Computational problems in matrix semigroups

This thesis deals with computational problems that are defined on matrix semigroups, which playa pivotal role in Mathematics and Computer Science in such areas as control theory, dynamical systems, hybrid systems, computational geometry and both classical and quantum computing to name but a few. Properties that researchers wish to study in such fields often turn out to be questions regarding the structure of the underlying matrix semigroup and thus the study of computational problems on such algebraic structures in linear algebra is of intrinsic importance. Many natural problems concerning matrix semigroups can be proven to be intractable or indeed even unsolvable in a formal mathematical sense. Thus, related problems concerning physical, chemical and biological systems modelled by such structures have properties which are not amenable to algorithmic procedures to determine their values. With such recalcitrant problems we often find that there exists a tight border between decidability and undecidability dependent upon particular parameters of the system. Examining this border allows us to determine which properties we can hope to derive algorithmically and those problems which will forever be out of our reach, regardless of any future advances in computational speed. There are a plethora of open problems in the field related to dynamical systems, control theory and number theory which we detail throughout this thesis. We examine undecidability in matrix semigroups for a variety of different problems such as membership and vector reachability problems, semigroup intersection emptiness testing and freeness, all of which are well known from the literature. We also formulate and survey decidability questions for several new problems such as vector ambiguity, recurrent matrix problems, the presence of any diagonal matrix and quaternion matrix semigroups, all of which we feel give a broader perspective to the underlying structure of matrix semigroups

07441 Abstracts Collection -- Algorithmic-Logical Theory of Infinite Structures

From 28.10. to 02.11.2007, the Dagstuhl Seminar 07441 ``Algorithmic-Logical Theory of Infinite Structures\u27\u27 was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available

Complexity classifications for nonmonotonic reasoning and enumeration

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Representation and Reality by Language: How to make a home quantum computer?

A set theory model of reality, representation and language based on the relation of completeness and incompleteness is explored. The problem of completeness of mathematics is linked to its counterpart in quantum mechanics. That model includes two Peano arithmetics or Turing machines independent of each other. The complex Hilbert space underlying quantum mechanics as the base of its mathematical formalism is interpreted as a generalization of Peano arithmetic: It is a doubled infinite set of doubled Peano arithmetics having a remarkable symmetry to the axiom of choice. The quantity of information is interpreted as the number of elementary choices (bits). Quantum information is seen as the generalization of information to infinite sets or series. The equivalence of that model to a quantum computer is demonstrated. The condition for the Turing machines to be independent of each other is reduced to the state of Nash equilibrium between them. Two relative models of language as game in the sense of game theory and as ontology of metaphors (all mappings, which are not one-to-one, i.e. not representations of reality in a formal sense) are deduced

A Mathematical Model of Quantum Computer by Both Arithmetic and Set Theory

A practical viewpoint links reality, representation, and language to calculation by the concept of Turing (1936) machine being the mathematical model of our computers. After the Gödel incompleteness theorems (1931) or the insolvability of the so-called halting problem (Turing 1936; Church 1936) as to a classical machine of Turing, one of the simplest hypotheses is completeness to be suggested for two ones. That is consistent with the provability of completeness by means of two independent Peano arithmetics discussed in Section I. Many modifications of Turing machines cum quantum ones are researched in Section II for the Halting problem and completeness, and the model of two independent Turing machines seems to generalize them. Then, that pair can be postulated as the formal definition of reality therefore being complete unlike any of them standalone, remaining incomplete without its complementary counterpart. Representation is formal defined as a one-to-one mapping between the two Turing machines, and the set of all those mappings can be considered as “language” therefore including metaphors as mappings different than representation. Section III investigates that formal relation of “reality”, “representation”, and “language” modeled by (at least two) Turing machines. The independence of (two) Turing machines is interpreted by means of game theory and especially of the Nash equilibrium in Section IV. Choice and information as the quantity of choices are involved. That approach seems to be equivalent to that based on set theory and the concept of actual infinity in mathematics and allowing of practical implementations

Mind and information processing: some considerations

The paper is devoted to the problem of relations between abilities of human mind and automatical information processing. Here we can use the notion of artificial inteligence as the abbrevation of information processing realized by mechanical devices. For the purpose of an analysis of the mentioned problem some classical notions and arguments from computer science are presented

Developing Email Interview Practices in Qualitative Research

This article describes using email as a kind of interview. In a sociological study of professional career transition into law, on several occasions in that study, interview participants suggested using emails rather than face-to-face interviews. This \'irregularity\' set off reflection whether email interviews counted as \'proper\' interviews. Discussing examples of email interviews clarifies differences from other uses of email in research, and assists exploration of advantages and disadvantages of email interviews as a qualitative research method. A preliminary framework is suggested for evaluation the suitability of email interviews. Present-day limitations point to continuing development in this area of social research. Current indications are that emergent media technologies such as email interviews, like other new media innovations, do not diminish older forms, but rather enrich the array of investigatory tools available for social research today.Email Interview; Email Research; Interview Methodology; Mixed Method

Chasing Bits Across Borders

As computer crime becomes more widespread, countries increasingly confront difficulties in securing evidence stored in electronic form outside of their borders. These difficulties have prompted two related responses. Some states have asserted a broad power to conduct remote cross-border searches - that is, to use computers within their territory to access and examine data physically stored outside of their territory. Other states have pressed for recognition of a remote cross-border search power in international fora, arguing that such a power is an essential weapon in efforts to combat computer crime. This Article explores these state responses and develops a framework for evaluating the legality of cross-border searches, both as a matter of international law and as a matter of U.S. law. The Article argues that remote cross-border searches are problematic as a matter of international law, and that U.S. adoption of bilateral or multilateral agreements authorizing remote-cross border searches on foreign law standards lower than those of the Fourth Amendment would be problematic as a matter of U.S. constitutional law. The Article also situates the remote cross-border search issue within the context of the larger theoretical debate over the power of geographically based sovereigns to exercise jurisdiction over internet activities. That debate ordinarily focuses on a sovereign\u27s jurisdiction to prescribe legal rules governing internet conduct; by broadening the inquiry to focus on a sovereign\u27s enforcement jurisdiction, a study of remote cross-border searches highlights certain normative bases for refining our understanding of how principles of territorial sovereignty apply in the internet context

Chasing Bits Across Borders

As computer crime becomes more widespread, countries increasingly confront difficulties in securing evidence stored in electronic form outside of their borders. These difficulties have prompted two related responses. Some states have asserted a broad power to conduct remote cross-border searches - that is, to use computers within their territory to access and examine data physically stored outside of their territory. Other states have pressed for recognition of a remote cross-border search power in international fora, arguing that such a power is an essential weapon in efforts to combat computer crime. This Article explores these state responses and develops a framework for evaluating the legality of cross-border searches, both as a matter of international law and as a matter of U.S. law. The Article argues that remote cross-border searches are problematic as a matter of international law, and that U.S. adoption of bilateral or multilateral agreements authorizing remote-cross border searches on foreign law standards lower than those of the Fourth Amendment would be problematic as a matter of U.S. constitutional law. The Article also situates the remote cross-border search issue within the context of the larger theoretical debate over the power of geographically based sovereigns to exercise jurisdiction over internet activities. That debate ordinarily focuses on a sovereign\u27s jurisdiction to prescribe legal rules governing internet conduct; by broadening the inquiry to focus on a sovereign\u27s enforcement jurisdiction, a study of remote cross-border searches highlights certain normative bases for refining our understanding of how principles of territorial sovereignty apply in the internet context