Context-Free Languages via Coalgebraic Trace Semantics

Abstract

In this paper we identify context-free grammars as coalgebras. To obtain the associated context-free languages (consisting of only finite-length strings) we introduce a general and novel technique of finite trace semantics for coalgebras. It builds on top of the (possibly infinite) trace semantics introduced earlier by the second author, but extracts only finite behavior. Interestingly the finite trace is uniquely characterized corecursively and hence it yields a final coalgebra in a suitable Kleisli category, while the ordinary trace is not unique and yields a weakly final coalgebra. Additionally, the constructions of both finite and possibly infinite parse trees are shown to be monads. Hence our extension of the application domain of coalgebras identifies several new mathematical constructions and structures.

An introduction to (co)algebra and (co)induction

“…of actions occurring in such an execution. This section describes a systematic way to capture such traces coalgebraically, following [HJ05,HJS07], which is general and generic. Here we concentrate on the powerset case.…”

“…The following result (from [HJ05]) combines initial algebras and final coalgebras for a description of trace semantics.…”

Context-Free Languages, Coalgebraically

“…Here + denotes the coproduct (or disjoint union), P ω the finite power set, and (A + X) * is the set of all the strings of finite length over A and X. According to the above definition, CFGs are coalgebras for the functor P ω ((A + (−)) * ), and indeed a coalgebraic account of context-free grammars and context-free languages using the above functor (without the finiteness condition on the power set) is presented in [5]. There, the focus is mainly on finite skeletal parsed trees (i.e.…”

“…This may explain why not so much algebraic or coalgebraic work has been devoted to study the theory of context-free languages. The first, and only, coalgebraic treatment of context-free languages we are aware of, is presented in [5]. In this paper context-free languages are described indirectly, as the result of flattening finite skeletal parsed trees.…”

Testing Semantics: Connecting Processes and Process Logics

“…In this paper, we will confine our presentation to the possibilistic setting, leaving the probabilistic setting for further work. For this setting the categorical trace semantics of finite state automata [8] and context-free languages [9] are clear examples, and are close conceptual predecessors of testing semantics. What appears to be new is our ability to bring Turing machines into the same setting.…”