Alessandro Ferrante

Enriched mu--calculi module checking,

The model checking problem for open systems has been intensively studied in the literature, for both finite-state (module checking) and infinite-state (pushdown module checking) systems, with respect to Ctl and Ctl*. In this paper, we further investigate this problem with respect to the \mu-calculus enriched with nominals and graded modalities (hybrid graded Mu-calculus), in both the finite-state and infinite-state settings. Using an automata-theoretic approach, we show that hybrid graded \mu-calculus module checking is solvable in exponential time, while hybrid graded \mu-calculus pushdown module checking is solvable in double-exponential time. These results are also tight since they match the known lower bounds for Ctl. We also investigate the module checking problem with respect to the hybrid graded \mu-calculus enriched with inverse programs (Fully enriched \mu-calculus): by showing a reduction from the domino problem, we show its undecidability. We conclude with a short overview of the model checking problem for the Fully enriched Mu-calculus and the fragments obtained by dropping at least one of the additional constructs.

Model Checking for Graded CTL

Recently, complexity issues related to the decidability of the μ-calculus, when the universal and existential quantifiers are augmented with graded modalities, have been investigated by Kupfermann, Sattler and Vardi ([19]). Graded modalities refer to the use of the universal and existential quantifiers with the added capability to express the concept of at least k or all but k, for a non-negative integer k. In this paper we study the Computational Tree Logic CTL, a branching time extension of classical modal logic, augmented with graded modalities and investigate the complexity issues with respect to the model-checking problem. We consider a system model represented by a Kripke structure K and give an algorithm to solve the model-checking problem running in time O(|K| · |p|) which is hence tight for the problem (here |p| is the number of temporal and boolean operators and does not include the values occurring in the graded modalities). In this framework, the graded modalities express the ability to generate a user-defined number of counterexamples to a specification p given in CTL. However, these multiple counterexamples can partially overlap, that is they may share some behavior. We have hence investigated the case when all of them are completely disjoint. In this case we prove that the model-checking problem is both NP-hard and coNP-hard and give an algorithm for solving it running in polynomial space. We have thus studied a fragment of graded-CTL, and have proved that the model-checking problem is solvable in polynomial time.

CTL Model-Checking with Graded Quantifiers

The use of the universal and existential quantifiers with the capability to express the concept of at leastkor all butk, for a non-negative integer k, has been thoroughly studied in various kinds of logics. In classical logic there are counting quantifiers, in modal logics graded modalities, in description logics number restrictions. Recently, the complexity issues related to the decidability of the μ-calculus, when the universal and existential quantifiers are augmented with graded modalities, have been investigated by Kupfermann, Sattler and Vardi. They have shown that this problem is ExpTime -complete. In this paper we consider another extension of modal logic, the Computational Tree Logic CTL, augmented with graded modalities generalizing standard quantifiers and investigate the complexity issues, with respect to the model-checking problem. We consider a system model represented by a pointed Kripke structure

Enriched µ-calculus pushdown module checking

\mathcal{K}

Enriched µ-calculi module checking

and give an algorithm to solve the model-checking problem running in time

A NuSMV extension for Graded-CTL model checking

O(|\mathcal{K}|\cdot |\varphi|)

Existence of Nash Equilibria in Selfish Routing Problems

which is hence tight for the problem (where |φ| is the number of temporal and boolean operators and does not include the values occurring in the graded modalities). In this framework, the graded modalities express the ability to generate a user-defined number of counterexamples (or evidences) to a specification φgiven in CTL . However these multiple counterexamples can partially overlap, that is they may share some behavior. We have hence investigated the case when all of them are completely disjoint. In this case we prove that the model-checking problem is both NP -hard and coNP -hard and give an algorithm for solving it running in polynomial space. We have thus studied a fragment of this graded- CTL logic, and have proved that the model-checking problem is solvable in polynomial time.

Improvements for truthful mechanisms with verifiable one-parameter selfish agents

The model checking problem for open systems (called module checking) has been intensively studied in the literature, both for finite-state and infinite-state systems. In this paper, we focus on pushdown module checking with respect to µ-calculus enriched with graded and nominals (hybrid graded µ-calulus). We show that this problem is decidable and solvable in double-exponential time in the size of the formula and in exponential time in the size of the system. This result is obtained by exploiting a classical automata-theoretic approach via pushdown nondeterministic parity tree automata. In particular, we reduce in exponential time our problem to the emptiness problem for these automata, which is known to be decidable in Exptime. As a key step of our algorithm, we show an exponential improvement of the construction of a nondeterministic parity tree automaton accepting all models of a formula of the considered logic. This result, not only allows our algorithm to match the known lower bound, but it is also interesting by itself, since it allows investigating decision problems related to enriched µ-calculus formulas in a greatly simplified manner. We conclude the paper with a discussion on the model checking w.r.t. µ-calculus formulas enriched with backward modalities as well.

Mixed Nash equilibria in selfish routing problems with dynamic constraints

The model checking problem for open finite-state systems (called module checking) has been intensively studied in the literature with respect to CTL and CTL*. In this paper, we focus on module checking with respect to the fully enriched µ-calculus and some of its fragments. Fully enriched µ-calculus is the extension of the propositional µ-calculus with inverse programs, graded modalities, and nominals. The fragments we consider here are obtained by dropping at least one of the additional constructs. For the full calculus, we show that module checking is undecidable by using a reduction from the domino problem. For its fragments, instead, we show that module checking is decidable and EXPTIME-complete. This result is obtained by using, for the upper bound, a classical automata-theoretic approach via Forest Enriched Automata and, for the lower bound, a reduction from the module checking problem for CTL, known to be EXPTIME-hard.

ON THE VERTEX-CONNECTIVITY PROBLEM FOR GRAPHS WITH SHARPENED TRIANGLE INEQUALITY

Graded-CTL is an extension of CTL with graded quantifiers which allow to reason about either at least or all but any number of possible futures In this paper we show an extension of the NuSMV model-checker implementing symbolic algorithms for graded-CTL model checking The implementation is based on the CUDD library, for BDDs and ADDs manipulation, and includes also an efficient algorithm for multiple counterexamples generation.