Papers

Consider a directed graph G and the (generally infinite) set Walks(G) of the walks in that graph. We are interested in ordering Walks(G) as w1, w2,... in a way that ensures that each walk is obtained from the previous one by a bounded number of elementary operations, namely, push operations (which add a new edge at either end of the walk) and pop operations (which remove the first or last edge of the walk). Our goal is to optimize the aforementioned bound, when it exists, in particular when imposing the additional requirement that walks must be ordered in increasing order of length. To this end, we introduce a condition on graphs inspired by earlier work, called inner-connectedness, which is testable in linear time; we show that inner-connected graphs are precisely those which admit bounded walk orderings. Focusing on such graphs, we then introduce and study the pp-rank and the pp-increasing-rank. Namely, the pp-rank of a graph is the best possible bound on the number of push-pop operations in a bounded walk ordering for that graph; and the pp-increasing-rank is the same where we further impose that shorter walks appear before longer walks in the ordering. We show that the pp-rank of inner-connected graphs is in fact always bounded by 3. The pp-increasing-rank of inner-connected graphs is also always finite, but it can be up to linear in the input graph. More precisely, we show that the pp-increasing-rank is related to a new parameter we introduce, the peripherality of graphs. In particular, for cyclic graphs, peripherality is related to the pp-increasing-rank up to multiplicative and additive constants. We last study how to compute these parameters, and show that while peripherality can be computed in polynomial time, the pp-increasing-rank itself is generally NP-hard to compute.

Greibach and Ginsburg (1969) studied abstract families of languages or AFLs. A full AFL is a class of languages closed under the following operations: homomorphism, inverse homomorphism, intersection with regular languages, union, concatenation, and Kleene star. Greibach (1970) used a restricted substitution, called nested iterated substitution, to introduce the notion of super AFLs, which are full AFLs that are further closed under nested iterated substitution. The smallest super AFL was shown to be the class of all context-free languages. From this fact, we focus on Greibach's limited, univariable application of nested iterated substitution and further generalize it into k-variate. This introduces the notion of nested iterated $k$-variate substitution whose definition does not rely on "machines". Recently, a "two-tape" version of Hibbard's (1967) rewriting systems, which were later reformulated by Pighizzini and Pisoni (2014) as k-limited automata with "single" input/work tapes, has been studied under the name of depth-k storage automata. This work unearths a few characteristics of the languages recognized by those automata under a restriction known as frozen blank sensitivity in terms of full AFLs and nested iterated k-variate substitution. This will promote a further abstract treatment of those languages.

Parikh's famous theorem relates to the semi-linearity of context-free languages. We seek for expanding the scope of the semi-linear property from context-free languages to languages beyond them. Of those languages, this work focuses on the languages that are particularly recognized by one-way nondeterministic depth-k storage automata under a natural restriction of frozen blank sensitivity. Those languages (which are called $k$-storable languages) have been already proven to satisfy a natural generalization of the well-known pumping lemmas for regular and context-free languages. The semi-linearity of k-storable languages generalizes Parikh's theorem on the semi-linearity of context-free languages. As for bounded languages, Ginsburg and Spanier's well-known theorem characterizes context-free languages in terms of stratified semi-linearity. This notion is further generalized and is used to characterize the k-storable languages as well. Such a generalization is interesting in its own right. This helps us take an abstract treatment of k-storable languages without relying on "machines". Our results strengthen the belief that those languages are "natural" generalizations of context-free languages.

Jumbled scattered factors are a generalisation of scattered factors (or subsequences) where a jumbled scattered factor is obtained from another word by deleting and rearranging letters. For limiting the number of rearrangements, we distinguish $\ell$-jumbled scattered factors where $\ell$ limits the number of letters that may occur rearranged. For instance, $\mathtt{car}$, $\mathtt{start}$, $\mathtt{rat}, \mathtt{cat}$ are jumbled scattered factors of $\mathtt{abstract}$ even though $\mathtt{car}$, $\mathtt{start}$ and $\mathtt{cat}$ are no scattered factors and each need one jumble. An example for a jumbled scattered factor with $2$ jumbles is $\mathtt{tatar}$. In this work, we investigate two problems on jumbled scattered factors from an algorithmic perspective. First, we give algorithms for testing jumbled Simon's equivalence where two words are equivalent if they share the same $\ell$-jumbled scattered factors up to length $k$. Second, we investigate how to count the possible occurrences of a jumbled scattered factor – the jumbled binomial coefficient. After defining the jumbled binomial coefficient, we study jumbled embeddings and factorisations of such as well.

