arXiv subjects

Stefan Kiefer

Publications and source records attributed to Stefan Kiefer.

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

Perturbation equivalence in labelled Markov chains

Behavioural equivalences, such as language equivalence and probabilistic bisimilarity, are fundamental techniques for reducing the size of probabilistic models. However, these equivalences are sensitive to the precise values of transition probabilities, making them unsuitable in applications where probabilities are subject to approximation. Motivated by settings in which the support graph of a labelled Markov chain is known but the transition probabilities are uncertain, we study robust variants of these equivalences. We introduce universal (perturbation) equivalence, which captures a variant of equivalence that is resilient to all perturbations of transition probabilities: two states or distributions are universally equivalent if they remain equivalent under every assignment of transition probabilities consistent with the support graph. We also consider the dual notion of existential (perturbation) equivalence, which holds whenever there exists an assignment of transition probabilities that yields equivalence. We establish that, for states, universal language equivalence coincides with universal probabilistic bisimilarity and develop a characterisation that yields a polynomial-time partition refinement algorithm. We implement the algorithm and demonstrate experimentally that it is effective as a technique for robust model reduction. We further show that universal language equivalence for distributions is closely related to the state case, and prove NL-completeness of deciding universal equivalence for both states and distributions. We prove that, for states, existential language equivalence coincides with existential probabilistic bisimilarity and deterministic witness transition functions always suffice, leading to an NP-completeness result. In contrast, we show that existential language equivalence for distributions is complete for the existential theory of the reals.

cs.LO↗

arXiv2026

Spectral and combinatorial methods for efficiently computing the rank of unambiguous finite automata

A zero-one matrix is a matrix with entries from $\{0, 1\}$. We study monoids containing only such matrices. A finite set of zero-one matrices generating such a monoid can be seen as the matrix representation of an unambiguous finite automaton, an important generalisation of deterministic finite automata which shares many of their good properties. Let $\mathcal{A}$ be a finite set of $n \times n$ zero-one matrices generating a monoid of zero-one matrices, and $m$ be the cardinality of $\mathcal{A}$. We study the computational complexity of computing the minimum rank of a matrix in the monoid generated by $\mathcal{A}$. By using linear-algebraic techniques, we show that this problem is in $\textsf{NC}$ and can be solved in $\mathcal{O}(mn^4)$ time and $\mathcal{O}(n^2)$ space. We also provide a combinatorial algorithm finding a matrix of minimum rank in $\mathcal{O}(mn^4)$ time and $\mathcal{O}(n^3)$ space. As a byproduct, we show a very weak version of a generalisation of the Černý conjecture: there always exists a straight line program of size $\mathcal{O}(n^2)$ describing a product resulting in a matrix of minimum rank. For the special case corresponding to total DFAs (that is, for the case where all matrices have exactly one 1 in each row), the minimum rank is the size of the smallest image of the set of all states under the action of a word. Our combinatorial algorithm finds a matrix of minimum rank in time $\mathcal{O}(n^3 + mn^2)$ in this case.

cs.FL↗

arXiv2026

On the Continuity of the Probabilistic Bisimilarity Distance

The probabilistic bisimilarity distance provides a quantitative measure of behavioural difference for labelled Markov chains, but it may be discontinuous under perturbations of the transition probabilities. This lack of continuity undermines its applicability to empirically derived models, where transition probabilities are often approximations. Recently, we (CAV 2025) introduced robust probabilistic bisimilarity as a sufficient condition for continuity at distance zero. In this paper, we show that it is also a necessary condition, that is, two states are robustly probabilistic bisimilar if and only if their probabilistic bisimilarity distance is small for any small enough perturbation of the transition probabilities. We further extend robustness to non-bisimilar state pairs to establish a complete characterization for continuity of the probabilistic bisimilarity distance. Based on this characterization, we develop a polynomial time algorithm to decide continuity. Finally, we complement our theoretical contributions with an experimental evaluation demonstrating the proposed approach in practice. Our results show that the extra step of deciding continuity requires minimal additional cost when compared to computing the probabilistic bisimilarity distance.

cs.LO↗

arXiv2026

Robust Probabilistic Bisimilarity for Labelled Markov Chains

Despite its prevalence, probabilistic bisimilarity suffers from a lack of robustness under minuscule perturbations of the transition probabilities. This can lead to discontinuities in the probabilistic bisimilarity distance function, undermining its reliability in practical applications where transition probabilities are often approximations derived from experimental data. Motivated by this limitation, we introduce the notion of robust probabilistic bisimilarity for labelled Markov chains, which ensures the continuity of the probabilistic bisimilarity distance function. We also propose an efficient algorithm for computing robust probabilistic bisimilarity and show that it performs well in practice, as evidenced by our experimental results.

