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

We continue the investigation of pushdown automata which are allowed to make a non-deterministic decision if and only if their pushdown content forms a string belonging to a given control language. We prove that if the control language is linear and non-regular, then the power of pushdown automata regulated in this way is increased to the power of Turing machines. From a practical point of view, however, it is inefficient to check the form of the pushdown content in each computational step. Therefore, we prove that only two checks of the pushdown content are of interest for these machines to be computationally complete. Based on this observation, we introduce and discuss a new model of regulated pushdown automata.

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Regulated Nondeterminism in Pushdown Automata: The Non-Regular Case

This paper discusses the cooperation and its effect on the
generative power of rewriting systems with some kind of simple
context-dependency checking represented by the mechanism of
random context grammars. Specifically, it discusses the
cooperation in terms of cooperating distributed grammar systems
with respect to all basic types of cooperation protocols, and
proves that although the cooperation is powerful enough to
increase the generative power of both permitting and forbidding
random context grammars, it has no effect on the generative
power of random context grammars. It also discusses two
possible definitions of the relation of the direct derivation
step used in the literature.

A Note on the Cooperation in Rewriting Systems with Context-Dependency Checking.

In this paper, we discuss pushdown automata which can make a
nondeterministic decision only if the pushdown content forms a
string that belongs to a given control language. It proves that
if the control language is linear and non-regular, then the
computational power of pushdown automata regulated in this way
is increased to the power of Turing machines. Naturally,
checking the pushdown content in each computational step is not
practically efficient. Therefore, we also prove that two checks
of the form of the pushdown content during any computation are
sufficient for these automata to be computationally complete.
Finally, some descriptional complexity results are discussed.

Regulated Nondeterminism in PDAs: The Non-Regular Case

The infimal prefix-closed, controllable, and observable superlanguage plays an essential role in the relationship between controllability, observability, and co-observability—the central notions of supervisory control theory. Existing algorithms for its computation are exponential and it is not known whether a polynomial algorithm exists. We answer the question by studying the state complexity of this language. State complexity of a language is the number of states of its minimal deterministic finite automaton (DFA). For a language with state complexity $n$ , we show that the upper bound state complexity on the infimal prefix-closed and observable superlanguage is $2^n + 1$ and that this bound is asymptotically tight. Hence, there is no algorithm computing a DFA of the infimal prefix-closed and observable superlanguage in polynomial time. Our construction shows that such a DFA can be computed in time $O(2^n)$ . The construction involves nondeterministic finite automata (NFAs) and a computation of the supremal prefix-closed sublanguage. We study the computation of supremal prefix-closed sublanguages and show that there is no polynomial-time algorithm computing an NFA of the supremal prefix-closed sublanguage of a language given as an NFA even if the language is unary.

Complexity of Infimal Observable Superlanguages

This paper introduces and discusses pure multi-pushdown automata that remove symbols from their pushdowns only by performing complete pushdown pops. This. means that during a pop operation, the entire pushdown is compared with a prefix of the input, and if they match, the whole contents of the pushdown is erased and the input is advanced by the prefix. The paper proves that these automata define an infinite hierarchy of language families identical with the infinite hierarchy of language families resulting from right linear simple matrix grammars. In addition, this paper discusses some other extensions of these automata with respect to operations they can perform with their pushdowns. More specifically, it discusses pure multi-pushdown automata that perform complete pushdown pops that are allowed to join two pushdowns and/or create a new pushdown.

Tomáš Masopust

Papers

Regulated Nondeterminism in Pushdown Automata: The Non-Regular Case

A Note on the Cooperation in Rewriting Systems with Context-Dependency Checking.

Regulated Nondeterminism in PDAs: The Non-Regular Case

Complexity of Infimal Observable Superlanguages

On pure multi-pushdown automata that perform complete pushdown pops