Logic and Proof

Lecturer: Dr L.C. Paulson (lcp@cl.cam.ac.uk)

No. of lectures: 12

This course is a prerequisite for Specification and Verification I (Part II).

Aims

This course will teach logic, especially the predicate calculus. It will present the basic principles and definitions, then describe a variety of different formalisms and algorithms that can be used to solve problems in logic. Putting logic into the context of Computer Science, the course will show how the programming language Prolog arises from the automatic proof method known as resolution. It will introduce topics that are important in mechanical verification, such as ordered binary decision diagrams (OBDDs) and modal logic.

Lectures

• Introduction to logic. Schematic statements. Interpretations and validity. Logical consequence. Inference.

• Propositional logic. Basic syntax and semantics. Equivalences. Normal forms. Tautology checking using CNF.

• The sequent calculus. A simple (Hilbert-style) proof system. Natural deduction systems. Sequent calculus rules. Sample proofs.

• Ordered binary decision diagrams. General concepts. Fast canonical form algorithm. Optimisations. Applications.

• First order logic. Basic syntax. Quantifiers. Semantics (truth definition).

• Formal reasoning in FOL. Free versus bound variables. Substitution. Equivalences for quantifiers. Sequent calculus rules. Examples.

• Clausal proof methods. Clause form. The Davis-Putnam procedure. The resolution rule. Examples. Refinements.

• Skolem functions and Herbrand's theorem. Prenex normal form. Skolemisation. Herbrand models and their properties.

• Unification. Composition of substitutions. Most general unifiers. A unification algorithm. Applications and variations.

• Prolog. Binary resolution. Factorisation. Example of Prolog execution. Proof by model elimination.

• Modal logics. Possible worlds semantics. Truth and validity. A Hilbert-style proof system. Sequent calculus rules.

• Tableaux methods. Simplifying the sequent calculus. Examples. Adding unification. Skolemisation. The world's smallest theorem prover?

Objectives

At the end of the course students should

• be able to manipulate logical formulas and to notice their own errors

• be able to prove logical theorems, either using a formal calculus or by applying standard laws of Boolean algebra

• be able to construct a small OBDD

• understand the relationships between the various clause methods (Davis-Putnam, resolution and Prolog)

• be able to describe the contrasting uses of the two main methods for propositional logic (OBDDs and Davis-Putnam)

• be able to apply the unification algorithm and to describe its uses

• know how to express and prove statements in modal logic

• understand the relationship between the sequent calculus and tableaux methods

• be able to describe the distinction between clausal and non-clausal methods

Recommended books

Chang, C-L. & Lee, R.C-T. (1973). Symbolic Logic and Mechanical Theorem Proving. Academic Press.
Galton, A. (1990). Logic for Information Technology. Wiley.
Reeves, S. & Clarke, M. (1990). Logic for Computer Science. Addison-Wesley.