cs.LO↗

arXiv2026

The asymptotic size of finite irreducible semigroups of rational matrices

In this paper we investigate the maximum size of finite semigroups of rational $n \times n$ matrices, with the goal of shedding more light on their structure. Such semigroups provide a rich generalisation of transition monoids of unambiguous (and, in particular, deterministic) finite automata. While in general such semigroups can be arbitrarily large in terms of $n$, a classical result of Schützenberger from 1962 implies an upper bound of $2^{O(n^2 \log n)}$ for irreducible semigroups. A semigroup of rational matrices is called irreducible if the only subspaces of $\mathbb{Q}^n$ that are invariant for all matrices in the semigroup are $\mathbb{Q}^n$ and the subspace consisting only of the zero vector. Irreducible matrix semigroups can be viewed as the building blocks of general matrix semigroups, and as such play an important role in mathematics and computer science. From the point of view of automata theory, they can be seen as a generalisation of strongly connected weighted automata. Using a very different technique from that of Schützenberger, we improve the upper bound on the cardinality to $3^{n^2}$. This is the main result of the paper. The bound is in some sense tight, as we show that there exists, for every $n$, a finite irreducible semigroup with $3^{\lfloor n^2/4 \rfloor}$ rational matrices. Our main result also leads to an improvement of a bound, due to Almeida and Steinberg, on the mortality threshold of finite semigroups of rational matrices. The mortality threshold is a number $\ell$ such that if the zero matrix is in the semigroup, then the zero matrix can be written as a product of at most $\ell$ matrices from any subset that generates the semigroup.

cs.FL↗

arXiv2025

The complexity of reachability problems in strongly connected finite automata

Several reachability problems in finite automata, such as completeness of NFAs and synchronisation of total DFAs, correspond to fundamental properties of sets of nonnegative matrices. In particular, the two mentioned properties correspond to matrix mortality and ergodicity, which ask whether there exists a product of the input matrices that is equal to, respectively, the zero matrix and a matrix with a column of strictly positive entries only. The case where the input automaton is strongly connected (that is, the corresponding set of nonnegative matrices is irreducible) frequently appears in applications and often admits better properties than the general case. In this paper, we address the existence of such properties from the computational complexity point of view, and develop a versatile technique to show that several NL-complete problems remain NL-complete in the strongly connected case. In particular, we show that deciding if a binary total DFA is synchronising is NL-complete even if it is promised to be strongly connected, and that deciding completeness of a binary unambiguous NFA with very limited nondeterminism is NL-complete under the same promise.

cs.FL↗

arXiv2025

The complexity of computing the period and the exponent of a digraph

The period of a strongly connected digraph is the greatest common divisor of the lengths of all its cycles. The period of a digraph is the least common multiple of the periods of its strongly connected components. These notions play an important role in the theory of Markov chains and the analysis of powers of nonnegative matrices. While the time complexity of computing the period is well-understood, little is known about its space complexity. We show that the problem of computing the period of a digraph is NL-complete, even if all its cycles are contained in the same strongly connected component. However, if the digraph is strongly connected, we show that this problem becomes L-complete. For primitive digraphs (that is, strongly connected digraphs of period one), there always exists a number $m$ such that there is a path of length exactly $m$ between every two vertices. We show that computing the smallest such $m$, called the exponent of a digraph, is NL-complete. The exponent of a primitive digraph is a particular case of the index of convergence of a nonnegative matrix, which we also show to be computable in NL, and thus NL-complete.

cs.DM↗

arXiv2024

Efficient Analysis of Unambiguous Automata Using Matrix Semigroup Techniques

We introduce a novel technique to analyse unambiguous Büchi automata quantitatively, and apply this to the model checking problem. It is based on linear-algebra arguments that originate from the analysis of matrix semigroups with constant spectral radius. This method can replace a combinatorial procedure that dominates the computational complexity of the existing procedure by Baier et al. We analyse the complexity in detail, showing that, in terms of the set $Q$ of states of the automaton, the new algorithm runs in time $O(|Q|^4)$, improving on an efficient implementation of the combinatorial algorithm by a factor of $|Q|$.

