Logic and Proof: 20102011
Lecturer 

Degrees 

Term 
Hilary Term 2011 (16 lectures) 
Overview
The main aim of the course is to give a first introduction to formal logic for computer scientists.
 Introduction to propositional logic. Syntax of propositional logic. Truth tables. Sequent Calculus. Resolution. Notions of soundness and completeness. Proofs by induction on the structure of formulas.
 Introduction to structures. Examples. Introduction to finite state transition systems.
 Introduction to firstorder logic. Syntax of firstorder logic. Semantics of firstorder logic. Gamebased semantics. Examples. Sequent Calculus. Notions of soundness and completeness.Logic as database query language.
 Introduction to temporal logics, especially Linear Temporal Logic (LTL). Examples.
Learning outcomes
At the end of the course students are expected to:
 Understand and be able to explain and illustrate the meaning of given logical formulas, to translate such formulas into English and viceversa.
 Construct simple, but rigorous, formal proofs for some given theorems, in a given proof system.
 Be able to express and formalize in a logical language useful properties of models such as transition systems, and be able to determine the truth or falsity of such formulas in a given model.
Synopsis
Approximately 16 lectures.
Propositional logic (7 Lectures).
 Introduction. Syntax of propositional logic. Examples. Recursive definitions of functions over formulas.
 Semantics of propositional logic. Validity and satisfiability of formulas. Truth tables. Modern SATsolving.
 Equivalence of formulas. Substitution. Normal forms.
 Resolution.
 Sequent calculus.
 Sequent calculus (ctd.). Soundness and completeness.
Firstorder logic (6 Lectures).
 Structures and examples for structures. Substructures.
 Introduction. Syntax of firstorder logic. Examples.
 Semantics.
 Examples. Satifiability and validity. Evaluation of formulas.
 Sequent Calculus.
 Sequent Calculus (ctd.). Examples. Soundsness and completeness.
Linear Temporal Logic [LTL] (3 lectures).
 Introduction. Syntax of LTL. Examples.
 Semantics of LTL.
 Model checking LTL formulas.
Syllabus
Syntax and Semantics of propositional and firstorder logic. Proofs by induction on the structure of formulas. Formal proofs using sequent calculus. Resolution. Brief discussion of issues of soundness and completeness. Mathematical models of computation, especially Kripke structures. Temporal logics: Linear Temporal Logic (LTL). Determining the truth of a temporal logic formula in a given model.Reading list
Primary text: Mathematical logic, 2nd edition, by H.D. Ebbinghaus, J. Flum and W. Thomas (Springer 1996).
 Logic in computer science: modelling and reasoning about systems,2nd edition, by M. Huth and M. Ryan (Cambridge University Press, Cambridge 2004).
 Proof and Disproof in Formal Logic, by Richard Bornat (Oxford University Press, 2005).
 Logic for Computer Science, by S. Reeves and M. Clarke (International Computer Science Series, OMW, University of London, AddisonWesley, 1990).
 The language of firstorder logic: including the Mackintosh version of Tarski's World 4.0, by J. Barwise and J. Etchemendy (CSLI Lecture Notes no. 23, CSLI Publications, Stanford 1993).
 Hyperproof by J. Barwise and J. Etchemendy (CSLI Lecture Notes no. 42, CSLI Publications, Stanford 1994).