Graded computation tree logic

Abstract

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.

Extended Graded Modalities in Strategy Logic

“…It also introduces a graded extension of SL, called GSL. However, in contrast with our work and [4] (and by extending an idea of counting paths in [5,10]), GSL provides a quite intricate way of counting strategies: it gives a semantic definition of equivalence of strategies, and the graded modalities count non-equivalent strategies. While this approach is sound, it heavily complicates the model-checking problem.…”

Reasoning on expressive description logics with arithmetic constraints

“…Graded modalities are the equivalent to qualified number restrictions in Description Logics. EXPTIME bounds for the Computational Tree Logic extended with graded modalities are proven by [13]. In [9], it is shown that the fully enriched μ-calculus is undecidable.…”

“…Other expressive logics with counting (arithmetic) constraints have been recently studied [10], [11], [9], [12], [13], [14], [15]. In [10], basic modal logic extended with Presburger constraints is shown to be in PSPACE.…”

Pushdown module checking with imperfect information

“…The formal definition of semi-alternating pushdown tree automata follows. 6 A semi-alternating pushdown tree automaton is a tuple A = Σ, D, Γ, Q , q 0 , , δ, F where Σ is a finite input alphabet, D is a finite set of directions, Γ is a finite pushdown store alphabet, Q is a finite set of states, q 0 ∈ Q is the initial state, / ∈ Γ is the pushdown store bottom symbol, and F is an acceptance condition, to be defined later. δ is a finite transition function δ : (d, q, β), where d is a direction, q is a state, and β is a string of pushdown store symbols.…”

“…The only subtle point involves handling the binary-encoded graded modalities without increasing the complexity with respect to the propositional μ-calculus. For more details see[5,6].…”