Weighted automata and multi-valued logics over arbitrary bounded lattices

AbstractWe show that L-weighted automata, L-rational series, and L-valued monadic second order logic have the same expressive power, for any bounded lattice L and for finite and infinite words. We also prove that aperiodicity, star-freeness, and L-valued first-order and LTL-definability coincide. This extends classical results of Kleene, Büchi–Elgot–Trakhtenbrot, and others to arbitrary bounded lattices, without any distributivity assumption that is fundamental in the theory of weighted automata over semirings. In fact, we obtain these results for large classes of strong bimonoids which properly contain all bounded lattices

Parity Automata for Quantitative Linear Time Logics

We initiate a study of automata-based model checking for previously proposed quantitative linear time logics interpreted over coalgebras. Our results include: (i) an automata-theoretic characterisation of the semantics of these logics, based on a notion of extent of a quantitative parity automaton, (ii) a study of the expressive power of Buchi variants of such automata, with implications on the expressiveness of fragments of the logics considered, and (iii) a naive algorithm for computing extents, under additional assumptions on the domain of truth values

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

Model checking ω-regular properties for quantum Markov chains

© Yuan Feng, Ernst Moritz Hahn, Andrea Turrini, and Shenggang Ying. Quantum Markov chains are an extension of classical Markov chains which are labelled with super-operators rather than probabilities. They allow to faithfully represent quantum programs and quantum protocols. In this paper, we investigate model checking !-regular properties, a very general class of properties (including, e.g., LTL properties) of interest, against this model. For classical Markov chains, such properties are usually checked by building the product of the model with a language automaton. Subsequent analysis is then performed on this product. When doing so, one takes into account its graph structure, and for instance performs different analyses per bottom strongly connected component (BSCC). Unfortunately, for quantum Markov chains such an approach does not work directly, because super-operators behave differently from probabilities. To overcome this problem, we transform the product quantum Markov chain into a single super-operator, which induces a decomposition of the state space (the tensor product of classical state space and the quantum one) into a family of BSCC subspaces. Interestingly, we show that this BSCC decomposition provides a solution to the issue of model checking ω-regular properties for quantum Markov chains

A note on the gap between rank and border rank

We study the tensor rank of the tensor corresponding to the algebra of n-variate complex polynomials modulo the dth power of each variable. As a result we find a sequence of tensors with a large gap between rank and border rank, and thus a counterexample to a conjecture of Rhodes. At the same time we obtain a new lower bound on the tensor rank of tensor powers of the generalised W-state tensor. In addition, we exactly determine the tensor rank of the tensor cube of the three-party W-state tensor, thus answering a question of Chen et al.Comment: To appear in Linear Algebra and its Application

Decidability for Sturmian Words

We show that the first-order theory of Sturmian words over Presburger arithmetic is decidable. Using a general adder recognizing addition in Ostrowski numeration systems by Baranwal, Schaeffer and Shallit, we prove that the first-order expansions of Presburger arithmetic by a single Sturmian word are uniformly ?-automatic, and then deduce the decidability of the theory of the class of such structures. Using an implementation of this decision algorithm called Pecan, we automatically reprove classical theorems about Sturmian words in seconds, and are able to obtain new results about antisquares and antipalindromes in characteristic Sturmian words