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Content available remote On the Borel Complexity of MSO Definable Sets of Branches
An infinite binaryword can be identified with a branch in the full binary tree. We consider sets of branches definable in monadic second-order logic over the tree, where we allow some extra monadic predicates on the nodes. We show that this class equals to the Boolean combinations of sets in the Borel class Σ over the Cantor discontinuum. Note that the last coincides with the Borel complexity of ω-regular languages.
Content available remote Expressing Cardinality Quantifiers in Monadic Second-Order Logic over Trees
We study an extension of monadic second-order logic of order with the uncountability quantifier "there exist uncountably many sets". We prove that, over the class of finitely branching trees, this extension is equally expressive to plain monadic second-order logic of order. Additionally we find that the continuum hypothesis holds for classes of sets definable in monadic second-order logic over finitely branching trees, which is notable for not all of these classes are analytic. Our approach is based on Shelah’s composition method and uses basic results from descriptive set theory. The elimination result is constructive, yielding a decision procedure for the extended logic.
As it is known for increasing of properties (YTS ≥ 380 MPa) of cast steels it is effective to increase content of substitutional alloying elements, (Si, Mn, Cr, Ni). However it leads to rising in price of Steel ton. Increasing of Si and Mn content only is limited by decreasing of ductility and weld ability. As a rule Silicon content at these steels is not higher than 0.4-0.6% and Si:Mn ratio is not higher than 1:2. Now for grain refinement uses inoculation of steel by nitrogen and elements with high chemical affinity to nitrogen. Mostly vanadium is used, however niobium sometime is used. Disadvantages of this are high cost of alloying elements and low thermodynamic stability of vanadium and niobium nitrides. Particles of V(C, N) and Nb(C, N) dissolve during heating for heat treatment or during welding. It leads to decreasing of grain refinement effect. Adaptation of this microalloying strategy for casts producing for freight railway cars let estimate possibility of application these casts in a new generation freight railway cars.
Celem przedstawionych badań jest analiza możliwości zastosowania utwardzania węglikowo-azotkowego staliwa zawierającego 0,2% C w celu zapewnienia możliwości zastosowania tego materiału do celów wykonania odpowiedzialnych odlewów kolejowych. Metoda wykorzystania mikrododatków stopowych, znana w produkcji wyrobów walcowanych, została zaadaptowana do produkcji odlewów kolejowych części wagonów towarowych nowej generacji. Wykonano serię wytopów doświadczalnych, na podstawie których otrzymano zakresy stężeń pierwiastków bazowych i dodatków modyfikujących.
Content available remote Composition Theorem for Generalized Sum
Composition theorems are tools which reduce sentences about some compound structure to sentences about its parts. A seminal example of such a theorem is the Feferman-Vaught Theorem [3] which reduces the first-order theory of generalized products to the first order theory of its factors and the monadic second-order theory of index structure. Shelah [23] used the composition theorem for linear orders as one of the main tools for obtaining very strong decidability results for the monadic second-order theory of linear orders. The main technical contribution of our paper is (1) a definition of a generalized sum of structures and (2) a composition theorem for first-order logic over the generalized sum. One of our objectives is to emphasize the work-out of the composition method.
Content available remote Synchronous Circuits over Continuous Time: Feedback Reliability and Completeness
To what mathematical models do digital computer circuits belong? In particular: (i) (Feedback reliability.) Which cyclic circuits should be accepted? In other words, under which conditions is causally faithful the propagation of signals along closed cycles of the circuit? (ii) (Comparative power and completeness.) What are the appropriate primitives upon which circuits may be (or should be) assembled? There are well-known answers to these questions for circuits operating in discrete time, and they point on the exclusive role of the unit-delay primitive. For example: (i) If every cycle in the circuit N passes through a delay, then N is feedback reliable. (ii) Every finite-memory operator F is implementable in a circuit over unit-delay and pointwise boolean gates. In what form, if any, can such phenomena and results be extended to circuits operating in continuous time? This is the main problem considered (and, hopefully, solved to some extent) in this paper. In order to tackle the problems one needs more insight into specific properties of continuous time signals and operators that are not visible at discrete time.
Content available remote Logics for Real Time: Decidability and Complexity
Over the last fifteen years formalisms for reasoning about metric properties of computations were suggested and discussed. First as extensions of temporal logic, ignoring the framework of classical predicate logic, and then, with the authors' work, within the framework of monadic logic of order. Here we survey our work on metric logic comparing it to the previous work in the field. We define a quantitative temporal logic that is based on a simple modality within the framework of monadic predicate Logic. Its canonical model is the real line (and not an w-sequence of some type). It can be interpreted either by behaviors with finite variability or by unrestricted behaviors. For finite variability models it is as expressive as any logic suggested in the literature. For unrestricted behaviors our treatment is new. In both cases we prove decidability and complexity bounds using general theorems from logic (and not from automata theory).
Content available remote Succinctness gap between monadic logic and duration calculus
In [8, 11] the expressive completeness of the Propositional fragment of Duration Calculus relative to monadic first-order logic of order was established. In this paper we show that there is at least an exponential blow-up in every meaning preserving translation from monadic logic to PDC. Hence, there exists an exponential gap between the succinctness of monadic logic and that of duration calculus.
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