# Course descriptions

## Fundamentals of metalogic

**Content: Completeness theorems, Model theory and proof theory.**

**Course description**: This course provides an introduction to the metatheory of elementary logic. Following a "refresher" on the basics of notation and the use of classical logic as a representation language, we concentrate on the twin notions of models and proof. An axiomatic system of first order logic is introduced and proved complete for the standard
semantics, and then we give a very brief overview of the basic concepts of proof theory and of formal set theory. The material in this course is presupposed by other courses in the Summer School, which is why it is presented first.

**Pre-requisites: ** it is assumed that students are at least familiar with logical notation for connectives and quantifiers, and can manipulate truth tables, some kind of proof system or semantic tableaux or the like. Normally they will have done a first logic course at tertiary level. Some basic mathematical competence is also presupposed, to be comfortable with the notion of a formal language, follow proofs by induction, be OK with basic set-theoretic notation and the like.

**Lecturer: **
John Slaney is the founder and convenor of the Logic Summer school. He originated in England, ever so long ago, and discovered Australia in 1977. Undeterred by the fact that it had apparently been discovered before, he gradually moved here, joining the Automated Reasoning Project in 1988 on a three-year contract because he could not think of anywhere better to be a logician. He is still here and still can't. His research interests pretty much cover the waterfront, including non-classical logics, automated deduction and all sorts of logic-based AI. He was formerly the leader of the Logic and Computation group at the ANU, and of the NICTA program of the same name.

## Introduction to modal logic

**Content: Kripke models, Hilbert calculi, Frame correspondences, Tableaux-based decisions procedures and
Propositional linear temporal logic**

**Course description: ** We cover the syntax, Kripke semantics, correspondence theory and tableaux-style proof theory of propositional modal and temporal logics. These logics have important applications in a diverse range of fields incuding Artificial Intelligence, Theoretical Computer Science and Hybrid Systems.

**Pre-requisites: ** You should have a good grounding in classical propositional and first-order logic, including material covered in the "Fundamentals of Metalogic" course. Some very basic notions from graph theory would also be useful, but are not essential.

**Lecturer: **
Rajeev Gore is currently a a professor in the ANU, and leader of the Logic and Computation research group. He was an Australian Research Council Queen Elizabeth II Fellow at the ANU from 1997-2001, a Research Fellow and later Senior Fellow at the ANU from 1994-2011, and a Research Associate at the University of Manchester from 1992-1993. He obtained his PhD in Modal Theorem Proving from the University of Cambridge in 1992.

## Overview of automated reasoning

**Content: Automated propositional theorem proving; automated first-order theorem proving; Interactive theorem proving**

**Course description: ** In many applications, we expect computers to reason logically. We might naively expect this to be what computers are good at, but in fact they find it extremely difficult. In this overview course, we look at various methods to automate logical reasoning, which are needed to support a variety of application domains.

Automated reasoning procedures are parametrized by the logic they are capable of reasoning with. We distinguish between propositional logic and first-order logic. Development and application of propositional logic procedures, also called SAT solvers, received considerable attention in the last ten years, e.g., for solving constraint satisfaction problems, applications in hardware design, verification, and planning and scheduling. Regarding automated deduction in first-order logic, we discuss applications, standard deductive procedures such as resolution, and basic concepts, such as unification. We also examine the dual problem of theorem proving, viz., generating models of a given theory, which has applications to finding counterexamples for non-theorems, and we provide an overview of some recent trends (instance-based methods, satisfiability modulo theories). Finally, we describe in some depth so-called quantifier elimination methods for theorem proving in decidable fragments of arithmetics over the reals and the integers.

**Pre-requisites: ** Elementary logic, as provided by the "Fundamentals of Metalogic" course of this Summer School.

**Lecturer: ** TBA

## Computability and incompleteness

**Contents: Computability; recursive Functions and Turing machines; diagonalisation; Peano arithmetic and Gödel numbering; undecidability of first-order logic; incompleteness of Peano arithmetic.**

**Course Description: ** We begin with two formal accounts of the intuitive notion of computability: recursive functions and Turing machines. They turn out to be the same, hence Church's Thesis: functions that can be computed by any means are precisely the partial recursive functions. Then we revisit the old Cretan who says that all Cretans always lie, and other forms of diagonalisation argument such as Halting Problem. Next we look at an axiomatic theory of arithmetic, known as Peano Arithmetic (PA), and show how we can represent all recursive functions in PA. This will lead to Goedel numbering: a neat trick enabling us to effectively encode notions like "theorem", "proof" and "provability in PA" within PA itself. We spend a while discussing Diagonalisation Lemma and Derivability Conditions. Finally, in one fell swoop we prove undecidability of first-order logic (Church's Theorem), undefinability of truth (Tarski's Theorem), incompleteness of PA given consistency of PA (First Goedel's Theorem) and unprovability of consistency of PA given consistency of PA (Second Goedel's Theorem).

**Pre-requisites: ** Foundations of first-order logic

**Background reading: ** G. Boolos and R. Jeffrey, *Computability and Logic*.

**Lecturer: **
Michael Norrish is a Pricipal Researcher in the Canberra Laboratory of Data61 (CSIRO). He is originally from Wellington, New Zealand, and is very happy to be back in the southern hemisphere, after an extended stint in Cambridge, England. It was there that he did his PhD, and then spent three years as a research fellow at St. Catharine's College. His research interests include interactive theorem-proving, and the application of this technology to areas of theoretical computer science, particularly the semantics of programming languages.