Predicate calculus

This section details the different aspects of the representation we have adopted to describe the problem solutions and the new control knowledge generated by the learning mechanism. Throughout the section we will continue to use the flowshop scheduling problem as an illustration. The section starts by discussing the motives for selecting the horn clause form of first-order predicate calculus, and then proceeds to show how the representation supports both the synthesis of problem solutions and their analysis. The section concludes with a description of how the sufficient... 

In predicate calculus, where we are allowed function and predicate symbols that take one or more arguments, the form is... 

First-order predicate calculus admits proof techniques that can be shown to be sound and complete (Lloyd, 1987). The soundness of the proof technique is important because it ensures that our methodology will not deduce results that are invalid. We are less concerned with completeness, because in most cases, although the proof technique will be complete, the theory of dominance or equivalence we have available will be incomplete for most problems. Restricting the first-order logic to be of horn clause form, enables the employment of SLD resolution, a simpler... 

More specifically, the basic notions of a Turing Machine, of computable functions and of undecidable properties are needed for Chapter VI (Decision Problems) the definitions of recursive, primitive recursive and partial recursive functions are helpful for Section F of Chapter IV and two of the proofs in Chapter VI. The basic facts regarding regular sets, context-free languages and pushdown store automata are helpful in Chapter VIII (Monadic Recursion Schemes) and in the proof of Theorem 3.14. For Chapter V (Correctness and Program Verification) it is useful to know the basic notation and ideas of the first order predicate calculus a highly abbreviated version of this material appears as Appendix A. 

Appendix A contains a brief summary of sane relevant ideas of satisfiability and validity of well-formed formulas in the predicate calculus. Using these ideas it gives a definition of partial and total correctness of a scheme with respect to a well-formed formula as output criterion. The treatment is cursory and nonrigorous. Readers who have not seen these ideas before should examine this appendix before we return to the treatment of correctness and program verification in Chapter V, and finally conclude this treatment in Chapjter VII. 

In this chapter we discuss techniques for program verification and their mathematical justification. The basic idea behind these methods was originally presented by Floyd mathematical formulations and logical justifications were developed by Cooper and Manna, and others, and continued in King s Ph.D. thesis in which he presented the development of a partial implementation for these techniques. A sanewhat different axiomatic approach has been pursued by Hoare et al. The reader who has never made acquaintance with the formalism of the first order predicate calculus should at this point turn to Appendix A for a brief and unrigorous exposition of the material relevant to this chapter. 

APPENDIX A - PROGRAM SCHEMES AND THE FIRST ORDER PREDICATE CALCULUS... 

We wish to present a few of the basic ideas of the first order predicate calculus, using as a starting point the ideas of interpretations of schemes which we have already encountered. 

Manna, Zohar, "Properties of Programs and the First-Order Predicate Calculus,"... 

A Multivalued Logic Predicate Calculus Approach to Synthesis Pla ining... 

In this paper we describe the need for planning, and then develop the predicate calculus we used and the choice of multi-valued logic. Finally we briefly describe the QED program, a few rules, and an example analysis. Other papers in the QED series will cover the program and chemical results in detail. 

WIPKE AND DOLATA A Predicate Calculus Approach to Synthesis Planning 189... 

We chose the first order predicate calculus (PC) as our language for representing synthetic principles. The first order predicate calculus (PC) is a "formal" system of logic.(11)(12)(13) In this context, formal means that it is the form of the arguments that is important, not the actual content. The term "calculus" comes from the meaning "a method of calculation", and does not refer to Newton s differential calculus. 

But in chemistry, where a typical molecule will have 20 - 30 atoms, as many bonds, several rings, stereocenters, hetereoatoms, etc., a theory expressed in the sentential calculus would require thousands of statements. Thus chemistry had to await development of the predicate calculus,(14) to axiomatize the theory. 

The multi-valued predicate calculus logic as implemented in QED has been demonstrated to be suitable for cleanly representing strategic axioms of chemical synthesis. QED is a powerful tool for exploring inference in the planning of synthesis strategies. QED helped us elucidate key strategic concepts and their interdependence and... 

These three statements merge two tables, partition a table into two based on a predicate, and copy a table, respectively. Each statement and its inverse can be represented as a logical formula in predicate calculus as well as SQL statements that... 

Copper has an atomic line at 324.8 nm In predicate calculus, we write atomic line(copper, 324.8 nm)... 

Complementary forms of knowledge representation are based on semantic nets and frames (Figure 8.2). Often, they represent just another form to input knowledge. The internal representation is usually based on the predicate calculus. The latter can also be interpreted as a relation of objects ... 

Clause Alternative formulation of statements expressed in first-order predicate calculus, having the form q, ... 

Predicate calculus Formal language to express and reason with statements over a domain of discourse. 

Clauses are alternative notations of statements expressed in first-order predicate calculus. They have the form of a logical implication. A clause is expressed as a pair of sets of terms in the form... 

Horn clauses do not have the expressive power of the full clausal form of logic and, therefore do not have the power of first-order predicate calculus. The main disadvantage is that it is impossible to represent negative information like Siblings are not married with Horn clauses. 

The knowledge base holds facts, which may comprise short-term and long-term information, and rules, which can be considered as long-term information, on how to generate new facts. Within the knowledge base information can be represented in a number of ways, including predicate calculus, frames, semantic networks, and production rules. 

Predicate calculus is derived from propositional logic, in which propositions or statements can result in one of two values, e.g., true or false. Several logical operators exist to enable propositions to be developed. Some of the most important are ... 

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