On the logical definability of certain graph and poset languages

We show that it is equivalent, for certain sets of finite graphs, to be definable in CMS (counting monadic second-order logic, a natural extension of monadic second-order logic), and to be recognizable in an algebraic framework induced by the notion of modular decomposition of a finite graph. More precisely, we consider the set $F\_\infty$ of composition operations on graphs which occur in the modular decomposition of finite graphs. If $F$ is a subset of $F\_{\infty}$, we say that a graph is an \calF-graph if it can be decomposed using only operations in $F$. A set of $F$-graphs is recognizable if it is a union of classes in a finite-index equivalence relation which is preserved by the operations in $F$. We show that if $F$ is finite and its elements enjoy only a limited amount of commutativity -- a property which we call weak rigidity, then recognizability is equivalent to CMS-definability. This requirement is weak enough to be satisfied whenever all $F$-graphs are posets, that is, transitive dags. In particular, our result generalizes Kuske's recent result on series-parallel poset languages

Modular functionals and perturbations of Nakano spaces

We settle several questions regarding the model theory of Nakano spaces left open by the PhD thesis of Pedro Poitevin \cite{Poitevin:PhD}. We start by studying isometric Banach lattice embeddings of Nakano spaces, showing that in dimension two and above such embeddings have a particularly simple and rigid form. We use this to show show that in the Banach lattice language the modular functional is definable and that complete theories of atomless Nakano spaces are model complete. We also show that up to arbitrarily small perturbations of the exponent Nakano spaces are $\aleph_0$-categorical and $\aleph_0$-stable. In particular they are stable

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

Generic Expression Hardness Results for Primitive Positive Formula Comparison

We study the expression complexity of two basic problems involving the comparison of primitive positive formulas: equivalence and containment. In particular, we study the complexity of these problems relative to finite relational structures. We present two generic hardness results for the studied problems, and discuss evidence that they are optimal and yield, for each of the problems, a complexity trichotomy

Logic and Automata

Mathematical logic and automata theory are two scientific disciplines with a fundamentally close relationship. The authors of Logic and Automata take the occasion of the sixtieth birthday of Wolfgang Thomas to present a tour d'horizon of automata theory and logic. The twenty papers in this volume cover many different facets of logic and automata theory, emphasizing the connections to other disciplines such as games, algorithms, and semigroup theory, as well as discussing current challenges in the field

Context-aware Trace Contracts

The behavior of concurrent, asynchronous procedures depends in general on the call context, because of the global protocol that governs scheduling. This context cannot be specified with the state-based Hoare-style contracts common in deductive verification. Recent work generalized state-based to trace contracts, which permit to specify the internal behavior of a procedure, such as calls or state changes, but not its call context. In this article we propose a program logic of context-aware trace contracts for specifying global behavior of asynchronous programs. We also provide a sound proof system that addresses two challenges: To observe the program state not merely at the end points of a procedure, we introduce the novel concept of an observation quantifier. And to combat combinatorial explosion of possible call sequences of procedures, we transfer Liskov's principle of behavioral subtyping to the analysis of asynchronous procedures