Symbolic and automatic differentiation of languages

Formal languages are usually defined in terms of set theory. Choosing type theory instead gives us languages as type-level predicates over strings. Applying a language to a string yields a type whose elements are language membership proofs describing how a string parses in the language. The usual building blocks of languages (including union, concatenation, and Kleene closure) have precise and compelling specifications uncomplicated by operational strategies and are easily generalized to a few general domain-transforming and codomain-transforming operations on predicates. A simple characterization of languages (and indeed functions from lists to any type) captures the essential idea behind language “differentiation” as used for recognizing languages, leading to a collection of lemmas about type-level predicates. These lemmas are the heart of two dual parsing implementations—using (inductive) regular expressions and (coinductive) tries—each containing the same code but in dual arrangements (with representation and primitive operations trading places). The regular expression version corresponds to symbolic differentiation, while the trie version corresponds to automatic differentiation. The relatively easy-to-prove properties of type-level languages transfer almost effortlessly to the decidable implementations. In particular, despite the inductive and coinductive nature of regular expressions and tries respectively, we need neither inductive nor coinductive/bisimulation arguments to prove algebraic properties.

On describing the regular closure of the linear languages with graph-controlled insertion-deletion systems

A graph-controlled insertion-deletion (GCID) system has several components and each component contains some insertion-deletion rules. A transition is performed by any applicable rule in the current component on a string and the resultant string is then moved to the target component specified in the rule. The language of the system is the set of all terminal strings collected in the final component. When resources are very limited (especially, when deletion is demanded to be context-free and insertion to be one-sided only), then GCID systems are not known to describe the class of recursively enumerable languages. Hence, it becomes interesting to explore the descriptional complexity of such GCID systems of small sizes with respect to language classes below RE and even below CF. To this end, we consider so-called closure classes of linear languages defined over the operations concatenation, Kleene star and union. We show that whenever GCID systems (with certain syntactical restrictions) describe all linear languages (LIN) with t components, we can extend this to GCID systems with just one more component to describe, for instance, the concatenation of two languages from the language family that can be described as the Kleene closure of linear languages. With further addition of one more component, we can extend the construction to GCID systems that describe the regular closure of LIN.

CLOSURES IN FORMAL LANGUAGES AND KURATOWSKI'S THEOREM

A famous theorem of Kuratowski states that, in a topological space, at most 14 distinct sets can be produced by repeatedly applying the operations of closure and complement to a given set. We re-examine this theorem in the setting of formal languages, where by "closure" we mean either Kleene closure or positive closure. We classify languages according to the structure of the algebras they generate under iterations of complement and closure. There are precisely 9 such algebras in the case of positive closure, and 12 in the case of Kleene closure. We study how the properties of being open and closed are preserved under concatenation. We investigate analogues, in formal languages, of the separation axioms in topological spaces; one of our main results is that there is a clopen partition separating two words if and only if the words do not commute. We can decide in quadratic time if the language specified by a DFA is closed, but if the language is specified by an NFA, the problem is PSPACE-complete.

POWERS OF REGULAR LANGUAGES

In this paper we prove that it is decidable whether the set pow (L), which we get by taking all the powers of all the words in some regular language L, is regular or not. The problem was originally posed by Calbrix and Nivat in 1995. Partial solutions have been given by Cachat for unary languages and by Horváth et al. for various kinds of exponent sets for the powers and regular languages which have primitive roots satisfying certain properties. We show that the regular languages which have a regular power are the ones which are 'almost' equal to their Kleene-closure.