arXiv subjects

Pierre Ohlmann

Publications and source records attributed to Pierre Ohlmann.

26 records · Page 2Linked to original sources

arXiv2021

Controlling a random population

Bertrand et al. introduced a model of parameterised systems, where each agent is represented by a finite state system, and studied the following control problem: for any number of agents, does there exist a controller able to bring all agents to a target state? They showed that the problem is decidable and EXPTIME-complete in the adversarial setting, and posed as an open problem the stochastic setting, where the agent is represented by a Markov decision process. In this paper, we show that the stochastic control problem is decidable. Our solution makes significant uses of well quasi orders, of the max-flow min-cut theorem, and of the theory of regular cost functions. We introduce an intermediate problem of independence interest called the sequential flow problem and study its complexity.

cs.FL↗

arXiv2021

Scaling Neural Program Synthesis with Distribution-based Search

We consider the problem of automatically constructing computer programs from input-output examples. We investigate how to augment probabilistic and neural program synthesis methods with new search algorithms, proposing a framework called distribution-based search. Within this framework, we introduce two new search algorithms: Heap Search, an enumerative method, and SQRT Sampling, a probabilistic method. We prove certain optimality guarantees for both methods, show how they integrate with probabilistic and neural techniques, and demonstrate how they can operate at scale across parallel compute environments. Collectively these findings offer theoretical and applied studies of search algorithms for program synthesis that integrate with recent developments in machine-learned program synthesizers.

cs.LG↗

arXiv2021

The GKK Algorithm is the Fastest over Simple Mean-Payoff Games

We study the algorithm of Gurvich, Khachyian and Karzanov (GKK algorithm) when it is ran over mean-payoff games with no simple cycle of weight zero. We propose a new symmetric analysis, lowering the $O(n^2 N)$ upper-bound of Pisaruk on the number of iterations down to $N + Ep + Em$, which is smaller than $nN$, where $n$ is the number of vertices, $N$ is the largest absolute value of a weight, and $Ep$ and $Em$ are respectively the largest finite energy and dual-energy values of the game. Since each iteration is computed in $O(m)$, this improves on the state of the art pseudopolynomial $O(mnN)$ runtime bound of Brim, Chaloupka, Doyen, Gentilini and Raskin, by taking into account the structure of the game graph. We complement our result by showing that the analysis of Dorfman, Kaplan and Zwick also applies to the GKK algorithm, which is thus also subject to the state of the art combinatorial runtime bound of $O(m 2^{n/2})$.

cs.GT↗

arXiv2021

New Algorithms for Combinations of Objectives using Separating Automata

The notion of separating automata was introduced by Bojanczyk and Czerwinski for understanding the first quasipolynomial time algorithm for parity games. In this paper we show that separating automata is a powerful tool for constructing algorithms solving games with combinations of objectives. We construct two new algorithms: the first for disjunctions of parity and mean payoff objectives, matching the best known complexity, and the second for disjunctions of mean payoff objectives, improving on the state of the art. In both cases the algorithms are obtained through the construction of small separating automata, using as black boxes the existing constructions for parity objectives and for mean payoff objectives.

cs.GT↗

arXiv2020

A symmetric attractor-decomposition lifting algorithm for parity games

Progress-measure lifting algorithms for solving parity games have the best worst-case asymptotic runtime, but are limited by their asymmetric nature, and known from the work of Czerwiński et al. (2018) to be subject to a matching quasi-polynomial lower bound inherited from the combinatorics of universal trees. Parys (2019) has developed an ingenious quasi-polynomial McNaughton- Zielonka-style algorithm, and Lehtinen et al. (2019) have improved its worst-case runtime. Jurdziński and Morvan (2020) have recently brought forward a generic attractor-based algorithm, formalizing a second class of quasi-polynomial solutions to solving parity games, which have runtime quadratic in the size of universal trees. First, we adapt the framework of iterative lifting algorithms to computing attractor-based strategies. Second, we design a symmetric lifting algorithm in this setting, in which two lifting iterations, one for each player, accelerate each other in a recursive fashion. The symmetric algorithm performs at least as well as progress-measure liftings in the worst-case, whilst bypassing their inherent asymmetric limitation. Thirdly, we argue that the behaviour of the generic attractor-based algorithm of Jurdzinski and Morvan (2020) can be reproduced by a specific deceleration of our symmetric lifting algorithm, in which some of the information collected by the algorithm is repeatedly discarded. This yields a novel interpretation of McNaughton-Zielonka-style algorithms as progress-measure lifting iterations (with deliberate set-backs), further strengthening the ties between all known quasi-polynomial algorithms to date.

