Abstract. In modal logics, graded (worlds) modalities have been deeply investigated as a useful framework for generalizing standard existential and universal modalities, in such a way they can express statements about a given number of immediately accessible worlds. These modalities have been recently investigated with respect to the propositional µ-calculus, which provide formulas exponentially more succinct, without affecting the computational complexity for the extended logic, i.e., the satisfiability problem remains solvable in EXPTIME. A natural question that arises is how less powerful logics than µ-calculus or more complex graded modalities affect the decidability of the logic. In this paper, we investigate this question in the case of the branching-time temporal logic CTL by introducing graded path modalities. These modalities naturally extend to (minimal) paths the generalization induced to successor worlds by classical graded modalities, i.e., they allow to express properties such as "there exist at least n minimal paths satisfying a given formula". As interesting results, we show that CTL extended with graded path modalities is more expressive than CTL, it retains the tree and the finite model properties, and its satisfiability problem remains decidable in EXPTIME. The latter result is obtained by exploiting an automata-theoretic approach. In particular, we introduce the class of partitioning alternating Büchi tree automata and show that the emptiness problems for them is EXPTIME-COMPLETE.
Abstract. In modal logics, graded (worlds) modalities have been deeply investigated as a useful framework for generalizing standard existential and universal modalities, in such a way they can express statements about a given number of immediately accessible worlds. These modalities have been recently investigated with respect to the propositional µ-calculus, which provide formulas exponentially more succinct, without affecting the computational complexity for the extended logic, i.e., the satisfiability problem remains solvable in EXPTIME. A natural question that arises is how less powerful logics than µ-calculus or more complex graded modalities affect the decidability of the logic. In this paper, we investigate this question in the case of the branching-time temporal logic CTL by introducing graded path modalities. These modalities naturally extend to (minimal) paths the generalization induced to successor worlds by classical graded modalities, i.e., they allow to express properties such as "there exist at least n minimal paths satisfying a given formula". As interesting results, we show that CTL extended with graded path modalities is more expressive than CTL, it retains the tree and the finite model properties, and its satisfiability problem remains decidable in EXPTIME. The latter result is obtained by exploiting an automata-theoretic approach. In particular, we introduce the class of partitioning alternating Büchi tree automata and show that the emptiness problems for them is EXPTIME-COMPLETE.
Half positionality is the property of a language of infinite words to admit positional winning strategies, when interpreted as the goal of a two-player game on a graph. Such problem applies to the automatic synthesis of controllers, where positional strategies represent efficient controllers. As our main result, we present a novel sufficient condition for half positionality, more general than what was previously known. Moreover, we compare our proposed condition with several others, proposed in the recent literature, outlining an intricate network of relationships, where only few combinations are sufficient for half positionality.
Graded path quantifiers have been recently introduced and investigated as a useful framework for generalizing standard existential and universal path quantifiers in the branching-time temporal logic CTL (GCTL), in such a way that they can express statements about a minimal and conservative number of accessible paths. These quantifiers naturally extend to paths the concept of graded world modalities, which has been deeply investigated for the mu-Calculus (Gmu-Calculus) where it allows to express statements about a given number of immediately accessible worlds. As for the “non-graded” case, it has been shown that the satisfiability problem for GCTL and the Gmu-Calculus coincides and, in particular, it remains solvable in ExpTime. However, GCTL has been only investigated w.r.t. graded numbers coded in unary, while Gmu-Calculus uses for this a binary coding, and it was left open the problem to decide whether the same result may or may not hold for binary GCTL. In this paper, by exploiting an automata theoretic-approach, which involves a model of alternating automata with satellites, we answer positively to this question. We further investigate the succinctness of binary GCTL and show that it is at least exponentially more succinct than Gmu-Calculus. Finally, we discuss the extension of the exploited technique to the case of binary graded CTL* (GCTL*). In this case, the satisfiability problem becomes solvable in 3ExpTime
Abstract. We consider finite graphs whose edges are labeled with elements, called colors, taken from a fixed finite alphabet. We study the problem of determining whether there is an infinite path where either (i) all colors occur with the same asymptotic frequency, or (ii) there is a constant which bounds the difference between the occurrences of any two colors for all prefixes of the path. These two notions can be viewed as refinements of the classical notion of fair path, whose simplest form checks whether all colors occur infinitely often. Our notions provide stronger criteria, particularly suitable for scheduling applications based on a coarse-grained model of the jobs involved. We show that both problems are solvable in polynomial time, by reducing them to the feasibility of a linear program.
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