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

The subject of this monograph is divided into two
parts-regulated and reduced formal models. The first part
introduces and studies self-regulating finite and pushdown
automata. In essence, these automata regulate the use of their
rules by a sequence of rules applied during the previous moves.
A special attention is paid to turns defined as moves during
which a self-regulating automaton starts a new self-regulating
sequence of moves. Based on the number of turns, two infinite
hierarchies of language families resulting from two variants of
these automata are established (see Sections 4.1.1 and 4.1.2).
Section 4.1.1 demonstrates that in case of self-regulating
finite automata these hierarchies coincide with the hierarchies
resulting from parallel right linear and right linear simple
matrix grammars, so the self-regulating finite automata can be
viewed as the automaton counterparts to these grammars.
Finally, both infinite hierarchies are compared. In addition,
Section 4.1.2 discusses some results and open problems
concerning self-regulating pushdown automata. The second part
studies descriptional complexity of partially parallel grammars
(Section 5.1) and grammars regulated by context conditions
(Section 5.2) with respect to the number of nonterminals and a
special type of productions. Specifically, Chapter 5 proves
that every recursively enumerable language is gen erated (i) by
a scattered context grammar with no more than four
non-context-free productions and four nonterminals, (ii) by a
multisequential grammar with no more than two selectors and two
nonterminals, (iii) by a multicontinuous grammar with no more
than two selectors and three nonterminals, (iv) by a
context-conditional grammar of degree (2, 1) with no more than
six conditional productions and seven nonterminals, (v) by a
simple context-conditional grammar of degree (2, 1) with no
more than seven conditional productions and eight nonterminals,
(vi) by a generalized forbidding grammar of degree two and
index six with no more than ten conditional productions and
nine nonterminals, (vii) by a generalized forbidding grammar of
degree two and index four with no more than eleven conditional
productions and ten nonterminals, (viii) by a generalized
forbidding grammar of degree two and index nine with no more
than eight conditional productions and ten nonterminals, (ix)
by a generalized forbidding grammar of degree two and unlimited
index with no more than nine conditional productions and eight
nonterminals, (x) by a semi-conditional grammar of degree (2,
1) with no more than seven conditional productions and eight
nonterminals, and (xi) by a simple semi-conditional grammar of
degree (2, 1) with no more than nine conditional productions
and ten nonterminals. Chapter 2 contains basic definitions and
the notation used during this monograph. Chapter 3 then
summarizes the previous known results related to the studied
formal models; regulated automata and descriptional complexity
of partially parallel grammars and grammars regulated by
context conditions. Chapter 4 studies self-regulating automata,
and Chapter 5 presents the new results concerning descriptional
complexity of partially parallel grammars and grammars
regulated by context conditions.

Formal Models: Regulation and Reduction

We continue the investigation of union-free regular languages that are described by regular expressions without the union operation. We also define deterministic union-free languages as languages recognized by one-cycle-free-path deterministic finite automata, and show that they are properly included in the class of union-free languages. We prove that (deterministic) union-freeness of languages does not accelerate regular operations, except for the reversal in the nondeterministic case.

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Complexity in union-free regular languages

We discuss a supervisory control problem of modular discrete-event systems that admits a distributed computation of supervisors. We provide a characterization and an algorithm to compute the supervisors. If the specification cannot be reached, we make use of a relaxation of coordination control to compute a sublanguage of the specification for which the supervisors can be computed in a distributed way.

On a distributed computation of supervisors in modular supervisory control

This paper studies cooperating distributed grammar systems
working in the terminal derivation mode where the components
are variants of permitting grammars. It proves that although
the family of permitting languages is strictly included in the
family of random context languages, the families of random
context languages and languages generated by permitting
cooperating distributed grammar systems in the above mentioned
derivation mode coincide. Moreover, if the components are
so-called left-permitting grammars, then cooperating
distributed grammar systems in the terminal mode characterize
the class of context-sensitive languages, or if erasing rules
are allowed, the class of recursively enumerable languages.
Descriptional complexity results are also presented. It is
shown that the number of permitting components can be bounded,
in the case of left-permitting components with erasing rules
even together with the number of nonterminals.

Cooperating Distributed Grammar Systems with PermittingGrammars as Components

Opacity is an information flow property characterizing whether a system reveals its secret to a passive observer. Several notions of opacity have been introduced in the literature. We study the notions of language-based opacity, current-state opacity, initial-state opacity, initial-and-final-state opacity, K-step opacity, and infinite-step opacity. Comparing the notions is a natural question that has been investigated and summarized by Wu and Lafortune, who provided transformations among current-state opacity, initial-and-final-state opacity, and language-based opacity, and, for prefix-closed languages, also between language-based opacity and initial-state opacity. We extend these results by showing that all the discussed notions of opacity are transformable to each other. Besides a deeper insight into the differences among the notions, the transformations have applications in complexity results. In particular, the transformations are computable in polynomial time and preserve the number of observable events and determinism, and hence the computational complexities of the verification of the notions coincide. We provide a complete and improved complexity picture of the verification of the discussed notions of opacity, and improve the algorithmic complexity of deciding language-based opacity, infinite-step opacity, and K-step opacity.

Tomáš Masopust

Papers

Formal Models: Regulation and Reduction

Complexity in union-free regular languages

On a distributed computation of supervisors in modular supervisory control

Cooperating Distributed Grammar Systems with PermittingGrammars as Components

Comparing the notions of opacity for discrete-event systems