Paul Hunter

Mean-payoff games (MPGs) are infinite duration two-player zero-sum games played on weighted graphs. Under the hypothesis of perfect information, they admit memoryless optimal strategies for both players and can be solved in NP-intersect-coNP. MPGs are suitable quantitative models for open reactive systems. However, in this context the assumption of...

Mean-payoff games are important quantitative models for open reactive systems. They have been widely studied as games of full observation. In this paper we investigate the algorithmic properties of several sub-classes of mean-payoff games where the players have asymmetric information about the state of the game. These games are in general undecidab...

Two-player, zero-sum games of infinite duration and their quantitative versions are often used in verification to model the interaction between a controller (Eve) and an antagonistic environment (Adam). The question usually addressed is that of the existence (and computability) of a strategy for Eve that can maximize her payoff against any strategy...

In this invited contribution, we summarize new solution concepts useful for the synthesis of reactive systems that we have introduced in several recent publications. These solution concepts are developed in the context of non-zero sum games played on graphs. They are part of the contributions obtained in the inVEST project funded by the European Re...

In this paper we look at the problem of minimizing regret in discounted-sum games. We give algorithms for the general problem of computing the minimal regret of the controller (Eve) as well as several variants depending on which strategies the environment (Adam) is permitted to use. We also consider the problem of synthesizing regret-free strategie...

Mean-payoff games (MPGs) are infinite duration two-player zero-sum games played on weighted graphs. Under the hypothesis of perfect information, they admit memoryless optimal strategies for both players and can be solved in . MPGs are suitable quantitative models for open reactive systems. However, in this context the assumption of perfect informat...

We consider two-player games with reachability objectives played on transition systems of succinct one-counter machines, that is, machines where the counter is incremented or decremented by a value given in binary. We show that the winner-determination problem is EXPSPACE-complete regardless of whether transitions are guarded by constraints on the...

A natural framework for real-time specification is monadic first-order logic over the structure (ℝ, < , + 1)—the ordered real line with unary + 1 function. Our main result is that (ℝ, < , + 1) has the 3-variable property: every monadic first-order formula with at most 3 free variables is equivalent over this structure to one that uses 3 variables i...

Mean-payoff games are important quantitative models for open reactive systems. They have been widely studied as games of perfect information. In this paper we investigate the algorithmic properties of several subclasses of mean-payoff games where the players have asymmetric information about the state of the game. These games are in general undecid...

A natural framework for real-time specification is monadic first-order logic over the structure $(\mathbb{R},<,+1)$---the ordered real line with unary $+1$ function. Our main result is that $(\mathbb{R},<,+1)$ has the 3-variable property: every monadic first-order formula with at most 3 free variables is equivalent over this structure to one that u...

Traditionally quantitative games such as mean-payoff games and discount sum games have two players -- one trying to maximize the payoff, the other trying to minimize it. The associated decision problem, "Can Eve (the maximizer) achieve, for example, a positive payoff?" can be thought of as one player trying to attain a payoff in the interval $(0,\i...

Mean-payoff games are important quantitative models for open reactive systems. They have been widely studied as games of perfect information. In this paper we investigate the algorithmic properties of several subclasses of mean-payoff games where the players have asymmetric information about the state of the game. These games are in general undecid...

A seminal result of Kamp is that over the reals Linear Temporal Logic (LTL) has the same expressive power as first-order logic with binary order relation < and monadic predicates. A key question is whether there exists an analogue of Kamp's theorem for Metric Temporal Logic (MTL)-a generalization of LTL in which the Until and Since modalities are a...

We introduce a variant of the classic node search game called LIFO-search where searchers are assigned different numbers. The additional rule is that a searcher can be removed only if no searchers of lower rank are in the graph at that moment. We show that all common variations of the game require the same number of searchers. We then introduce the...

Metric Temporal Logic (MTL) is a generalisation of Linear Temporal Logic in which the Until and Since modalities are annotated with intervals that express metric constraints. A seminal result of Hirshfeld and Rabinovich shows that over the reals, first-order logic with binary order relation < and unary function +1 is strictly more expressive than M...

Tree-width is a well-known metric on undirected graphs that measures how tree-like a graph is and gives a notion of graph decomposition that proves useful in algorithm development. Tree-width can be characterised by a graph searching game where a number of cops attempt to capture a robber. We consider the natural adaptation of this game to directed...

We consider the extension of the last-in-first-out graph searching game of Giannopoulou and Thilikos to digraphs. We show that all common variations of the game require the same number of searchers, and the minimal number of searchers required is one more than the cycle-rank of the digraph. We also obtain a tight duality theorem, giving a precise m...

A fundamental problem in numerical computation and computational geometry is to determine the sign of arithmetic expressions in radicals. Here we consider the simpler problem of deciding whether å m i=1 CiA Xi i is zero for given rational numbers Ai, Ci, Xi. It has been known for almost twenty years that this can be decided in polynomial time (2)....

We consider various well-known, equivalent complexity measures for graphs such as elimination orderings, k-trees and cops and robber games and study their natural translations to digraphs. We show that on digraphs the translations of these measures are also equivalent and induce a natural connectivity measure. We introduce a decomposition for digra...

Tree-width is a well-known metric on undirected graphs that measures how tree-like a graph is and gives a notion of graph decomposition that proves useful in algorithm development. Tree-width is characterised by a game known as the cops-and-robber game where a number of cops chase a robber on the graph. We consider the natural adaptation of this ga...

We consider the complexity of infinite games played on finite graphs. We establish a framework in which the expressiveness and succinctness of different types of winning conditions can be compared. We show that the problem of deciding the winner in Muller games is PSPACE-complete. This is then used to establish PSPACE-completeness for Emerson-Lei g...

We consider the complexity of infinite games played on finite graphs. We establish a framework in which the expressiveness and succinctness of different types of winning conditions can be compared. We show that the problem of deciding the winner in Muller games is PSPACE-complete. This is then used to establish PSPACE-completeness for Emerson-Lei g...

We consider the complexity of infinite games played on finite graphs. We estab-lish a framework in which the expressiveness and succinctness of different types of winning conditions can be compared. We show that the problem of deciding the winner in Muller games is Pspace-complete. This is then used to establish Pspace-completeness for Emerson-Lei...