arXiv subjects

Patrick Totzke

Publications and source records attributed to Patrick Totzke.

At least 19 recordsLinked to original sources

arXiv2026

Strategy Complexity of Büchi and Transience Objectives in Concurrent Stochastic Games

We study 2-player stochastic games on countable graphs. Players Max and Min seek respectively to maximize and minimize the probability of satisfying the game objective. The Büchi objective is to visit a given set of states infinitely often. The Transience objective is to visit no state infinitely often. In Büchi games there exist $\varepsilon$-optimal Max strategies that use just a step counter plus 1 bit of public memory. This upper bound holds for all countable graphs, but is a new result even for finite graphs. It is tight, since Max strategies that use just a step counter, or just finite memory, are not sufficient even on finite game graphs. This upper bound follows from a slightly stronger new result: $\varepsilon$-optimal Max strategies for the combined Büchi and Transience objective require exactly 1 bit of public memory. Moreover, $\varepsilon$-optimal Max strategies for the Transience objective alone can be chosen as memoryless.

cs.GT↗

arXiv2026

The Emptiness Problem for Quantum Finite Automata with Classical States

Quantum Finite Automata with Classical states (QFACs) are nondeterministic finite automata over a finite alphabet of quantum operations. We study expressiveness of this model on finite words and the corresponding emptiness problem. We show that regular languages are incomparable with those definable by Quantum Finite Automata (QFAs) and that both are strictly subsumed by QFAC-definable languages. We show that the emptiness problem for a QFAC can be reduced to the emptiness of the language intersection of a QFA and a finite automaton. This intersection is known to be decidable for strict thresholds but undecidable for non-strict cases. Furthermore, we consider the problem for flat QFACs, a restriction where the underlying automata contain no nested loops, and relate it to the higher-dimensional orbit problem, a long-standing open challenge in dynamical systems. Finally, we propose a sound and semi-complete witness searching procedure to verify the non-emptiness of one-loop QFACs, which are sufficiently expressive to represent some prominent quantum algorithms, such as Grover's search and quantum random walks.

cs.FL↗

arXiv2026

History-Constrained Systems

We study verification problems for history-constrained systems (HCS), a model of guarded computation that uses nested systems. An outer system describes the process architecture in which a sequence of actions represents the communication between sub-systems through a global bus. Actions are either permitted or blocked locally by guards; these guards read and decide based on the sequence of actions so far in the global bus. When HCS have both the outer systems and the local guard controllers modelled by finite automata, we show they have the same expressive power as regular languages and finite automata, but they are exponentially more succinct. We also analyse games on this model, representing the interaction between environment and controller, and show that solving such games is EXPTIME-complete, where the lower bound already holds for reachability/safety games and the upper bound holds for any $ω$-regular winning condition. Finally, we consider HCS with guards of greater expressive power, Vector Addition Systems with States (VASS). We show that with deterministic coverability-VASS guards the reachability problem is EXPSPACE-complete, while with reachability-VASS the problem is undecidable.

cs.FL↗

arXiv2026

Resolving Nondeterminism by Chance

History-deterministic automata are those in which nondeterministic choices can be correctly resolved stepwise: there is a strategy to select a continuation of a run given the next input letter so that if the overall input word admits some accepting run, then the constructed run is also accepting. Motivated by checking qualitative properties in probabilistic verification, we consider the setting where the resolver strategy can randomize and only needs to succeed with lower-bounded probability. We study the expressiveness of such stochastically-resolvable automata as well as consider the decision questions of whether a given automaton has this property. In particular, we show that it is undecidable to check if a given NFA is $λ$-stochastically resolvable. This problem is decidable for finitely-ambiguous automata. We also present complexity upper and lower bounds for several well-studied classes of automata for which this problem remains decidable.

cs.FL↗

arXiv2026

Optimally Controlling a Random Population

The population control problem is a parameterised problem where a controller sends messages to a whole population of identical finite-state agents, aiming to eventually move them all into a target state. The decision problem asks whether this can be achieved for arbitrarily large finite populations. We focus on the randomised version of this problem, where every agent is a copy of the same finite Markov Decision Process and non-determinism in the {global} action chosen by the controller is resolved independently and uniformly at random. Colcombet, Fijalkow and Ohlmann showed that this problem is decidable, but without any complexity upper bound. We show that the random population control problem is in fact EXPTIME-complete.

