Proof in Theorem Proving

A resolution in which one of the two parent clauses is a unit clause is called a unit resolution, whereas a resolution in which one of the two parent clauses is an original input clause is called an input resolution. A unit (input) proof is a deduction of the empty clause □ such that every resolution in the deduction is a unit (input) resolution. It is proved in the paper that a set S of clauses containing its unit factors has a unit proof if and only if S has an input proof. A LISP program implementing unit resolution is described and results of experiments are given.

An admissible and optimal algorithm for searching AND/OR graphs

Abstract An AND/OR graph is a graph which represents a problem-solving process. A solution graph is a subgraph of the AND/OR graph which represents a derivation for a solution of the problem. Therefore, solving a problem can be viewed as searching for a solution graph in an AND/OR graph. A “cost” is associated with every solution graph. A minimal solution graph in a solution graph with minimal cost. In this paper, an algorithm for searching for a minimal solution graph in an AND/OR graph is described. If the “lower bound” condition is satisfied, the algorithm is guaranteed to find a minimal solution graph when one exists. Furthermore, the “optimality” of the algorithm is also proved.

Pattern Recognition by Piecewise Linear Discriminant Functions

A piecewise linear function is represented in terms of a set of linear functions through the use of the maximum and minimum functions. A procedure for finding piecewise linear discriminant functions for pattern recognition is described. The procedure iteratively uses the accelerated relaxation method to find every linear function in a piecewise linear function. The procedure was implemented by a Fortran program. Experimental results with the program showed that the procedure is promising for obtaining piecewise linear discriminant functions.

Herbrand's Theorem

This chapter focuses on Herbrands theorem. By definition, a valid formula is a formula that is true under all interpretations. Herbrand developed an algorithm to find an interpretation that can falsify a given formula. However, if the given formula is indeed valid, no such interpretation can exist and his algorithm will halt after a finite number of trials. Herbrands method is the basis for most modern automatic proof procedures. By definition, a set S of clauses is unsatisfiable if and only if it is false under all interpretations over all domains. When the atom set A of a set S of clauses is infinite, any complete semantic tree for S will be infinite. A complete semantic tree for S corresponds to an exhaustive survey of all possible interpretations for S . If S is unsatisfiable, then S fails to be true in each of these interpretations. A semantic tree T is said to be closed if and only if every branch of T terminates at a failure node.

The Equality Relation

This chapter focuses on the equality relation. Equality is a very important relation, and many theorems can be easily symbolized through its use. For example, one can employ only one predicate, that of equality, one function, the successor function, and one constant 0, to formalize number theory. The equality relation has many special properties—it is reflexive, symmetric, and transitive. Also, one can substitute equals for equals. When the equality relation is used to symbolize a theorem, besides axioms for the specific theorem itself, one usually needs a collection of extra axioms describing these properties of equality. Paramodulation is an inference rule for the equality relation. As in resolution, the purpose of refining paramodulation is to increase efficiency. The refinements of paramodulation are somewhat weaker than those of resolution. Hyperresolution and linear resolution can be extended to paramodulation. The chapter discusses hyperparamodulation, and reviews unit, input, and linear paramodulations. An input paramodulation is a paramodulation in which one of the two parent clauses is an input clause.

The Propositional Logic

This chapter discusses the simplest symbolic logic—the propositional logic or the propositional calculus. A proposition is a declarative sentence that is either true or false, but not both. The “true” or “false” assigned to a proposition is called the truth value of the proposition. From propositions, compound propositions can be built by using logical connectives. A formula is said to be valid if and only if it is true under all its interpretations. A formula is said to be invalid if and only if it is not valid. A formula is said to be inconsistent or unsatisfiable, if and only if it is false under all its interpretations. A formula is said to be consistent or satisfiable, if and only if it is not inconsistent. In the propositional logic, since the number of interpretations of a formula is finite, one can always decide whether a formula in the propositional logic is valid—inconsistent—by exhaustively examining all of its possible interpretations.

The Specialization of Programs by Theorem Proving

Suppose a program P is written to accept a set of inputs I. If we are only interested in a nonempty subset