Slides 00 1 Advanced Logics Part 1: Lambda Calculus and Type Theory Lecturer: Lim Yohanes Stefanus Part 2: Modal Logic Lecturer: Gregory Wheeler

Slides 00 2 course webpage

Slides 00 3 Part 1

Slides 00 4 References J. Roger Hindley. Basic Simple Type Theory. Cambridge University Press, H. P. Barendregt. Lambda Calculi with Types, in Handbook of Logic in Computer Science, Vol. 2, ed. S. Abramsky et al., CLarendon Press, 1992, pp /Vorlesungen/Typentheorie/SS98/barendregt.html

Slides 00 5 Marking Scheme  [20%] one programming assignment in Prolog, Haskell, or Java (the principle-type algorithm in Hindley's book chapter 3); last day for presentation: Friday March 31 st, 2006  [10% each] three exercises  [50%] written exam (Friday April 21 st, 2006)

Slides 00 6 Overview Different kinds of logic have been developed to express various situations. Not only the formal power of different logics that is at issue, but also how useful they are at representing different real life situations. Here are just a few of the well-known logics: classical logic, intuitionistic logic, temporal logic, modal logic, linear logic.

Slides 00 7 We can think of these as a set of tools that can be applied to different situations, depending on which concept we are modeling. Classical logic fits with most of our daily reasoning; it is based on the notion of truth. Intuitionistic logic is more about proof than truth – a sentence is true when we can provide a constructive proof of the statement.

Slides 00 8 An example that distinguishes intuitionistic logic from classical logic is the sentence "A or not A". In classical logic, this is always true, since whatever A is (true or false), the sentence is true. In intuitionistic logic we are unable to prove this sentence, since in general it is not the case that we can find a proof of either "A" or "not A".

Slides 00 9 Temporal logic captures the notion of time in a proof and allows us to express ideas such that sentences become true at a certain point in time. More generally, modal logics explore alternative modes of truth, where truth may depend on the state we are in, whether it be time, a set of beliefs or the current state of a machine.

Slides Linear logic is a refined logic that captures the notion of resources in a proof. A proof of a sentence in linear logic requires that we have correctly used all the assumptions.

Slides All of these logics have applications in computer science, for example: intuitionistic logic is used for type systems in functional programming; temporal logic is used for concurrent system specification / verification and model checking; modal logic is used for AI; and linear logic is used for automated theorem proving.

Slides Why Type Theory? It provides a framework which brings together logic and programming languages in an elegant way such that  program development and verification can proceed within a single system.  programs can be extracted from proofs in the logic.

Slides Focus We study one very neat and special polymorphic type system called TA (for "type-assignment"). Its types contain type-variables and arrows but nothing else, and its terms are built by λ-abstraction and application from term-variables and nothing else. Its expressive power is close to that of the system called simple type theory, developed by Alonzo Church.

Slides TA has no ∀-types and hence it is weaker than the strong polymorphic theories. However, it lies at the core of nearly every one of them. TA is an excellent training ground for learning the techniques of type-theory as a whole. TA's methods and algorithms are not trivial; and many complex techniques for analyzing structures in the stronger type- theories, appear in TA in a very clean and neat stripped-down form.

Slides By learning the basic techniques of type theory from TA, you will acquire a very good foundation for the study of other type systems.