math.OC↗

arXiv2026

Optimal Sequential Flows

We provide a new algebraic technique to solve the sequential flow problem in polynomial space. The task is to maximise the flow through a graph where edge capacities can be changed over time by choosing a sequence of capacity labelings from a given finite set. Our method is based on a novel factorization theorem for finite semigroups that, applied to a suitable flow semigroup, allows to derive small witnesses. This generalises to multiple in/output vertices, as well as regular constraints.

math.OC↗

arXiv2025

Temporal Explorability Games

Temporal graphs extend ordinary graphs with discrete time that affects the availability of edges. We consider solving games played on temporal graphs where one player aims to explore the graph, i.e., visit all vertices. The complexity depends majorly on two factors: the presence of an adversary and how edge availability is specified. We demonstrate that on static graphs, where edges are always available, solving explorability games is just as hard as solving reachability games. In contrast, on temporal graphs, the complexity of explorability coincides with generalized reachability (NP-complete for one-player and PSPACE- complete for two player games). We further show that if temporal graphs are given symbolically, even one-player reachability and thus explorability and generalized reachability games are PSPACE-hard. For one player, all these are also solvable in PSPACE and for two players, they are in PSPACE, EXP and EXP, respectively.

cs.GT↗

arXiv2025

HyperLTL Satisfiability Is Highly Undecidable, HyperCTL$^*$ is Even Harder

Temporal logics for the specification of information-flow properties are able to express relations between multiple executions of a system. The two most important such logics are HyperLTL and HyperCTL*, which generalise LTL and CTL* by trace quantification. It is known that this expressiveness comes at a price, i.e. satisfiability is undecidable for both logics. In this paper we settle the exact complexity of these problems, showing that both are in fact highly undecidable: we prove that HyperLTL satisfiability is $Σ_1^1$-complete and HyperCTL* satisfiability is $Σ_1^2$-complete. These are significant increases over the previously known lower bounds and the first upper bounds. To prove $Σ_1^2$-membership for HyperCTL*, we prove that every satisfiable HyperCTL* sentence has a model that is equinumerous to the continuum, the first upper bound of this kind. We also prove this bound to be tight. Furthermore, we prove that both countable and finitely-branching satisfiability for HyperCTL* are as hard as truth in second-order arithmetic, i.e. still highly undecidable. Finally, we show that the membership problem for every level of the HyperLTL quantifier alternation hierarchy is $Π_1^1$-complete.

cs.LO↗

arXiv2024

History-deterministic Timed Automata

We explore the notion of history-determinism in the context of timed automata (TA) over infinite timed words. History-deterministic (HD) automata are those in which nondeterminism can be resolved on the fly, based on the run constructed thus far. History-determinism is a robust property that admits different game-based characterisations, and HD specifications allow for game-based verification without an expensive determinization step. We show that the class of timed $ω$-languages recognized by HD timed automata strictly extends that of deterministic ones, and is strictly included in those recognised by fully non-deterministic TA. For non-deterministic timed automata it is known that universality is already undecidable for safety/reachability TA. For history-deterministic TA with arbitrary parity acceptance, we show that timed universality, inclusion, and synthesis all remain decidable and are EXPTIME-complete. For the subclass of TA with safety or reachability acceptance, one can decide (in EXPTIME) whether such an automaton is history-deterministic. If so, it can effectively determinized without introducing new automaton states.

