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A family A P(!) is called countably splitting if for every countable F [!] , some element of A splits every member of F . We de ne a notion of a splitting tree, by means of which we prove that every analytic countably splitting family contains a closed countably splitting family. An application of this notion solves a problem of Blass.

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Contents Introduction 2 1 !-Automata 7 1.1 Notations and Basic Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2 Transition Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3 Acceptance Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.4 Regular Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.5 Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.6 Duality and Complementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2 Modes of Transition Functions 25 2.1 Safra's Determinization Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.2 Optimality of Safra's Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.3 The Breakpoint Construction . . . . . . . . . . . . . . .

Abstract. We develop a theory of labellings for infinite trees, define the notion of a combinatorial labelling, and show that ∆ 0 2 is the largest boldface pointclass in which every set admits a combinatorial labelling. 1

We generalize the operators of classical linear time temporal logic to partial orders, such as the ones used in domain theory. This relates denotaional semantics and temporal logic. We put this into the general perspective of modal logic. 2 and 3 are viewed as standard modalities, which gives "half " of the standard axioms of LTL. Moroever, we show that the next-time operator fl can be defined as a combination of two modalities. We show the role of the standard LTL axioms in narrowing the underlying partial orders to linear ones generated by the immediatesuccessor relation. We distinguish between modal and temporal validity of formulas and investigate their relation. 1 Introduction In [11] a stream has been identified with the set of its finite prefixes. Based on this, we have used a special way of characterising sets of streams through sets of relevant finite "snapshots". Given a set P ` A where A is a set of atomic actions, states or data, we define str P def = f( v Q) : Q ` ...

The Linguistic Geometry (LG) approach to discrete systems was introduced by B. Stilman in early 80s. It employed competing/cooperating agents for modeling and controlling of discrete systems. The approach was applied to a variety of problems with huge state spaces including control of aircraft, battlefield robots, and chess. One of the key innovations of LG is the use of almost winning strategies, rather than truly winning strategies for the participating agents. There are many cases where the winning strategies have so high time complexity that they are not computable in practice, whereas the almost winning strategies can be applied and they beat the opposing agent almost guaranteed. Independently of LG the idea of competing/ cooperating agents was employed in the late 80s by A. Nerode, A. Yakhnis, and V. Yakhnis (NYY) within their approach to modeling concurrent systems and, more recently, within the “Strategy Approach to Hybrid Systems ” developed for continuous systems by A. Nerode, W. Kohn, A. Yakhnis, and others.

We provide some mathematical properties of behaviours of systems, where the individual elements of a behaviour are modeled by ideals of a suitable partial order. It is well-known that the associated ideal completion provides a simple way of constructing algebraic cpos. An ideal can be viewed as a set of consistent finite or compact approximations of an object which itself may even be infinite. A special case is the domain of streams where the finite approximations are the finite prefixes of a stream. We introduce a special way of characterising behaviours through sets of relevant approximations. This is a generalisation of the technique used earlier for the case of streams. Given a set P ` M of a partial order (M;), we define ide P: = fQ: Q ` P directedg; where Q

: Finite deterministic 1-acceptor accepting both finite and infinite words over a finite alphabet is introduced. It is shown that 1-regular languages can be defined as sets of 1-words accepted by an 1-acceptor. A limit operator on regular languages is used to define a special class of 1-regular languages. In 1-acceptors accepting these languages incidence relations between the sets used for acceptance are determined. 1. Introduction The paper deals with special classes of 1-regular languages. If a finite alphabet \Sigma is given, then by an 1-regular language over \Sigma we understand the union of a regular and an !-regular language over \Sigma. In this paper we show that it is possible to define such languages by means of a single finite-state device. A deterministic machine capable of constructing both finite and infinite sequences is first introduced in [7]. In [4] the structure of the sets constructed in [7] is investigated and in [5] a non-deterministic version of such machines ...

We introduce operators and laws of an algebra of...