In this short note, we prove that any closed interval in a well-ordered and discrete space is finite. The coincidence between finiteness and boundedness can then be established for transition systems where the state space is well-ordered and discrete. Finally, this result is also proven when the state space is a Cartesian product of $d$ discrete and well-ordered sets.

We study a formulation of regular language membership as a promise problem, which we dub conditional membership: we fix a promise language $P$ and a target language $T \subseteq P$, we are given a word $w$ which is promised to belong to $P$, and we must determine whether $w$ belongs to $T$. We study the deterministic worst-case query complexity of this task, when the length $n$ of $w$ is known in advance and where each query examines the $i$-th position of $w$ for some $1 \leq i \leq n$ and returns the $i$-th letter $w_i$ of $w$. Our goal is to characterize the asymptotic complexity as a function of the word length $n$, depending on the fixed languages $P$ and $T$. Our main result is to show that the complexity of this task admits a trichotomy between three regimes: $O(1)$ queries, $\Theta(\log n)$ queries, and $\Theta(n)$ queries. We further show that the trichotomy criterion is decidable, namely, it is PSPACE-complete to decide which regime applies, given nondeterministic automata $A_P$ and $A_T$ respectively representing $P$ and $T$.

Using the formalism of presheaf automata, we introduce new classes of automata for modelling concurrency that extend higher-dimensional automata and symmetric higher-dimensional automata. We study their languages and characterise these languages in terms of posetal cospans.

We prove that reachability for constraint automata over finite binary words is undecidable. Eight registers suffice, with conjunctive guards using only equality, proper prefix, proper suffix, and infix incomparability, without word constants or concatenation. Finite-run nonemptiness, Buechi nonemptiness, and bounded Buechi nonemptiness are undecidable as consequences. For Boolean guards over equality, prefix, suffix, and prefix and suffix incomparability, Buechi nonemptiness is PSPACE-complete with arbitrarily many registers and word constants. The upper bound also allows positive infix atoms and yields PSPACE-completeness of prefix–suffix LTL satisfiability.

In 2022 Dassow and Jecker started investigations on the operational complexity of the pumping constant w.r.t.~two pumping lemmata. Since then, the research on pumping lemmata emerged to several variants for regular as well as context-free languages. Within this work, we further investigate the witnesses of regular languages (firstly introduced by Rauch) w.r.t.~different pumping lemmata: For a language $L$ and a minimal pumping constant $K$, a witness of a language is a word that cannot be pumped by $K$ w.r.t.~the given pumping lemma. Here, the pumping constant is minimal if for all smaller numbers, the pumping lemma condition is not satisfied. Alongside combinatorial investigations on (unary) witness languages, we introduce and study the measure of scattered factor distance, which is related to minimal pumping constants. We present bounds on minimal pumping constants by the scattered factor distance and show that deciding, given two regular languages, whether one of them is equal to the witness language of the second is \coNP-complete.

Rabin games have their roots in Michael Rabin's seminal work on automata over infinite trees and form a classical model for infinite-round games with ω-regular winning conditions. They play a central role in automata theory and temporal synthesis: translating an LTL specification into an equivalent deterministic Rabin automaton reduces synthesis to solving a Rabin game. In practice, however, Rabin games are typically solved indirectly, via parity games. One possible reason is that many direct algorithms have a factorial dependence on the Rabin index k, the number of Rabin pairs. In this work, we revisit Rabin's 1972 recursive algorithm. The algorithm is implicit in Rabin's proof and intertwined with finite-state determinacy. First, we "excavate" it as an explicit arena-based solver using standard reachability- and Büchi-game solving as basic building blocks, thus avoiding the exponential dependence on the alphabet in Rabin's original presentation. Second, we show that memoization removes the factorial term from the recursion-tree analysis. Finally, we introduce a variation that recurses on the Rabin index. These results provide a modern analysis of Rabin's recursive method and a basis for studying direct Rabin-game solvers for temporal synthesis.

