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Logic and Proof:  2008-2009

Lecturer

Degrees

ModerationsComputer Science

Part AMathematics and Computer Science

Term

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. Natural deduction. Notions of soundness and completeness.
  • Introduction to first-order logic. Syntax of first-order logic. Semantics of first-order logic. Examples. Natural deduction. Notions of soundness and completeness, and brief discussion of incompleteness and undecidability.
  • Introduction to some finite models for computation, in particular Kripke structures. Examples.
  • Introduction to temporal logics, especially Linear Temporal Logic (LTL) and Computation Tree Logic (CTL).
  • Time permitting: discussion of model-checking algorithms.

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 vice-versa.
  • 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 Kripke structures, and be able to determine the truth or falsity of such formulas in a given model.

Synopsis

Approximately 16 lectures.

Propositional logic (4 Lectures).

  1. Introduction. Syntax of propositional logic. Examples.
  2. Truth tables. Modern SAT-solving.
  3. Natural deduction.
  4. Natural deduction (ctd.). Soundness and completeness.

First-order logic (5 Lectures).

  1. Introduction. Syntax of first-order logic. Examples.
  2. Semantics.
  3. Natural deduction.
  4. Natural deduction.
  5. Soundness and completeness; incompleteness.

Models of computation (1 lecture).

  1. Kripke structures and brief mention of other models.

Linear Temporal Logic [LTL] (3 lectures).

  1. Introduction. Syntax of LTL. Examples.
  2. Semantics of LTL.
  3. Model checking LTL formulas.

Computation Tree Logic [CTL] (3 lectures).

  1. Introduction. Syntax of CTL. Examples.
  2. Semantics of CTL.
  3. Model checking CTL formulas

Syllabus

Syntax and Semantics of propositional and first-order logic. Formal proofs using natural deduction. Brief discussion of issues of soundness, completeness, and incompleteness. Mathematical models of computation, especially Kripke structures. Temporal logics: Linear Temporal Logic (LTL) and Computation Tree Logic (CTL). Determining the truth of a temporal logic formula in a given model.

Reading list

Primary text:
  •  Logic in computer science: modelling and reasoning about systems,2nd Editions, by M. Huth and M. Ryan (Cambridge University Press, Cambridge 2004).
Other texts to be consulted
  • 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, Addison-Wesley, 1990).
  • Mathematical Logic by H.D. Ebbinghaus, J. Flum, W. Thomas (Undergraduate Texts in Mathematics, Springer Verlag 1984).
  • The language of first-order 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).