2. Clausal logic and resolution: theoretical backgrounds¶

In this chapter we develop a more formal view of Logic Programming by means of a rigorous treatment of clausal logic and resolution theorem proving. Any such treatment has three parts: syntax, semantics, and proof theory. Syntax defines the logical language we are using, i.e. the alphabet, different kinds of ‘words’, and the allowed ‘sentences’. Semantics defines, in some formal way, the meaning of words and sentences in the language. As with most logics, semantics for clausal logic is truth-functional, i.e. the meaning of a sentence is defined by specifying the conditions under which it is assigned certain truth values (in our case: true or false). Finally, proof theory specifies how we can obtain new sentences (theorems) from assumed ones (axioms) by means of pure symbol manipulation (inference rules).

Of these three, proof theory is most closely related to Logic Programming, because answering queries is in fact no different from proving theorems. In addition to proof theory, we need semantics for deciding whether the things we prove actually make sense. For instance, we need to be sure that the truth of the theorems is assured by the truth of the axioms. If our inference rules guarantee this, they are said to be sound. But this will not be enough, because sound inference rules can be actually very weak, and unable to prove anything of interest. We also need to be sure that the inference rules are powerful enough to eventually prove any possible theorem: they should be complete.

Concepts like soundness and completeness are called meta-theoretical, since they are not expressed in the logic under discussion, but rather belong to a theory about that logic (‘meta’ means above). Their significance is not merely theoretical, but extends to logic programming languages like Prolog. For example, if a logic programming language is unsound, it will give wrong answers to some queries; if it is incomplete, it will give no answer to some other queries. Ideally, a logic programming language should be sound and complete; in practice, this will not be the case. For instance, in the next chapter we will see that Prolog is both unsound and incomplete. This has been a deliberate design choice: a sound and complete Prolog would be much less efficient. Nevertheless, any Prolog programmer should know exactly the circumstances under which Prolog is unsound or incomplete, and avoid these circumstances in her programs.

The structure of this chapter is as follows. We start with a very simple (propositional) logical language, and enrich this language in two steps to full clausal logic. For each of these three languages, we discuss syntax, semantics, proof theory, and meta-theory. We then discuss definite clause logic, which is the subset of clausal logic used in Prolog. Finally, we relate clausal logic to Predicate Logic, and show that they are essentially equal in expressive power.