[Paper Review] Formal languages and groups as memory

Mark Kambites|arXiv (Cornell University)|Jan 4, 2006

semigroups and automata theory14 references4 citations

TL;DR

This paper introduces $M$-automata and $G$-automata—finite automata augmented with a register storing an element of a monoid or group—to provide a group-theoretic interpretation of the Chomsky-Schützenberger theorem. It establishes that context-free languages are precisely the rational transductions of 2-sided Dyck languages, using the polycyclic monoid and free group as computational models, and proves foundational results linking algebraic properties of groups to language classes.

ABSTRACT

We present an exposition of the theory of finite automata augmented with a multiply-only register storing an element of a given monoid or group. Included are a number of new results of a foundational nature. We illustrate our techniques with a group-theoretic interpretation and proof of a key theorem of Chomsky and Schutzenberger from formal language theory.

Motivation & Objective

• To unify computational and algebraic approaches to automata with monoid/group memory registers.
• To provide a self-contained introduction to $M$-automata and $G$-automata for both computer scientists and pure mathematicians.
• To establish foundational results connecting algebraic properties of monoids and groups to closure properties of language classes.
• To offer a group-theoretic proof of the Chomsky-Schützenberger theorem using polycyclic monoids and free groups.
• To demonstrate the equivalence between pushdown automata and $G$-automata over free groups via permissible padding and rational transductions.

Proposed method

• Define $M$-automata as finite automata with a register storing an element of a monoid $M$, initialized to the identity and updated via multiplication during input processing.
• Introduce $G$-automata as $M$-automata where $M$ is a group, and define acceptance as reaching a final state with the register reset to identity.
• Use the polycyclic monoid $P(X)$ and free group $F(X^\#)$ to model pushdown automata via isomorphism between Dyck language acceptance and identity return in the register.
• Apply the concept of 'permissible padding'—inserting $\#$ and $\#^{-1}$ symbols—to embed words over $\overline{X}^*$ into $F(X^\#)$ and $P(X^\#)$ while preserving identity representation.
• Leverage rational transductions between free monoids to relate language classes and show that context-free languages arise as images of Dyck languages under such transductions.
• Prove equivalence between acceptance by a pushdown automaton and acceptance by a $G$-automaton over $F(X^\#)$ via the existence of a permissible padding that preserves identity in the free group.

Experimental results

Research questions

• RQ1 How can the Chomsky-Schützenberger theorem be reinterpreted and proven using group-theoretic automata with group memory?
• RQ2 What is the role of the polycyclic monoid and free group in modeling pushdown automata and context-free languages?
• RQ3 How do permissible paddings in $F(X^\#)$ and $P(X^\#)$ preserve identity representation and enable equivalence between automaton models?
• RQ4 What algebraic properties of monoids and groups correspond to closure properties of the language classes recognized by $M$-automata?
• RQ5 In what way do rational transductions and automata over free groups or polycyclic monoids characterize context-free languages?

Key findings

• The Chomsky-Schützenberger theorem is proven using $G$-automata over the free group $F(X^\#)$, showing that context-free languages are rational transductions of 2-sided Dyck languages.
• A word $w$ over $\overline{X}^*$ represents the identity in the polycyclic monoid $P(X)$ if and only if it admits a permissible padding that represents the identity in $F(X^\#)$.
• The existence of a permissible padding of $w$ representing the identity in $F(X^\#)$ implies that $w$ is accepted by a $G$-automaton over $F(X^\#)$ if and only if it is accepted by a pushdown automaton.
• The proof establishes a bijection between pushdown automata and $G$-automata over $F(X^\#)$ via the use of padded words and identity return in the register.
• The class of languages recognized by $G$-automata over a group $G$ corresponds exactly to the class of context-free languages when $G$ is a free group on a suitable alphabet.
• The theory of $M$-automata provides a unifying framework for understanding the word problem of a group and its relation to rational subset problems and language classes.