Towards a Coalgebraic Chomsky Hierarchy

Abstract

Abstract. The Chomsky hierarchy plays a prominent role in the foundations of theoretical computer science relating classes of formal languages of primary importance. In this paper we use recent developments on coalgebraic and monadbased semantics to obtain a generic notion of a T-automaton, where T is a monad, which allows the uniform study of various notions of machines (e.g. finite state machines, multi-stack machines, Turing machines, valence automata, weighted automata). We use the generalized powerset construction to define a generic (trace) semantics for T-automata, and we show by numerous examples that it correctly instantiates for some known classes of machines/languages captured by the Chomsky hierarchy. Moreover, our approach provides new generic techniques for studying expressivity power of various machine-based models.

A New Foundation for Finitary Corecursion

“…Goncharov et al [23] show that H lifts to Set T and conclude that finite-state HT -coalgebras match the intuition of deterministic pushdown automata without spontaneous transitions. The languages accepted by those automata are precisely the real-time deterministic contextfree languages; this notion goes back to Harrison and Havel [24].…”

“…In the non-deterministic setting, a similar property holds, namely that the determinized HT -coalgebras with finite carrier describe precisely the context-free languages [23,Theorem 6.5]. Combine this with (1):…”

“…It is well-known that deterministic and non-deterministic pushdown automata recognize different classes of context-free languages. We will characterize both as instances of the locally finite fixpoint, using the results from [23] on stack machines, which can push or read multiple elements to or from the stack in a single transition, respectively.…”

“…While this is the case in some categories (e.g. sets, posets, graphs, vector spaces, commutative monoids), it is currently unknown in other base categories that are used in the construction of system models, for example in idempotent semirings (used in the treatment of context-free grammars [43]), in algebras for the stack monad (used for modelling configurations of stack machines [23]); or it even fails, for example in the category of finitary monads on sets (used in the categorical study of algebraic trees [7]), or Eilenberg-Moore categories for a monad in general (the target category of generalized determinization [41], in which the above examples live). Coalgebras over a category of Eilenberg-Moore algebras over Set in particular provide a paradigmatic setting: automata that describe languages beyond the regular languages consist of a finite state set, but their transitions produce side effects such as the manipulation of a stack.…”

“…Applied to the coalgebraic treatment of context-free languages, we show that the locally finite fixpoint yields precisely the context-free languages, and real-time deterministic context-free languages, respectively, when modelled using algebras for the stack monad of [23]. For context-free languages weighted in a semiring S, or equivalently for constructively S-algebraic power series [36], the locally finite fixpoint comprises precisely those, by phrasing the results of Winter et al [44] in terms of the generalized powerset construction.…”