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

Minimising the Probabilistic Bisimilarity Distance

A labelled Markov decision process (MDP) is a labelled Markov chain with nondeterminism; i.e., together with a strategy a labelled MDP induces a labelled Markov chain. The model is related to interval Markov chains. Motivated by applications to the verification of probabilistic noninterference in security, we study problems of minimising probabilistic bisimilarity distances of labelled MDPs, in particular, whether there exist strategies such that the probabilistic bisimilarity distance between the induced labelled Markov chains is less than a given rational number, both for memoryless strategies and general strategies. We show that the distance minimisation problem is ExTh(R)-complete for memoryless strategies and undecidable for general strategies. We also study the computational complexity of the qualitative problem about making the distance less than one. This problem is known to be NP-complete for memoryless strategies. We show that it is EXPTIME-complete for general strategies.

cs.FL↗

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

Markov Chains and Unambiguous Automata

Unambiguous automata are nondeterministic automata in which every word has at most one accepting run. In this paper we give a polynomial-time algorithm for model checking discrete-time Markov chains against ω-regular specifications represented as unambiguous automata. We furthermore show that the complexity of this model checking problem lies in NC: the subclass of P comprising those problems solvable in poly-logarithmic parallel time. These complexity bounds match the known bounds for model checking Markov chains against specifications given as deterministic automata, notwithstanding the fact that unambiguous automata can be exponentially more succinct than deterministic automata. We report on an implementation of our procedure, including an experiment in which the implementation is used to model check LTL formulas on Markov chains.

cs.LO↗

arXiv2023

On the Sequential Probability Ratio Test in Hidden Markov Models

We consider the Sequential Probability Ratio Test applied to Hidden Markov Models. Given two Hidden Markov Models and a sequence of observations generated by one of them, the Sequential Probability Ratio Test attempts to decide which model produced the sequence. We show relationships between the execution time of such an algorithm and Lyapunov exponents of random matrix systems. Further, we give complexity results about the execution time taken by the Sequential Probability Ratio Test.

math.PR↗

arXiv2022

Image-Binary Automata

We introduce a certain restriction of weighted automata over the rationals, called image-binary automata. We show that such automata accept the regular languages, can be exponentially more succinct than corresponding NFAs, and allow for polynomial complementation, union, and intersection. This compares favourably with unambiguous automata whose complementation requires a superpolynomial state blowup. We also study an infinite-word version, image-binary Büchi automata. We show that such automata are amenable to probabilistic model checking, similarly to unambiguous Büchi automata. We provide algorithms to translate $k$-ambiguous Büchi automata to image-binary Büchi automata, leading to model-checking algorithms with optimal computational complexity.

cs.FL↗

arXiv2022

On Complementing Unambiguous Automata and Graphs With Many Cliques and Cocliques

We show that for any unambiguous finite automaton with $n$ states there exists an unambiguous finite automaton with $\sqrt{n+1} \cdot 2^{n/2}$ states that recognizes the complement language. This builds and improves upon a similar result by Jirásek et al. [Int. J. Found. Comput. Sci. 29 (5) (2018)]. Our improvement is based on a reduction to and an analysis of a problem from extremal graph theory: we show that for any graph with $n$ vertices, the product of the number of its cliques with the number of its cocliques (independent sets) is bounded by $(n+1) 2^n$.

cs.FL↗

arXiv2022

The Big-O Problem

Given two weighted automata, we consider the problem of whether one is big-O of the other, i.e., if the weight of every finite word in the first is not greater than some constant multiple of the weight in the second. We show that the problem is undecidable, even for the instantiation of weighted automata as labelled Markov chains. Moreover, even when it is known that one weighted automaton is big-O of another, the problem of finding or approximating the associated constant is also undecidable. Our positive results show that the big-O problem is polynomial-time solvable for unambiguous automata, coNP-complete for unlabelled weighted automata (i.e., when the alphabet is a single character) and decidable, subject to Schanuel's conjecture, when the language is bounded (i.e., a subset of $w_1^*\dots w_m^*$ for some finite words $w_1,\dots,w_m$) or when the automaton has finite ambiguity. On labelled Markov chains, the problem can be restated as a ratio total variation distance, which, instead of finding the maximum difference between the probabilities of any two events, finds the maximum ratio between the probabilities of any two events. The problem is related to $\varepsilon$-differential privacy, for which the optimal constant of the big-O notation is exactly $\exp(\varepsilon)$.

cs.FL↗

arXiv2022

Lower Bounds for Unambiguous Automata via Communication Complexity

We use results from communication complexity, both new and old ones, to prove lower bounds for unambiguous finite automata (UFAs). We show three results. $\textit{Complement:}$ There is a language $L$ recognised by an $n$-state UFA such that the complement language $\overline{L}$ requires NFAs with $n^{\tildeΩ(\log n)}$ states. This improves on a lower bound by Raskin. $\textit{Union:}$ There are languages $L_1$, $L_2$ recognised by $n$-state UFAs such that the union $L_1\cup L_2$ requires UFAs with $n^{\tildeΩ(\log n)}$ states. $\textit{Separation:}$ There is a language $L$ such that both $L$ and $\overline{L}$ are recognised by $n$-state NFAs but such that $L$ requires UFAs with $n^{Ω(\log n)}$ states. This refutes a conjecture by Colcombet.

cs.FL↗