Quantum Finite Automata and Logic

Elektroniskā versija nesatur pielikumusAnotācija Atslēgas vārdi – kvantu automāti, loģika, automāti bezgalīgiem vārdiem. Matemātiskās loģikas un klasiskās skaitļošanas saistībai ir bijusi liela nozīme datorzinātnes attīstībā. Tas ir galvenais iemesls, kas raisījis interesi pētīt kvantu skaitļošanas un loģikas saistību. Promocijas darbs aplūko saistību starp galīgiem kvantu automātiem un loģiku. Pamatā pētījumi balstās uz galīgu kvantu automātu un tā dažādiem veidiem (galīgu kvantu automātu ar mērījumu beigās, galīgu kvantu automātu ar mērījumu katrā solī, galīgo "latviešu" kvantu automātu), precīzāk, valodām, ko akceptē dažādie kvantu automātu modeļi, un to saistību ar valodām, ko apraksta dažādie loģikas veidi ( pirmās kārtas loģika, modulārā loģika u.c.). Darbā ir arī aplūkoti galīgi kvantu automāti, kas akceptē bezgalīgus vārdus.Abstract Keywords – quantum automata, logic, automata over infinite words The connection between the classical computation and mathematical logic has had a great impact in the computer science which is the main reason for the interest in the connection between the quantum computation and mathematical logic. The thesis studies a connection between quantum finite state automata and logic. The main research area is a quantum finite state automaton and its different notations (measure-once quantum finite automaton, measure-many quantum finite automaton, and Latvian quantum finite automaton), more precisely, the languages accepted by the various models of the quantum finite state automaton and its connection to languages described by the different kinds of logic. Additionally, a quantum finite state automaton over infinite words is introduced

Turing machines based on unsharp quantum logic

In this paper, we consider Turing machines based on unsharp quantum logic. For a lattice-ordered quantum multiple-valued (MV) algebra E, we introduce E-valued non-deterministic Turing machines (ENTMs) and E-valued deterministic Turing machines (EDTMs). We discuss different E-valued recursively enumerable languages from width-first and depth-first recognition. We find that width-first recognition is equal to or less than depth-first recognition in general. The equivalence requires an underlying E value lattice to degenerate into an MV algebra. We also study variants of ENTMs. ENTMs with a classical initial state and ENTMs with a classical final state have the same power as ENTMs with quantum initial and final states. In particular, the latter can be simulated by ENTMs with classical transitions under a certain condition. Using these findings, we prove that ENTMs are not equivalent to EDTMs and that ENTMs are more powerful than EDTMs. This is a notable difference from the classical Turing machines.Comment: In Proceedings QPL 2011, arXiv:1210.029

Computational universes

Suspicions that the world might be some sort of a machine or algorithm existing ``in the mind'' of some symbolic number cruncher have lingered from antiquity. Although popular at times, the most radical forms of this idea never reached mainstream. Modern developments in physics and computer science have lent support to the thesis, but empirical evidence is needed before it can begin to replace our contemporary world view.Comment: Several corrections of typos and smaller revisions, final versio

Łukasiewicz-Moisil Many-Valued Logic Algebra of Highly-Complex Systems

A novel approach to self-organizing, highly-complex systems (HCS), such as living organisms and artificial intelligent systems (AIs), is presented which is relevant to Cognition, Medical Bioinformatics and Computational Neuroscience. Quantum Automata (QAs) were defined in our previous work as generalized, probabilistic automata with quantum state spaces (Baianu, 1971). Their next-state functions operate through transitions between quantum states defined by the quantum equations of motion in the Schroedinger representation, with both initial and boundary conditions in space-time. Such quantum automata operate with a quantum logic, or Q-logic, significantly different from either Boolean or Łukasiewicz many-valued logic. A new theorem is proposed which states that the category of quantum automata and automata--homomorphisms has both limits and colimits. Therefore, both categories of quantum automata and classical automata (sequential machines) are bicomplete. A second new theorem establishes that the standard automata category is a subcategory of the quantum automata category. The quantum automata category has a faithful representation in the category of Generalized (M,R)--Systems which are open, dynamic biosystem networks with defined biological relations that represent physiological functions of primordial organisms, single cells and higher organisms

Quantum Turing automata

A denotational semantics of quantum Turing machines having a quantum control is defined in the dagger compact closed category of finite dimensional Hilbert spaces. Using the Moore-Penrose generalized inverse, a new additive trace is introduced on the restriction of this category to isometries, which trace is carried over to directed quantum Turing machines as monoidal automata. The Joyal-Street-Verity Int construction is then used to extend this structure to a reversible bidirectional one.Comment: In Proceedings DCM 2012, arXiv:1403.757

Empirical logic of finite automata: microstatements versus macrostatements

We compare the two approaches to the empirical logic of automata. The first, called partition logic (logic of microstatements), refers to experiments on individual automata. The second one, the logic of simulation (logic of macrostatements), deals with ensembles of automata.Comment: late

Finite automata models of quantized systems: conceptual status and outlook

Since Edward Moore, finite automata theory has been inspired by physics, in particular by quantum complementarity. We review automaton complementarity, reversible automata and the connections to generalized urn models. Recent developments in quantum information theory may have appropriate formalizations in the automaton context.Comment: 12 pages, prepared for the Sixth International Conference on Developments in Language Theory, Kyoto, Japan, September 18-21, 200