Verification of real-time systems with multiple components controlled by multiple parties is a challenging task due to its computational complexity. We present an on-the-fly algorithm for verifying timed alternating-time temporal logic (TATL), a branching-time logic with quantifiers over outcomes that results from coalitions of players in such systems. We combine existing work on games and timed CTL verification in the abstract dependency graph (ADG) framework, which allows for easy creation of on-the-fly algorithms that only explore the state space as needed. In addition, we generalize the conventional inclusion check to the ADG framework which enables dynamic reductions of the dependency graph. Using the insights from the generalization, we present a novel abstraction that eliminates the need for inclusion checking altogether in our domain. We implement our algorithms in Uppaal and our experiments show that while inclusion checking considerably enhances performance, our abstraction provides even more significant improvements, almost two orders of magnitude faster than the naive method. In addition, we outperform Uppaal Tiga, which can verify only a strict subset of TATL. After implementing our new abstraction in Uppaal Tiga, we also improve its performance by almost an order of magnitude.

We study distributed games played on non-deterministic asynchronous automata which feature a central decision maker process that participates in all key decision making tasks. In these partial-information games, processes use their causal past to respond to scheduling choices made by the scheduler and cooperatively strategize as a team to achieve the winning objective. We show that the problem of deciding the existence of a distributed winning strategy is efficiently solvable for global safety and local parity objectives. We provide algorithmic solutions that match their computational hardness. We formulate the notion of a finite-state distributed strategy which allows to quantify its distributed memory requirements. For the aforementioned objectives, we establish that finite-state distributed winning strategies always exist. In fact, we provide novel constructions of such winning strategies which are shown to have almost optimal amount of distributed memory. We also show that a natural extension of the model with two decision making processes is undecidable.

We propose a local, past-oriented fragment of propositional dynamic logic to reason about concurrent scenarios modelled as Mazurkiewicz traces, and prove it to be expressively complete with respect to regular trace languages. Because of locality, specifications in this logic are efficiently translated into asynchronous automata, in a way that reflects the structure of formulas. In particular, we obtain a new proof of Zielonka's fundamental theorem and we prove that any regular trace language can be implemented by a cascade product of localized asynchronous automata, which essentially operate on a single process.

International audience

We develop new algebraic tools to reason about concurrent behaviours modelled as languages of Mazurkiewicz traces and asynchronous automata. These tools reflect the distributed nature of traces and the underlying causality and concurrency between events, and can be said to support true concurrency. They generalize the tools that have been so efficient in understanding, classifying and reasoning about word languages. In particular, we introduce an asynchronous version of the wreath product operation and we describe the trace languages recognized by such products (the so-called asynchronous wreath product principle). We then propose a decomposition result for recognizable trace languages, analogous to the Krohn-Rhodes theorem, and we prove this decomposition result in the special case of acyclic architectures. Finally, we introduce and analyze two distributed automata-theoretic operations. One, the local cascade product, is a direct implementation of the asynchronous wreath product operation. The other, global cascade sequences, although conceptually and operationally similar to the local cascade product, translates to a more complex asynchronous implementation which uses the gossip automaton of Mukund and Sohoni. This leads to interesting applications to the characterization of trace languages definable in first-order logic: they are accepted by a restricted local cascade product of the gossip automaton and 2-state asynchronous reset automata, and also by a global cascade sequence of 2-state asynchronous reset automata. Over distributed alphabets for which the asynchronous Krohn-Rhodes theorem holds, a local cascade product of such automata is sufficient and this, in turn, leads to the identification of a simple temporal logic which is expressively complete for such alphabets.

We develop a new algebraic framework to reason about languages of Mazurkiewicz traces. This framework supports true concurrency and provides a non-trivial generalization of the wreath product operation to the trace setting. A novel local wreath product principle has been established. The new framework is crucially used to propose a decomposition result for recognizable trace languages, which is an analogue of the Krohn-Rhodes theorem. We prove this decomposition result in the special case of acyclic architectures and apply it to extend Kamp's theorem to this setting. We also introduce and analyze distributed automata-theoretic operations called local and global cascade products. Finally, we show that aperiodic trace languages can be characterized using global cascade products of localized and distributed two-state reset automata.

We propose a seamless integration of the block product operation to the recently developed algebraic framework for regular languages of countable words. A simple but subtle accompanying block product principle has been established. Building on this, we generalize the well-known algebraic characterizations of first-order logic (resp. first-order logic with two variables) in terms of strongly (resp. weakly) iterated block products. We use this to arrive at a complete analogue of Schiitzenberger-McNaughton-Papert theorem for countable words. We also explicate the role of block products for linear temporal logic by formulating a novel algebraic characterization of a natural fragment.

Abstract This paper extends a recently proposed robust computational framework for constructing the boundary representation (brep) of the volume swept by a given smooth solid moving along a one parameter family h of rigid motions. Our extension allows the input solid to have sharp features, and thus it is a significant and useful generalization of that work. This naturally requires a precise description of the geometry of the surface generated by the sweep of a sharp edge supported by two intersecting smooth faces. We uncover the geometry along with the related issues like parametrization and singularities via a novel mathematical analysis. Correct trimming of such a surface is achieved by an analysis of the interplay between the cone of normals at a sharp point and its trajectory under h. The overall topology is explained by a key lifting theorem which allows us to compute the adjacency relations amongst entities in the swept volume by relating them to corresponding adjacencies in the input solid. Moreover, global issues related to body‐check such as orientation, singularities and self‐intersections are efficiently resolved. Examples from a pilot implementation illustrate the efficiency and effectiveness of our framework.

Given a rank-1 bimatrix game (A,B), i.e., where rank(A+B)=1, we construct a suitable linear subspace of the rank-1 game space and show that this subspace is homeomorphic to its Nash equilibrium correspondence. Using this homeomorphism, we give the first polynomial time algorithm for computing an exact Nash equilibrium of a rank-1 bimatrix game. This settles an open question posed by Kannan and Theobald (SODA'07). In addition, we give a novel algorithm to enumerate all the Nash equilibria of a rank-1 game and show that a similar technique may also be applied for finding a Nash equilibrium of any bimatrix game. Our approach also provides new proofs of important classical results such as the existence and oddness of Nash equilibria, and the index theorem for bimatrix games. Further, we extend the rank-1 homeomorphism result to a fixed rank game space, and give a fixed point formulation on [0,1]k for solving a rank-k game. The homeomorphism and the fixed point formulation are piece-wise linear and considerably simpler than the classical constructions.