Lecture Notes in Economics and Mathematical Systems

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Computers And Intractability A Guide To The Theory Of Np Completeness

This book studies language(s) and linguistic theories from a mathematical point of view. Starting with ideas already contained in Montague's work, it develops the mathematical foundations of present day linguistics. It equips the reader with all the background necessary to understand and evaluate theories as diverse as Montague Grammar, Categorial Grammar, HPSG and GB. The mathematical tools are mainly from universal algebra and logic, but no particular knowledge is presupposed beyond a certain mathematical sophistication that is in any case needed in order to fruitfully work within these theories. The presentation focuses on abstract mathematical structures and their computational properties, but plenty of examples from different natural languages are provided to illustrate the main concepts and results. In contrast to books devoted to so-called formal language theory, languages are seen here as semiotic systems, that is, as systems of signs. A language sign correlates form with meaning. Using the principle of compositionality it is possible to gain substantial insight into the interaction between form and meaning in natural languages.

Mathematics of Language

Verification of initial-state opacity in security applications of discrete event systems

Control and coordination in hierarchical systems

Critical observability is a property of cyber-physical systems to detect whether the current state belongs to a set of critical states. In safety-critical applications, critical states model operations that may be unsafe or of a particular interest. De Santis et al. introduced critical observability for linear switching systems, and Pola et al. adapted it for discrete-event systems, focusing on algorithmic complexity. We study the computational complexity of deciding critical observability for systems modeled as (networks of) finite-state automata and Petri nets. We show that deciding critical observability is (i) NL-complete for finite automata, that is, it is efficiently verifiable on parallel computers, (ii) PSPACE-complete for networks of finite automata, that is, it is very unlikely solvable in polynomial time, and (iii) undecidable for labeled Petri nets, but becoming decidable if the set of critical states (markings) is finite or co-finite, in which case the problem is as hard as the non-reachability problem for Petri nets.

Critical Observability for Automata and Petri Nets

This chapter presents a brief introduction to the theory of automata, formal languages, decidability, and complexity.

Languages, Decidability, and Complexity

The tight bound on the state complexity of the reverse of \({\mathcal R}\)-trivial and \({\mathcal J}\)-trivial regular languages of the state complexity n is 2 n − 1. The witness is ternary for \({\mathcal R}\)-trivial regular languages and (n − 1)-ary for \({\mathcal J}\)-trivial regular languages. In this paper, we prove that the bound can be met neither by a binary \({\mathcal R}\)-trivial regular language nor by a \({\mathcal J}\)-trivial regular language over an (n − 2)-element alphabet. We provide a characterization of tight bounds for \({\mathcal R}\)-trivial regular languages depending on the state complexity of the language and the size of its alphabet. We show the tight bound for \({\mathcal J}\)-trivial regular languages over an (n − 2)-element alphabet and a few tight bounds for binary \({\mathcal J}\)-trivial regular languages. The case of \({\mathcal J}\)-trivial regular languages over an (n − k)-element alphabet, for 2 ≤ k ≤ n − 3, is open.

On the State Complexity of the Reverse of \({\mathcal R}\)- and \({\mathcal J}\)-Trivial Regular Languages

In this paper, we place some left restrictions on derivations in CD grammar systems with phrase- structure grammars, controlled by the regular languages. The rst restriction requires that every production is always applied within the rst k nonterminals in every sentential form, for some k 1. The second restriction says how many blocks of non-terminals can be in each sentential form. The third restriction extends the second restriction and says how many blocks of non-terminals with limited length can be in each sentential form. We demonstrate that under these restrictions, the grammar systems generate dierent families of languages. Indeed, under the rst restriction, these systems generate only context-free languages. Under the second restriction, even one-component systems characterize the entire family of recursively enumerable languages. In the end, the family of languages generated by grammar systems under the third restriction is equal to the family of languages generated by programmed grammars with context-free rules without -rules of nite index.

Some Power-Decreasing Derivation Restrictions in Grammar Systems

A tower is a sequence of words alternating between two languages in such a way that every word is a subsequence of the following word. The height of the tower is the number of words in the sequence. If there is no infinite tower (a tower of infinite height), then the height of all towers between the languages is bounded. We study upper and lower bounds on the height of maximal finite towers between two regular languages with respect to the size of the NFA (respectively the DFA) representation. Our motivation to study the bounds on maximal finite towers comes from a method to compute a piecewise testable separator of two regular languages. We show that the upper bound is polynomial in the number of states and exponential in the size of the alphabet, and that it is asymptotically tight if the size of the alphabet is fixed. If the alphabet may grow, then, using an alphabet of size approximately the number of states of the automata, the lower bound on the height of towers is exponential with respect to that number. In this case, there is a gap between the lower and upper bound, and the asymptotically optimal bound remains an open problem. Since, in many cases, the constructed towers are sequences of prefixes, we also study towers of prefixes.

https://hal.inria.fr/hal-02269576/document

Tomáš Masopust

Papers

Critical Observability for Automata and Petri Nets

Languages, Decidability, and Complexity

On the State Complexity of the Reverse of \({\mathcal R}\)- and \({\mathcal J}\)-Trivial Regular Languages

Some Power-Decreasing Derivation Restrictions in Grammar Systems

On the Height of Towers of Subsequences and Prefixes