Lambda Calculus

Chair for Logic and Verification

Overview

Professor Prof. Tobias Nipkow
Time and Place Monday 16:15 ‐ 17:45 in MI Hörsaal 2
Live stream RBG Live
First Lecture 2021-10-25
Language English
TUMonline IN2358
Moodle Course Link
Discussion Forum Zulip

News

General notice

Exam

The exam will be in written form and take place on 2022-02-23 08:00 - 09:30 in MW 2001. In accordance with the hygiene provisions for exams, the 3G rule applies which means that unvaccinated people need to provide an official rapid certificate not older than 24 hours or PCR test certificate not older than 48 hours. Additionally, students are required to wear an FFP2 mask that they can take off while seated. You may bring an A4 sheet with hand-written notes on both sides.

Grading

Grade Min Points
1 36.5
1.3 34
1.7 31.5
2 29
2.3 26.5
2.7 24
3 21.5
3.3 19
3.7 16.5
4 14
4.3 11.5
4.7 9
5 0

Post-Exam Review

You have received a link for the online review together with the grade published on TUMonline. The exam review runs until Sunday midnight, i.e. 2022-03-14 00:15.

Exercises

Organiser Lukas Stevens
Time and Place Thursday 10:05 ‐ 11:35 in seminar room 00.08.038
First Tutorial 2021-10-28
TUMonline IN2358

Exercise Sheets

Exercise sheet Tutorial solution
Exercise 01 Tutorial Solutions 01
Exercise 02 Tutorial Solutions 02
Exercise 03 Tutorial Solutions 03, Homework 3.5 Solution
Exercise 04 Tutorial Solution 04
Exercise 05 Tutorial Solution 05
Exercise 06 Tutorial Solution 06
Exercise 07 Tutorial Solution 07
Exercise 08, Exercise 8.1 Template Tutorial Solution 08, Exercise 8.1 Solution
Exercise 09, Exercise 9.2 Template Tutorial Solution 09, Exercise 9.2 Solution
Exercise 10, Exercise 10.2 Template Tutorial Solution 10, Exercise 10.2 Solution
Exercise 11 Tutorial Solution 11
Exercise 12

Contents

The \(\lambda\)-calculus is a universal model of computation, i.e. it can simulate any Turing machine, that was introduced by Alonzo Church in the 1930s. Today it forms the basis of many functional programming languages such as Haskell or Idris. Due to the Curry-Howard correspondence terms of the \(\lambda\)-calculus can not only be interpreted as programs but also as proofs. In its simplest form, the \(\lambda\)-calculus only has three rules that dictate how a term can be constructed:

Syntax Name Description
\(x\) Variable A name representing a parameter or mathematical value.
\((\lambda x.\ t)\) Abstraction Function definition where \(t\) is a \(\lambda\)-term. The variable \(x\) becomes bound in the expression.
\((f\ t)\) Application Applying the function \(f\) to the argument \(t\). Both \(f\) and \(t\) are \(\lambda\)-terms.

In order to compute with \(\lambda\)-terms we define \(\beta\)-reduction: the term \((\lambda x.\ t)\ s\) reduces to \(t[s / x]\) which means that any occurence of the variable \(x\) in \(t\) is replaced by \(s\). The above rules formalise the basic untyped \(\lambda\)-calculus. In the lecture, we will discuss the theoretical properties of both untyped and (simply) typed lambda calculus. In particular, we will investigate the correspondence of programs and proofs in the second part of the lecture:

  1. Untyped Lambda calculus

    1. Syntax: Terms, notational conventions, Currying, static binding, free and bound variables, substitution, alpha-conversion.

    2. Beta-reduction: Definition of \(\beta\)-reduction. Proof that \(\beta\)-reduction is confluent.

    3. Eta-reduction: Motivation, definition and basic properties: termination and (local) confluence.

    4. Reduction strategies: Without proof: contraction of leftmost \(\beta\)-redexes leads to a normal form if one exists.

    5. Lambda calculus as a programming language: Booleans, pairs, Church numerals, fixed-point combinators.

  2. Typed Lambda calculus

    1. Simply typed lambda calculus: Simple types. Implicitly and explicitly typed terms. Type checking rules.

    2. Type inference: Type-correct terms no longer have a unique type but still a most general type. Proof by a concrete Prolog-like interpretation of the typing rules as backward computation rules.

    3. Let-polymorphism: Universally quantified type schemas. Typing rules for “let” and for type schemas. Syntax-directed typing rules with built-in quantifier handling.

    4. Curry-Howard correspondence: Types = propositions, lambda-terms = proofs, beta-reduction = proof-reduction. Proof of the subterm property of proofs in normal form. Proof of decidability of intuitionistic propositional logic via proofs in normal form.

Literature