The article continues our study of the genus of a regular language L, defined as the minimal genus among all genera of all finite deterministic automata recognizing L. Here we define and study two closely related tools on a directed graph: directed emulators and automatic relations. A directed emulator morphism essentially encapsulates at the graph-theoretic level an epimorphism onto the minimal deterministic automaton. An automatic relation is the graph-theoretic version of the Myhill-Nerode relation. We show that an automatic relation determines a directed emulator morphism and respectively, a directed emulator morphism determines an automatic relation up to isomorphism. Consider the set S of all directed emulators of the underlying directed graph of the minimal deterministic automaton for L. We prove that the genus of L is minG∈Sg(G). We also consider the more restrictive notion of directed cover and prove that the genus of L is reached in the class of directed covers of the underlying directed graph of the minimal deterministic automaton for L. This stands in sharp contrast to undirected emulators and undirected covers which we also consider. Finally we prove that if the problem of determining the minimal genus of a directed emulator of a directed graph has a solution then the problem of determining the minimal genus of an undirected emulator of an undirected graph has a solution.

Abstract This article continues the study of the genus of regular languages that the authors introduced in a 2013 paper (published in 2018). In order to understand further the genus g(L) of a regular language L, we introduce the genus size of |L|gen to be the minimal size of all finite deterministic automata of genus g(L) computing L.We show that the minimal finite deterministic automaton of a regular language can be arbitrarily far away from a finite deterministic automaton realizing the minimal genus and computing the same language, in terms of both the difference of genera and the difference in size. In particular, we show that the genus size |L|gen can grow at least exponentially in size |L|. We conjecture, however, the genus of every regular language to be computable. This conjecture implies in particular that the planarity of a regular language is decidable, a question asked in 1976 by R. V. Book and A. K. Chandra. We prove here the conjecture for a fairly generic class of regular languages having no short cycles. The methods developed for the proof are used to produce new genus-based hierarchies of regular languages and in particular, we show a new family of regular languages on a two-letter alphabet having arbitrary high genus.

The paper defines and studies the genus of finite state deterministic automata (FSA) and regular languages. Indeed, an FSA can be seen as a graph for which the notion of genus arises. At the same time, an FSA has a semantics via its underlying language. It is then natural to make a connection between the languages and the notion of genus. After we introduce and justify the the notion of the genus for regular languages, the following questions are addressed. First, depending on the size of the alphabet, we provide upper and lower bounds on the genus of regular languages: we show that under a relatively generic condition on the alphabet and the geometry of the automata, the genus grows at least linearly in terms of the size of the automata. Second, we show that the topological cost of the powerset determinization procedure is exponential. Third, we prove that the notion of minimization is orthogonal to the notion of genus. Fourth, we build regular languages of arbitrary large genus: the notion of genus defines a proper hierarchy of regular languages.

A very large amount of work in Natural Language Processing (NLP) use tree structure as the first class citizen mathematical structures to represent linguistic structures, such as parsed sentences or feature structures. However, some linguistic phenomena do not cope properly with trees; for instance, in the sentence ‘Max decides to leave,’ ‘Max’ is the subject of the both predicates ‘to_decide’ and ‘to_leave’. Tree-based linguistic formalisms generally use some encoding to manage sentences like the previous example. In former papers (Bonfante et al. 2011; Guillaume and Perrier 2012), we discussed the interest to use graphs rather than trees to deal with linguistic structures, and we have shown how Graph Rewriting could be used for their processing, for instance in the transformation of the sentence syntax into its semantics. Our experiments have shown that Graph Rewriting applications to NLP do not require the full computational power of the general Graph Rewriting setting. The most important observation is that all graph vertices in the final structures are in some sense ‘predictable’ from the input data, and so we can consider the framework of Non-size increasing Graph Rewriting. In our previous papers, we have formally described the Graph Rewriting calculus we used and our purpose here is to study the theoretical aspect of termination with respect to this calculus. Given that termination is undecidable in general, we define termination criterions based on weight, we prove the termination of weighted rewriting systems, and we give complexity bounds on derivation lengths for these rewriting systems.

Fighting malware involves analyzing large numbers of suspicious binary files. In this context, disassembly is a crucial task in malware analysis and reverse engineering. It involves the recovery of assembly instructions from binary machine code. Correct disassembly of binaries is necessary to produce a higher level representation of the code and thus allow the analysis to develop high-level understanding of its behavior and purpose. Nonetheless, it can be problematic in the case of malicious code, as malware writers often employ techniques to thwart correct disassembly by standard tools. In this paper, we focus on the disassembly of x86 self-modifying binaries with overlapping instructions. Current state-of-the-art disassemblers fail to interpret these two common forms of obfuscation, causing an incorrect disassembly of large parts of the input. We introduce a novel disassembly method, called concatic disassembly, that combines CONCrete path execution with stATIC disassembly. We have developed a standalone disassembler called CoDisasm that implements this approach.

Formal specifications and reasoning techniques in software modelling are needed to ensure the correctness of the system at the design phase. Event-B is a formal method with support tools that allows the stepwise development of reactive systems. Such systems include multi-agent systems as a subclass. In this paper, we propose an approach to specify capabilities of a number of software agents. We then verify whether these capabilities help the agents to accomplish a certain task using a supported tool for Event-B. We use the binary numeral system as a case study to illustrate our approach.

We study the effect of polynomial interpretation termination proofs of deterministic (resp. non-deterministic) algorithms defined by con uent (resp. non-con uent) rewrite systems over data structures which include strings, lists and trees, and we classify them according to the interpretations of the constructors. This leads to the definition of six function classes which turn out to be exactly the deterministic (resp. non-deterministic) polynomial time, linear exponential time and linear doubly exponential time computable functions when the class is based on con uent (resp. non-con uent) rewrite systems. We also obtain a characterisation of the linear space computable functions. Finally, we demonstrate that functions with exponential interpretation termination proofs are super-elementary.