cs.FL↗

arXiv2024

Strategy Complexity of Reachability in Countable Stochastic 2-Player Games

We study countably infinite stochastic 2-player games with reachability objectives. Our results provide a complete picture of the memory requirements of $\varepsilon$-optimal (resp. optimal) strategies. These results depend on the size of the players' action sets and on whether one requires strategies that are uniform (i.e., independent of the start state). Our main result is that $\varepsilon$-optimal (resp. optimal) Maximizer strategies require infinite memory if Minimizer is allowed infinite action sets. This lower bound holds even under very strong restrictions. Even in the special case of infinitely branching turn-based reachability games, even if all states allow an almost surely winning Maximizer strategy, strategies with a step counter plus finite private memory are still useless. Regarding uniformity, we show that for Maximizer there need not exist positional (i.e., memoryless) uniformly $\varepsilon$-optimal strategies even in the special case of finite action sets or in finitely branching turn-based games. On the other hand, in games with finite action sets, there always exists a uniformly $\varepsilon$-optimal Maximizer strategy that uses just one bit of public memory.

cs.GT↗

arXiv2024

The Power of Counting Steps in Quantitative Games

We study deterministic games of infinite duration played on graphs and focus on the strategy complexity of quantitative objectives. Such games are known to admit optimal memoryless strategies over finite graphs, but require infinite-memory strategies in general over infinite graphs. We provide new lower and upper bounds for the strategy complexity of mean-payoff and total-payoff objectives over infinite graphs, focusing on whether step-counter strategies (sometimes called Markov strategies) suffice to implement winning strategies. In particular, we show that over finitely branching arenas, three variants of limsup mean-payoff and total-payoff objectives admit winning strategies that are based either on a step counter or on a step counter and an additional bit of memory. Conversely, we show that for certain liminf total-payoff objectives, strategies resorting to a step counter and finite memory are not sufficient. For step-counter strategies, this settles the case of all classical quantitative objectives up to the second level of the Borel hierarchy.

cs.GT↗

arXiv2024

Bounded-Memory Strategies in Partial-Information Games

We study the computational complexity of solving stochastic games with mean-payoff objectives. Instead of identifying special classes in which simple strategies are sufficient to play $ε$-optimally, or form $ε$-Nash equilibria, we consider general partial-information multiplayer games and ask what can be achieved with (and against) finite-memory strategies up to a {given} bound on the memory. We show $NP$-hardness for approximating zero-sum values, already with respect to memoryless strategies and for 1-player reachability games. On the other hand, we provide upper bounds for solving games of any fixed number of players $k$. We show that one can decide in polynomial space if, for a given $k$-player game, $ε\ge 0$ and bound $b$, there exists an $ε$-Nash equilibrium in which all strategies use at most $b$ memory modes. For given $ε>0$, finding an $ε$-Nash equilibrium with respect to $b$-bounded strategies can be done in $FN[NP]$. Similarly for 2-player zero-sum games, finding a $b$-bounded strategy that, against all $b$-bounded opponent strategies, guarantees an outcome within $ε$ of a given value, can be done in $FNP[NP]$. Our constructions apply to parity objectives with minimal simplifications. Our results improve the status quo in several well-known special cases of games. In particular, for $2$-player zero-sum concurrent mean-payoff games, one can approximate ordinary zero-sum values (without restricting admissible strategies) in $FNP[NP]$.

cs.GT↗

arXiv2024

Parity Games on Temporal Graphs

Temporal graphs are a popular modelling mechanism for dynamic complex systems that extend ordinary graphs with discrete time. Simply put, time progresses one unit per step and the availability of edges can change with time. We consider the complexity of solving $ω$-regular games played on temporal graphs where the edge availability is ultimately periodic and fixed a priori. We show that solving parity games on temporal graphs is decidable in PSPACE, only assuming the edge predicate itself is in PSPACE. A matching lower bound already holds for what we call punctual reachability games on static graphs, where one player wants to reach the target at a given, binary encoded, point in time. We further study syntactic restrictions that imply more efficient procedures. In particular, if the edge predicate is in $P$ and is monotonically increasing for one player and decreasing for the other, then the complexity of solving games is only polynomially increased compared to static graphs.

cs.LO↗

arXiv2024

Memoryless Strategies in Stochastic Reachability Games

We study concurrent stochastic reachability games played on finite graphs. Two players, Max and Min, seek respectively to maximize and minimize the probability of reaching a set of target states. We prove that Max has a memoryless strategy that is optimal from all states that have an optimal strategy. Our construction provides an alternative proof of this result by Bordais, Bouyer and Le Roux, and strengthens it, as we allow Max's action sets to be countably infinite.

