Solution sets for equations over free groups are EDT0L languages

© World Scientific Publishing Company. We show that, given an equation over a finitely generated free group, the set of all solutions in reduced words forms an effectively constructible EDT0L language. In particular, the set of all solutions in reduced words is an indexed language in the sense of Aho. The language characterization we give, as well as further questions about the existence or finiteness of solutions, follow from our explicit construction of a finite directed graph which encodes all the solutions. Our result incorporates the recently invented recompression technique of Jez, and a new way to integrate solutions of linear Diophantine equations into the process. As a byproduct of our techniques, we improve the complexity from quadratic nondeterministic space in previous works to NSPACE(n log n) here

Generalized Results on Monoids as Memory

We show that some results from the theory of group automata and monoid automata still hold for more general classes of monoids and models. Extending previous work for finite automata over commutative groups, we demonstrate a context-free language that can not be recognized by any rational monoid automaton over a finitely generated permutable monoid. We show that the class of languages recognized by rational monoid automata over finitely generated completely simple or completely 0-simple permutable monoids is a semi-linear full trio. Furthermore, we investigate valence pushdown automata, and prove that they are only as powerful as (finite) valence automata. We observe that certain results proven for monoid automata can be easily lifted to the case of context-free valence grammars.Comment: In Proceedings AFL 2017, arXiv:1708.0622

On Pregroups, Freedom, and (Virtual) Conceptual Necessity

Pregroups were introduced in (Lambek, 1999), and provide a founda-tion for a particularly simple syntactic calculus. Buszkowski (2001) showed that free pregroup grammars generate exactly the -free context-free lan-guages. Here we characterize the class of languages generable by all pre-groups, which will be shown to be the entire class of recursively enumerable languages. To show this result, we rely on the well-known representation of recursively enumerable languages as the homomorphic image of the inter-section of two context-free languages (Ginsburg et al., 1967). We define an operation of cross-product over grammars (so-called because of its behaviour on the types), and show that the cross-product of any two free-pregroup grammars generates exactly the intersection of their respective languages. The representation theorem applies once we show that allowing ‘empty cat-egories ’ (i.e. lexical items without overt phonological content) allows us to mimic the effects of any string homomorphism.

Church-Rosser Systems, Codes with Bounded Synchronization Delay and Local Rees Extensions

What is the common link, if there is any, between Church-Rosser systems, prefix codes with bounded synchronization delay, and local Rees extensions? The first obvious answer is that each of these notions relates to topics of interest for WORDS: Church-Rosser systems are certain rewriting systems over words, codes are given by sets of words which form a basis of a free submonoid in the free monoid of all words (over a given alphabet) and local Rees extensions provide structural insight into regular languages over words. So, it seems to be a legitimate title for an extended abstract presented at the conference WORDS 2017. However, this work is more ambitious, it outlines some less obvious but much more interesting link between these topics. This link is based on a structure theory of finite monoids with varieties of groups and the concept of local divisors playing a prominent role. Parts of this work appeared in a similar form in conference proceedings where proofs and further material can be found.Comment: Extended abstract of an invited talk given at WORDS 201

On the Degree of Extension of Some Models Defining Non-Regular Languages

This work is a survey of the main results reported for the degree of extension of two models defining non-regular languages, namely the context-free grammar and the extended automaton over groups. More precisely, we recall the main results regarding the degree on non-regularity of a context-free grammar as well as the degree of extension of finite automata over groups. Finally, we consider a similar measure for the finite automata with translucent letters and present some preliminary results. This measure could be considered for many mechanisms that extend a less expressive one.Comment: In Proceedings AFL 2023, arXiv:2309.0112

Extended finite automata over groups

We consider a simple and natural extension of a finite automaton, namely an element of a given group is associated with each configuration. An input string is accepted if and only if the neutral element of the group is associated to a final configuration reached by the automaton. The accepting power is smaller when abelian groups are considered, in comparison with the non-abelian groups. We prove that this is due to the commutativity. Each language accepted by a finite automaton over an abelian group is actually a unordered vector language. We get a new characterization of the context-free languages as soon as the considered group is the binary free group. The result cannot be carried out in the deterministic case. Some remarks about other groups are also presented

Security-Policy Analysis with eXtended Unix Tools

During our fieldwork with real-world organizations---including those in Public Key Infrastructure (PKI), network configuration management, and the electrical power grid---we repeatedly noticed that security policies and related security artifacts are hard to manage. We observed three core limitations of security policy analysis that contribute to this difficulty. First, there is a gap between policy languages and the tools available to practitioners. Traditional Unix text-processing tools are useful, but practitioners cannot use these tools to operate on the high-level languages in which security policies are expressed and implemented. Second, practitioners cannot process policy at multiple levels of abstraction but they need this capability because many high-level languages encode hierarchical object models. Finally, practitioners need feedback to be able to measure how security policies and policy artifacts that implement those policies change over time. We designed and built our eXtended Unix tools (XUTools) to address these limitations of security policy analysis. First, our XUTools operate upon context-free languages so that they can operate upon the hierarchical object models of high-level policy languages. Second, our XUTools operate on parse trees so that practitioners can process and analyze texts at multiple levels of abstraction. Finally, our XUTools enable new computational experiments on multi-versioned structured texts and our tools allow practitioners to measure security policies and how they change over time. Just as programmers use high-level languages to program more efficiently, so can practitioners use these tools to analyze texts relative to a high-level language. Throughout the historical transmission of text, people have identified meaningful substrings of text and categorized them into groups such as sentences, pages, lines, function blocks, and books to name a few. Our research interprets these useful structures as different context-free languages by which we can analyze text. XUTools are already in demand by practitioners in a variety of domains and articles on our research have been featured in various news outlets that include ComputerWorld, CIO Magazine, Communications of the ACM, and Slashdot

Hairdressing in groups: a survey of combings and formal languages

A group is combable if it can be represented by a language of words satisfying a fellow traveller property; an automatic group has a synchronous combing which is a regular language. This article surveys results for combable groups, in particular in the case where the combing is a formal language.Comment: 17 pages. Published copy, also available at http://www.maths.warwick.ac.uk/gt/GTMon1/paper24.abs.htm