ABSTRACT. Much discussion exists about what is computation, but less about what is a computational problem. In the sequential setting, Turing's definition of computational problem was based on functions. In a concurrent setting, various notions of distributed problems have been used, under different models of computation, ranging from models equivalent to a Turing machine, to models where much heated discussion has taken place about whether interactive computation is fundamentally different from sequential computing. Often distributed problems for non-terminating interaction are considered. It is argued here that there is no need to go all the way to non-terminating computation, to appreciate how different distributed computation is from sequential computation. In a decision problem each process of a distributed system starts with an initial private input value, and after communicating with other processes in the system, produces a local output value. An input/output relation is needed, to specify which output values are legal for a particular assignment of input values to the processes. An overview is provided of some results that show how rich the topic of distributed decision problems can be, when asynchronous processes can fail. We are in a world very different from that of the functions of sequential computation; moving away from graphs representing binary relations, to the world of simplicial complexes.

Language Inclusion for Boundedly-Ambiguous Vector Addition Systems is Decidable

ABSTRACT. We consider the problems of language inclusion and language equivalence for Vector Addition Systems with States (VASSes) with the acceptance condition defined by the set of accepting states (and more generally by some upward-closed conditions). In general the problem of language equivalence is undecidable even for one-dimensional VASSes, thus to get decidability we investigate restricted subclasses. On one hand we show that the problem of language inclusion of a VASS in $k$-ambiguous VASS (for any natural k) is decidable and even in Ackermann. On the other hand we prove that the language equivalence problem is Ackermann-hard already for deterministic VASSes. These two results imply Ackermann-completeness for language inclusion and equivalence in several possible restrictions. Some of our techniques can be also applied in much broader generality in infinite-state systems, namely for some subclass of well-structured transition systems.

Complexity of Coverability in Depth-Bounded Processes

ABSTRACT. We consider the class of depth-bounded processes in $\pi$-calculus. These processes
are the most expressive fragment of $\pi$-calculus, for which verification problems
are known to be decidable. The decidability of the coverability problem for this class has been achieved by means of well-quasi orders. (Meyer, IFIP TCS 2008; Wies, Zufferey and Henzinger, FoSSaCS 2010). However, the precise complexity of this problem is not known so far, with only a known EXPSPACE-lower bound.

In this paper, we prove that coverability for depth-bounded processes is $\mathbf{F}_{\epsilon_0}$-complete, where $\mathbf{F}_{\epsilon_0}$ is a class
in the fast-growing hierarchy of complexity classes. This solves an open problem mentioned by Haase, Schmitz, and Schnoebelen (LMCS, Vol 10, Issue 4) and also addresses a question raised by Wies, Zufferey and Henzinger (FoSSaCS 2010).

ABSTRACT. One-Counter Nets (OCNs) are finite-state automata equipped with a counter that is not allowed to become negative, but does not have zero tests.
Their simplicity and close connection to various other models (e.g., VASS, Counter Machines and Pushdown Automata) make them an attractive model for studying the border of decidability for the classical decision problems.

The deterministic fragment of OCNs (DOCNs) typically admits more tractable decision problems, and while these problems and the expressive power of DOCNs have been studied, the determinization problem, namely deciding whether an OCN admits an equivalent DOCN, has not received attention.

We introduce four notions of OCN determinizability, which arise naturally due to intricacies in the model, and specifically, the interpretation of the initial counter value.
We show that in general, determinizability is undecidable under most notions, but over a singleton alphabet (i.e., 1 dimensional VASS) one definition becomes decidable, and the rest become trivial, in that there is always an equivalent DOCN.

ABSTRACT. An {\em energy game\/} is played between two players, modeling a resource-bounded system and its environment. The players take turns moving a token along a finite graph. Each edge of the graph is labeled by an integer, describing an update to the energy level of the system that occurs whenever the edge is traversed. The system wins the game if it never runs out of energy.
Different applications have led to extensions of the above basic setting. For example, addressing a combination of the energy requirement with behavioral specifications, researchers have studied richer winning conditions, and addressing systems with several bounded resources, researchers have studied games with multi-dimensional energy updates. All extensions, however, assume that the environment has no bounded resources.