cs.LO↗

arXiv2023

Handling of Past and Future with Phenesthe+

Writing temporal logic formulae for properties that combine instantaneous events with overlapping temporal phenomena of some duration is difficult in classical temporal logics. To address this issue, in previous work we introduced a new temporal logic with intuitive temporal modalities specifically tailored for the representation of both instantaneous and durative phenomena. We also provided an implementation of a complex event processing system, Phenesthe, based on this logic, that has been applied and tested on a real maritime surveillance scenario. In this work, we extend our temporal logic with two extra modalities to increase its expressive power for handling future formulae. We compare the expressive power of different fragments of our logic with Linear Temporal Logic and dyadic first-order logic. Furthermore, we define correctness criteria for stream processors that use our language. Last but not least, we evaluate empirically the performance of Phenesthe+, our extended implementation, and show that the increased expressive power does not affect efficiency significantly.

cs.LO↗

arXiv2023

History-deterministic Vector Addition Systems

We consider history-determinism, a restricted form of non-determinism, for Vector Addition Systems with States (VASS) when used as acceptors to recognise languages of finite words. History-determinism requires that the non-deterministic choices can be resolved on-the-fly; based on the past and without jeopardising acceptance of any possible continuation of the input word. Our results show that the history-deterministic (HD) VASS sit strictly between deterministic and non-deterministic VASS regardless of the number of counters. We compare the relative expressiveness of HD systems, and closure-properties of the induced language classes, with coverability and reachability semantics, and with and without $\varepsilon$-labelled transitions. Whereas in dimension 1, inclusion and regularity remain decidable, from dimension two onwards, HD-VASS with suitable resolver strategies, are essentially able to simulate 2-counter Minsky machines, leading to several undecidability results: It is undecidable whether a VASS is history-deterministic, or if a language equivalent history-deterministic VASS exists. Checking language inclusion between history-deterministic 2-VASS is also undecidable.

cs.FL↗

arXiv2021

Transience in Countable MDPs

The Transience objective is not to visit any state infinitely often. While this is not possible in finite Markov Decision Process (MDP), it can be satisfied in countably infinite ones, e.g., if the transition graph is acyclic. We prove the following fundamental properties of Transience in countably infinite MDPs. 1. There exist uniformly $ε$-optimal MD strategies (memoryless deterministic) for Transience, even in infinitely branching MDPs. 2. Optimal strategies for Transience need not exist, even if the MDP is finitely branching. However, if an optimal strategy exists then there is also an optimal MD strategy. 3. If an MDP is universally transient (i.e., almost surely transient under all strategies) then many other objectives have a lower strategy complexity than in general MDPs. E.g., $ε$-optimal strategies for Safety and co-Büchi and optimal strategies for $\{0,1,2\}$-Parity (where they exist) can be chosen MD, even if the MDP is infinitely branching.

math.PR↗

arXiv2021

HyperLTL Satisfiability is $Σ_1^1$-complete, HyperCTL* Satisfiability is $Σ_1^2$-complete

Temporal logics for the specification of information-flow properties are able to express relations between multiple executions of a system. The two most important such logics are HyperLTL and HyperCTL*, which generalise LTL and CTL* by trace quantification. It is known that this expressiveness comes at a price, i.e. satisfiability is undecidable for both logics. In this paper we settle the exact complexity of these problems, showing that both are in fact highly undecidable: we prove that HyperLTL satisfiability is $Σ_1^1$-complete and HyperCTL* satisfiability is $Σ_1^2$-complete. These are significant increases over the previously known lower bounds and the first upper bounds. To prove $Σ_1^2$-membership for HyperCTL*, we prove that every satisfiable HyperCTL* sentence has a model that is equinumerous to the continuum, the first upper bound of this kind. We prove this bound to be tight. Finally, we show that the membership problem for every level of the HyperLTL quantifier alternation hierarchy is $Π_1^1$-complete.

cs.LO↗