# Equational Logic

## News

• 10.07.: The slots for the oral exam have been allocated. Please find them in TUMonline. The exam will take place in Prof. Nipkow's office: MI 00.09.055.
• 10.06.: The oral exam will take place on July 21st, between 09:00 and 12:00. Slots will be allocated as soon as registration for the exams is closed.
• 24.04.: The exercise session on April 29th has been re-scheduled to start 14:15 in room MI 00.09.038.
• 15.04.: The first exercise sheet will be published in week 2 and is due in week 3.
• There are no lectures in week 1. To compensate for that, the lectures in week 2 jump right in, without lengthy motivation. Read these 4 pages and the Wikipedia pages on Rewriting and Lambda Calculus for some motivation.
• There will be weekly homework sheets, which will be graded. Students who achieve 50% of the homework points will receive a bonus mark of 0.3, provided that they also pass the final exam.

## Contents

This course exclusively treats equational logic instead of the usual first-order logic. It combines the topics term-rewriting and lambda-calculus with a logical view.

### Part I: Term Rewriting

Abstract reduction systems
Abstract reduction systems
Basic definitions Thm Church-Rosser iff confluent. Thm For convergent reductions, two elements are equivalent iff their normal forms are equal.
Well-founded Induction
Thm Well-founded induction is valid iff the reduction terminates. Application: König's Lemma.
Termination proofs
Thm A finitely branching relation terminates iff it can be embedded into (N,>).
Lexicographic orders
Thm The lexicographic order on AxB terminates if the order on A and B terminate. Thm The lexicographic order on A* terminates if the order on A terminates.
Multiset order
Thm (Dershowitz and Manna) The multiset order terminates if the underlying order terminates.
Equational Logic
Terms and Substitutions
Signature, term, position in term, subterm at position, replacement at position, substitution, composition of substitutions.
Term Rewriting and Equational Logic
Identities, term rewriting (→E), equational logic (s ≈E t and E |- s ≈ t). Thm s ↔*E t iff s ≈E t.
Note that I defined equational logic with rule K (if s ≈E t then u[s]pE u[t]p) but the book uses a different rule: if siE ti for i=1,...,n then f(s1,...,sn) ≈E f(t1,...,tn). It can be shown easily that both rules are equivalent (in the presence of the other rules).
Semantics
Algebra. Validity and semantic consequence (|=). Thm (Birkhoff) E |- s ≈ t iff E |= s ≈ t.
Equational Problems
Definition of word problem, unification problem and matching problem modulo E
Word problems
Undecidable for arbitrary E (Example: undecidable semigroup). Decidable for finite and convergent E.
Congruence closure
Thm If E is finite and ground and s ≈E t, then there is a proof of this fact that involves only subterms of E, s, t.
Hence the word problem is decidable (in polynomial time) if E is finite and ground.
Syntactic unification
Basic definitions: the "more general" quasiorder on substitutions, most general unifier, idempotent substitution.
Unification by transformation
Transformation rules, soundness and completeness. Exponential worst case complexity.
Termination
• Termination is undecidable because TRS can simulate Turing machines and the Halting Problem for Turing machines is undecidable. Still undecidable for 1-rule TRS, and for TRS of 3 or more rules if all function symbols are unary. Open problem: is it decidable for 1 or 2 rules if all function symbols are unary?
• Termination is decidable for finite TRS where all right-hand sides are ground.
• Reduction orders, the interpretation method and polynomial interpretations.
• There is an annual Termination Competition
Confluence
• Thm Confluence is undecidable for arbitrary finite TRSs.
• Newman's Lemma A terminating and localy confluent reduction is confluent.
• Critical Pair Lemma If all critical pairs of a TRS are joinable, the TRS is locally confluent.
• Knuth-Bendix Completion The completion algorithm. Example (groups): completion of (x.y).z → x.(y.z), 1.x → x, i(x).x → 1 generates the additional rules x.1 → x, i(1) → 1, x.i(x) → 1, i(x).(x.y) → y, x.(i(x).y) → y, i(i(x)) → x, i(x.y) → i(y).i(x).
• Thm Orthogonal TRSs (no critical pairs, left-linear) are confluent.

### Part II: Lambda Calculus

Untyped Lambda calculus
Syntax
Terms, notational conventions, Currying, static binding, free and bound variables, substitution, alpha-conversion.
Beta-reduction
Definition of beta-reduction. Definition of parallel reduction >. Proof that > has the diamond property. Because >* = →β* this implies that β-reduction is confluent. (See confluence proof for orthogonal TRS)
Eta-reduction
Motivation, definition and basic properties: termination and (local) confluence. No proofs.
Reduction strategies
Without proof: contraction of leftmost β-redexes leads to a normal form if one exists.
Lambda calculus as a programming language
Booleans, pairs, Church numerals, fixed-point combinators.
Typed Lambda calculus
Simply typed lambda calculus
Simple types. Implicitly and explicitly typed terms. Type checking rules. Properties: 1. beta-reduction preserves types ("subject reduction") (w/o proof). 2. beta-reduction terminates on type-correct terms (w/o proof). Thus beta-equivalence is decidable for type-correct terms (but has non-elementary complexity). Corollary The simply-typed lambda-calculus can only represent total functions. Thm Every computable functions is representable as a closed type-correct lambda-term whose only contants are additional fixed-point combinators (proof only sketched).
Type inference
The rules. 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.
Let-polymorphism
Universally quantified type schemas. Typing rules for "let" and for type schemas. Syntax-directed typing rules with built-in quantifier handling.
Curry-Howard isomorphism
Types = formulas, 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

Primary:
Additional:
• Jürgen Avenhaus. Reduktionssyteme, Springer, 1995
• Hendrik Pieter Barendregt. The Lambda Calculus, its Syntax and Semantics, North-Holland, 2nd edition, 1984
• Jean-Yves Girard, Yves Lafont, and Paul Taylor. Proofs and Types, Cambridge Tracts in Theoretical Computer Science, Cambridge University Press, 1989
• Chris Hankin. An Introduction to Lambda Calculi for Computer Scientists, King's College Publications, 2004
• J. Roger Hindley and Jonathan P. Seldin. Introduction to Combinators and Lambda Calculus, Cambridge University Press, 1986
• J. Roger Hindley and Jonathan P. Seldin. Lambda-Calculus and Combinators: An Introduction, Cambridge University Press, 2008