Two-player zero-sum games of infinite duration and their quantitative versions are used in verification to model the interaction between a controller (Eve) and its 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 of Adam. In this work, we are interested in strategies of Eve that minimize her regret, i.e. strategies that minimize the difference between her actual payoff and the payoff she could have achieved if she had known the strategy of Adam in advance. We give algorithms to compute the strategies of Eve that ensure minimal regret against an adversary whose choice of strategy is (1) unrestricted, (2) limited to positional strategies, or (3) limited to word strategies, and show that the two last cases have natural modelling applications. These results apply for quantitative games defined with the classical payoff functions $$\mathsf {Inf}$$ , $$\mathsf {Sup}$$ , $${\mathsf {LimInf}}$$ , $$\mathsf {LimSup}$$ , and mean-payoff. We also show that our notion of regret minimization in which Adam is limited to word strategies generalizes the notion of good for games introduced by Henzinger and Piterman, and is related to the notion of determinization by pruning due to Aminof, Kupferman and Lampert.

In this paper, we study the problem of minimizing regret in discounted-sum games played on weighted game graphs. 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 strategies for Eve in each of these scenarios.

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,infinity). In this paper we consider the more general problem of determining if a player can attain a payoff in a finite union of arbitrary intervals for various payoff functions (liminf/limsup, mean-payoff, discount sum, total sum). In particular this includes the interesting exact-value problem, "Can Eve achieve a payoff of exactly (e.g.) 0?"

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 annotated with intervals that express metric constraints. Hirshfeld and Rabinovich gave a negative answer, showing that first-order logic with binary order relation < and unary function +1 is strictly more expressive than MTL with integer constants. However, a recent result of Hunter, Ouaknine and Worrell shows that with rational timing constants, MTL has the same expressive power as first-order logic, giving a positive answer. In this paper we generalize these results by giving a precise characterization of those sets of constants for which MTL and first-order logic have the same expressive power. We also show that full first-order expressiveness can be recovered with the addition of counting modalities, strongly supporting the assertion of Hirshfeld and Rabinovich that Q2MLO is one of the most expressive decidable fragments of FO(

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. Hirshfeld and Rabinovich have shown that over the reals, firstorder logic with binary order relation <; and unary function +1 is strictly more expressive than MTL with integer constants. Indeed they prove that no temporal logic whose modalities are definable by formulas of bounded quantifier depth can be expressively complete for FO(<;, +1). In this paper we show that if we allow unary functions +q, q ∈ Q, in first-order logic and correspondingly allow rational constants in MTL, then the two logics have the same expressive power. This gives the first generalisation of Kamp's theorem on the expressive completeness of LTL for FO(<;) to the quantitative setting. The proof of this result involves a generalisation of Gabbay's notion of separation to the metric setting.

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 $\sum_{i=1}^m C_i A_i^{X_i}$ is zero for given rational numbers $A_i$, $C_i$, $X_i$. It has been known for almost twenty years that this can be decided in polynomial time. In this paper we improve this result by showing membership in uniform TC0. This requires several significant departures from Blömer's polynomial-time algorithm as the latter crucially relies on primitives, such as gcd computation and binary search, that are not known to be in TC0.