Abstract Algebraic Language Theory

Citation preview

Abstract Algebraic Language Theory Achim Blumensath

Abstract Algebraic Language Theory Achim Blumensath

ab BRNO 2022

Achim Blumensath [email protected]

This document was last updated 2022-12-11. The latest version can be found at www.fi.muni.cz/~blumens

Copyright 2022 Achim Blumensath This work is licensed under the Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.

Contents I Monads 1. An overview . . . . . . . 2. Discrete categories . . . . 3. Polynomial functors . . . 4. Monads . . . . . . . . . . 5. Eilenberg-Moore algebras 6. Lifting monads . . . . . . 7. Bialgebras . . . . . . . . .

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1 1 8 21 33 39 60 81

II Algebra 1. Factorisation systems 2. Subalgebras . . . . . . 3. Reducts . . . . . . . . 4. Congruences . . . . . 5. Varieties . . . . . . .

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93 93 104 116 119 127

III Languages 1. Weights . . . . . . . . . . . . 2. Languages . . . . . . . . . . 3. Minimal algebras . . . . . . . 4. Syntactic algebras . . . . . . 5. Varieties . . . . . . . . . . . 6. The profinitary term monad 7. Axiomatisations . . . . . . .

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IV Logic

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Contents

1. Abstract logics . . . 2. Compositionality . 3. Definable algebras . 4. Definable languages

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V Trees 1. Monads and logics for trees and forests 2. Finite forests . . . . . . . . . . . . . . . 3. Countable chains . . . . . . . . . . . . . 4. Counterexamples . . . . . . . . . . . . 5. MSO-definable algebras . . . . . . . . . 6. First-order logic . . . . . . . . . . . . .

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253 . 253 . 263 . 265 . 278 . 284 . 294

VI Temporal Logics 1. Temporal logics . . 2. Bisimulation . . . . 3. The logic EF . . . . 4. Wreath products . . 5. Distributive algebras 6. Path algebras . . . .

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Recommended Literature Bibliography Symbol Index Index

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I. Monads 1. An Overview There are many different formalisms to specify formal languages based on automata, grammars, regular expressions, homomorphisms, logics, and so on. The central topic of formal language theory is the study of such formalisms. In particular, we are interested in their expressive power and their algorithmic properties, i.e., which questions are decidable for them and what the respective complexity is. Several frameworks exist for answering such questions. Here we will adopt a very general category-theoretic point of view that covers many of them. Our focus will be on algebraic and logical approaches with a special emphasis on languages of infinite trees and their monadic second-order theories. Before starting to develop the general theory, let us shortly present some of the specific examples it is supposed to subsume. We will be rather succinct and intended mainly as a reminder to readers already familiar with the material. The reader is encouraged to ignore and/or skip any parts that look incomprehensible.

Finite Words The prototypical example of a formal language theory is that of finite words. A finite word over a given alphabet Σ is a finite sequence (possibly empty) of elements of Σ. We denote the set of all finite words by Σ∗ . (When not explicitly mentioned otherwise, we will assume alphabets to be finite.) A (formal) language is a set L ⊆ Σ∗ of such words. The main algebraic framework for such languages is based on monoids (or semigroups). A monoid M = ⟨M, ⋅ , e⟩ is

abstract algebraic language theory 2022-12-11

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I. Monads

a structure with universe M, a binary operation ⋅ ∶ M × M → M, and a constant e ∈ M such that ◆ ⋅ is associative: a ⋅ (b ⋅ c) = (a ⋅ b) ⋅ c ; ◆ e is a neutral element: e ⋅ a = a = a ⋅ e. Examples include ◆ the set {0, 1} with the usual multiplication and the neutral element 1; ◆ the natural numbers ⟨N, +, 0⟩ with addition; ◆ the natural numbers ⟨N, ⋅ , 1⟩ with multiplication; ◆ the set Σ∗ of all finite words with concatenation as the product and the empty sequence ⟨⟩ as the neutral element. The monoid ⟨Σ∗ , ⋅ , ⟨⟩⟩ is also called the free monoid since it has the following universal property: for every monoid M and every function f ∶ Σ → M, there exists a unique homomorphism φ ∶ Σ∗ → M that agrees with f on the elements of Σ. A homomorphism φ ∶ M → N of monoids is a function φ ∶ M → N between their universes that preserves products and the neutral element, that is φ(a ⋅ b) = φ(a) ⋅ φ(b)

and

φ(e) = e .

We can use a homomorphism φ ∶ Σ∗ → M from the free monoid to another (usually finite) monoid M to represent languages of Σ. We say that L ⊆ Σ∗ is recognised by φ if L = φ−1 [P], for some P ⊆ M. Examples. (a) The monoid ⟨{0, 1}, max, 0⟩ recognises the language Σ∗ aΣ∗ via the morphism mapping a to 1 and every other letter to 0. (b) The monoid Z/2Z recognises the language of all words with an even number of letters a via the morphism mapping a to 1 and every other letter ⌟ to 0. Note that we can encode every language L = φ−1 [P] recognised by some finite monoid M by specifying ◆ the multiplication table of M,

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1. An overview

◆ the neutral element of M, ◆ the set P, and ◆ the values φ(c), for c ∈ Σ. This is a finite amount of information. Hence, we can use this encoding for algorithms that take languages as input. Of course, not every language can be encoded this way: there are only countably many encodings but uncountably many languages (if the alphabet contains at least two letters). So, which languages can be recognised by a homomorphism in this way? It turns out that these are exactly the wellknown regular languages. Theorem 1.1. Let L ⊆ Σ∗ . The following statements are equivalent. (1) L is recognised by a finite automaton. (2) L is the value of a regular expression. (3) L is denoted by a linear grammar. (4) L is definable in monadic second-order logic. (5) L is recognised by a homomorphism to a finite monoid. (6) The syntactic congruence of L has only finitely many classes. Let us briefly explain the last item in the above characterisation. The syntactic congruence ∼L of a language L ⊆ Σ∗ is a binary relation on Σ∗ which is defined by u ∼L v

iff

(xuy ∈ L ⇔ xv y ∈ L) ,

for all x, y ∈ Σ∗ .

It turns out that ∼L forms a congruence on the free monoid ⟨Σ∗ , ⋅ , ⟨⟩⟩ and the quotient homomorphism Σ∗ → Σ∗ /∼L recognises L. Hence, if ∼L has only finitely many classes, the quotient Σ∗ /∼L forms a finite monoid recognising L

Infinite Words An ω-word over an alphabet Σ is an infinite sequence w = (c n )n