arXiv subjects

Quentin Guilmant

Publications and source records attributed to Quentin Guilmant.

5 recordsLinked to original sources

arXiv2026

On Periodic and Aperiodic Optimal Strategies in Solvency Games

Solvency games are a gambling problem on infinite-state Markov decision processes in which the state $n \in \mathbb{N}$ represents an investor's fortune. In every round, the investor chooses an action from a finite action set, and every action yields a distribution over integer-valued gains in an interval $\{-\ell,\ldots,m\}$. The risk-averse investor wants to minimise the probability of eventual ruin (reaching a fortune $\le 0$). It was shown in [Berger et al.] that memoryless deterministic optimal strategies exist, but they are not eventually constant in general. Even in the special case of gains in $\{-2,\ldots,1\}$, the optimal strategy may need to make use of two different actions at arbitrarily high fortunes. We show that optimal strategies in solvency games need not be ultimately periodic in general (thus disproving a 2012 conjecture of Kučera). Already in the case of gains in $\{-3,\ldots,1\}$, it is possible for the optimal strategy to be unique but aperiodic. For gains in $\{-2,\ldots,1\}$, there always exists an ultimately periodic optimal strategy whose tail is constant or alternates between two actions. Finally, we show that the optimal strategy is computable if it is unique. Moreover, (some) optimal strategy can always be computed in the case of gains in $\{-\ell,\ldots,1\}$ for any $\ell \in \mathbb{N}$. Computability in the general case however remains open.

cs.GT↗

arXiv2026

The Value Problem for Weighted Timed Games with Two Clocks is Undecidable

The Value Problem for weighted timed games (WTGs) consists in determining, given a two-player weighted timed game with a reachability objective and a rational threshold, whether or not the value of the game exceeds the threshold. This problem was shown to be undecidable some ten years ago for WTGs making use of at least three clocks, and is known to be decidable for single-clock WTGs. In this paper, we establish undecidability for two-clock WTGs making use of non-negative weights, even in a time-bounded setting, closing the last remaining major gap in our algorithmic understanding of WTGs.

cs.GT↗

arXiv2024

The 2-Dimensional Constraint Loop Problem is Decidable

A linear constraint loop is specified by a system of linear inequalities that define the relation between the values of the program variables before and after a single execution of the loop body. In this paper we consider the problem of determining whether such a loop terminates, i.e., whether all maximal executions are finite, regardless of how the loop is initialised and how the non-determinism in the loop body is resolved. We focus on the variant of the termination problem in which the loop variables range over $\mathbb{R}$. Our main result is that the termination problem is decidable over the reals in dimension~2. A more abstract formulation of our main result is that it is decidable whether a binary relation on $\mathbb{R}^2$ that is given as a conjunction of linear constraints is well-founded.

cs.LO↗

arXiv2022

Surreal fields stable under exponential, logarithmic, derivative and anti-derivative functions

The class of surreal numbers, denoted by $\textbf{No}$, initially proposed by Conway, is a universal ordered field in the sense that any ordered field can be embedded in it. They include in particular the real numbers and the ordinal numbers. They have strong relations with other fields such as field of transseries. Following Gonshor, surreal numbers can be seen as signs sequences of ordinal length, with some exponential and logarithmic functions that extend the usual functions over the reals. $\textbf{No}$ can actually be seen as an elegant (generalized) power series field with real coefficients, namely Hahn series with exponents in $\textbf{No}$ itself. Some years ago, Berarducci and Mantova considered derivation over the surreal numbers, seeing them as germs of functions, in correspondence to transseries. In this article, following our previous work, we exhibit a sufficient condition on the structure of a surreal field to be stable under all operations among exponential, logarithm, derivation and anti-derivation. Motivated, in the long term, by computability considerations, we also provide a non-trivial application of this theorem: the existence of a pretty reasonable field that only requires ordinals up to $ε_ω$, which is far smaller than $ω_1^{CK}$ (resp. $ω_1$), the first non-computable (resp. uncountable) ordinal.

math.LO↗

arXiv2022

Surreal fields stable under exponential and logarithmic functions

Surreal numbers, have a very rich and elegant theory. This class of numbers, denoted by No, includes simultaneously the ordinal numbers and the real numbers, and forms a universal huge real closed field: It is universal in the sense that any real closed field can be embedded in it. Following Gonshor, surreal numbers can also be seen as signs sequences of ordinal length, with some exponential and logarithmic functions that extend the usual functions over the reals. No can actually also be seen as an elegant particular (generalized) power series field with real coefficients, namely Hahn series with exponents in No itself. It can also be considered as a particular field of transseries, providing tools to do some analysis and asymptotic analysis for functions over the continuum, providing natural concepts for discussing hyperexponential or sublogarithm functions, and their asymptotics. In this article, we consider stability of subfields of No under exponential and logarithmic functions. Namely, we consider the set surreal numbers whose signs sequences have length less than some ordinal λ. Extending the discussion from van den Dries and Ehrlich, we show that is stable by exponential and logarithm iff λ is some ε-number. Motivated in a longer term by computability issues using ordinal machines, we consider subfields stables by exponential and logarithmic functions defined by Hahn series that does not require to go up to cardinal lengths and exponents. We prove that No can be expressed as a strict hierarchy of subfields stable by exponential and logarithmic functions. This provides many explicit examples of subfields of No stable by exponential and logarithmic functions, and does not require to go up to a cardinal λ to provide such examples.

math.LO↗