cs.LO↗

arXiv2019

On the Monniaux Problem in Abstract Interpretation

The Monniaux Problem in abstract interpretation asks, roughly speaking, whether the following question is decidable: given a program $P$, a safety (\emph{e.g.}, non-reachability) specification $φ$, and an abstract domain of invariants $\mathcal{D}$, does there exist an inductive invariant $I$ in $\mathcal{D}$ guaranteeing that program $P$ meets its specification $φ$. The Monniaux Problem is of course parameterised by the classes of programs and invariant domains that one considers. In this paper, we show that the Monniaux Problem is undecidable for unguarded affine programs and semilinear invariants (unions of polyhedra). Moreover, we show that decidability is recovered in the important special case of simple linear loops.

cs.LO↗

arXiv2019

The complexity of mean payoff games using universal graphs

We study the computational complexity of solving mean payoff games. This class of games can be seen as an extension of parity games, and they have similar complexity status: in both cases solving them is in $\textbf{NP} \cap \textbf{coNP}$ and not known to be in $\textbf{P}$. In a breakthrough result Calude, Jain, Khoussainov, Li, and Stephan constructed in 2017 a quasipolynomial time algorithm for solving parity games, which was quickly followed by two other algorithms with the same complexity. Our objective is to investigate how these techniques can be extended to the study of mean payoff games. The starting point is the notion of separating automata, which has been used to present all three quasipolynomial time algorithms for parity games and gives the best complexity to date. The notion naturally extends to mean payoff games and yields a class of algorithms for solving mean payoff games. The contribution of this paper is to prove tight bounds on the complexity of algorithms in this class. We construct two new algorithms for solving mean payoff games. Our first algorithm depends on the largest weight $N$ (in absolute value) appearing in the graph and runs in sublinear time in $N$, improving over the previously known linear dependence in $N$. Our second algorithm runs in polynomial time for a fixed number $k$ of weights. We complement our upper bounds by providing in both cases almost matching lower bounds, showing the limitations of the separating automata approach. We show that we cannot hope to improve on the dependence in $N$ nor break the linear dependence in the exponent in the number $k$ of weights. In particular, this shows that separating automata do not yield a quasipolynomial algorithm for solving mean payoff games.

cs.GT↗

arXiv2017

Semialgebraic Invariant Synthesis for the Kannan-Lipton Orbit Problem

The \emph{Orbit Problem} consists of determining, given a linear transformation $A$ on $\mathbb{Q}^d$, together with vectors $x$ and $y$, whether the orbit of $x$ under repeated applications of $A$ can ever reach $y$. This problem was famously shown to be decidable by Kannan and Lipton in the 1980s. In this paper, we are concerned with the problem of synthesising suitable \emph{invariants} $\mathcal{P} \subseteq \mathbb{R}^d$, \emph{i.e.}, sets that are stable under $A$ and contain $x$ and not $y$, thereby providing compact and versatile certificates of non-reachability. We show that whether a given instance of the Orbit Problem admits a semialgebraic invariant is decidable, and moreover in positive instances we provide an algorithm to synthesise suitable invariants of polynomial size. It is worth noting that the existence of \emph{semilinear} invariants, on the other hand, is (to the best of our knowledge) not known to be decidable.

cs.CC↗