We introduce and study {\em both-bounded energy games\/} (BBEGs), in which both the system and the environment have multi-dimensional energy bounds. In BBEGs, each edge in the game graph is labeled by two integer vectors, describing updates to the multi-dimensional energy levels of the system and the environment. A system wins a BBEG if it never runs out of energy or if its environment runs out of energy.
We show that BBEGs are determined, and that the problem of determining the winner in a given BBEG is decidable iff both the system and the environment have energy vectors of dimension~$1$. We also study how restrictions on the memory of the system and/or the environment as well as upper bounds on their energy levels influence the winner and the complexity of the problem

Half-Positional Objectives Recognized by Deterministic Büchi Automata

ABSTRACT. A central question in the theory of two-player games over graphs is to understand which objectives are half-positional, that is, which are the objectives for which the protagonist does not need memory to implement winning strategies. Objectives for which both players do not need memory have already been characterized (both in finite and infinite graphs). However, less is known about half-positional objectives. In particular, no characterization of half-positionality is known for the central class of omega-regular objectives.

In this paper, we characterize objectives recognizable by deterministic Büchi automata (a class of omega-regular objectives) that are half-positional, in both finite and infinite graphs. Our characterization consists of three natural conditions linked to the language-theoretic notion of right congruence. Furthermore, this characterization yields a polynomial-time algorithm to decide half-positionality of an objective recognized by a given deterministic Büchi automaton.

ABSTRACT. We consider two-player zero-sum games with winning objectives beyond regular languages, expressed as a parity condition in conjunction with a Boolean combination of boundedness conditions on a finite set of counters which can be incremented, reset to $0$, but not tested. A boundedness condition requires that a given counter is bounded along the play. Such games are decidable, though with non-optimal complexity, by an encoding into the logic WMSO with the unbounded and path quantifiers, which is known to be decidable over infinite trees. Our objective is to give tight or tighter complexity results for particular classes of counter games with boundedness conditions, and study their strategy complexity. In particular, counter games with conjunction of boundedness conditions are easily seen to be equivalent to Streett games, so, they are CoNP-c. Moreover, finite-memory strategies suffice for Eve and memoryless strategies suffice for Adam. For counter games with a disjunction of boundedness conditions, we prove that they are in solvable in NP and in CoNP, and in PTime if the parity condition is fixed. In that case memoryless strategies suffice for Eve while infinite memory strategies might be necessary for Adam. Finally, we consider an extension of those games with a max operation. In that case, the complexity increases: for conjunctions of boundedness conditions, counter games are EXPTIME-c.

Different strokes in randomised strategies: Revisiting Kuhn's theorem under finite-memory assumptions

ABSTRACT. Two-player (antagonistic) games on (possibly stochastic) graphs are a prevalent model in theoretical computer science, notably as a framework for reactive synthesis.

Optimal strategies may require randomisation when dealing with inherently probabilistic goals, balancing multiple objectives or in contexts of partial information. There is no unique way to define randomised strategies. For instance, one can use so-called mixed strategies or behavioural ones. In the most general settings, these two classes do not share the same expressiveness. A seminal result in game theory - Kuhn's theorem - asserts their equivalence in games of perfect recall.

This result crucially relies on the possibility for strategies to use infinite memory, i.e., unlimited knowledge of all the past of a play. However, computer systems are finite in practice. Hence it is pertinent to restrict our attention to finite-memory strategies, defined as automata with outputs. Randomisation can be implemented in these in different ways: the initialisation, outputs or transitions can be randomised or deterministic respectively. Depending on which aspects are randomised, the expressiveness of the corresponding class of finite-memory strategies differs.

In this work, we study two-player turn-based stochastic games and provide a complete taxonomy of the classes of finite-memory strategies obtained by varying which of the three aforementioned components are randomised. Our taxonomy holds both in settings of perfect and imperfect information.