We present asymptotically optimal translations from linear temporal logic with past to limit-deterministic Buchi automata and onwards to deterministic parity automata. In contrast to previously known translations involving past, the ones presented here are incremental: states and transitions can be constructed as needed on-the-fly, and they do not involve backtracking in order to correct for temporal inconsistencies. The past is naturally encoded in the formula by alternating between weak and strong versions of past operators. To handle the difficulties that arise from the alternation of least and greatest fix-point operators (such as between the "until" operator and its weak counterpart), we rewrite the given formula based on the infinitary behavior of a given word with respect to subformulae rooted with such operators.

Timing information leakage occurs whenever an attacker successfully deduces confidential internal information by observing some timed information such as the total duration of a run. Timed automata (TAs) are an extension of finite-state automata with a set of clocks evolving at the same rate and that can be tested or reset. Hence, this formalism is able to represent systems involving timing constraints reacting to the passage of time. Some of the transitions in a timed automaton may be controlled, i.e., enabled or disabled, to limit the runs that are allowed. Execution-time opacity (ET-opacity) aims at studying timing information leakage through the analysis of timed automata. More precisely, the ET-opacity problem consists in deciding whether an attacker can deduce whether some secret locations were visited by a run, solely from the run's duration. When control is allowed, the goal becomes to find a strategy that blocks the runs leaking information. This problem has been shown decidable for TAs, with and without control. However, these existing results only provide upper bounds on the complexity of the ET-opacity problem, sometimes largely above the tight bound. Our main contribution is to provide tight lower bounds for those two results. Moreover, for controllable TAs, two variants of ET-opacity (depending on whether detecting if a run did not visit a secret location represents an information leakage or not, and named full and weak ET-opacity respectively) had been separately studied; we prove here that these two notions are inter-reducible.

For every $n\ge22$, we construct a language over a fixed three-letter alphabet whose two-way deterministic state complexity is $n$ and whose complement has state complexity $2n$. Acceptance is at the left endmarker, and infinite computations reject. The lower bound holds for arbitrary deterministic recognizers. The proof uses the group of transformations induced by the witness to constrain the behavior of a smaller recognizer, then derives a contradiction from a permutation query problem. A matching halting complement follows from a $2n$-state construction for sources whose destination states determine the direction of each transition. Splitting states by incoming direction gives a $2n+2b$ upper bound for general sources, where $b$ counts states entered in both directions. In the acceptance-anywhere model of Geffert, Mereghetti, and Pighizzini, the same family has exact complexities $N$ and $2N-1$ for every $N\ge23$.

Twisted equations in free groups are equations in which a finite group of symmetries acts on the free group by automorphisms and anti-automorphisms, such as inversion and reversal; moreover, the variables may be subject to rational constraints. Algorithms from the theory of equations in virtually free groups solve such equations in PSPACE, but they are intricate. Our main result is an elementary algorithm for quadratic twisted equations, which extends the algorithm of Comerford and Edmunds (1981) for quadratic equations in free groups to the twisted setting and to rational constraints. Besides deciding solvability in PSPACE and yielding EDT0L descriptions of solution sets, our algorithm computes, for every quadratic expression, finitely many parametric forms describing its values as freely reduced words. As an application, we study the palindrome length of elements of free groups, that is, the least number of palindromes whose product they are. Guided by the parametric forms that our algorithm computes for products of palindromes, we found, and then proved directly, a characterization of the freely reduced words of palindrome length at most $n$ in terms of patterns describing how palindromes are nested by twisted conjugations, thereby confirming a Catalan structure conjectured by the second author. Consequently, these words form a context-free language, and palindrome length is computable in cubic time as well as in LOGCFL, answering a question of Bardakov, Shpilrain, and Tolstykh (2005).

We study concurrent stochastic games in which a team of agents plays against an adversarial environment. If the agents could access a shared source of randomness, then these games reduce to two-player games. Despite being common in the literature on coalition games, this assumption does not accurately model real-world systems. In this paper, we consider the setting where the agents only have access to individual sources of randomness, and must satisfy a given reachability or safety objective. The limit-sure problem asks whether the team can ensure that the objective is satisfied with probability at least $1-\lambda$ for every $\lambda>0$. For reachability objectives, we show that this problem is $\mathsf{NP}$-complete. For safety objectives, we show that limit-sure winning coincides with almost-sure winning, which is known to lie in $\mathsf{P}$. As a corollary of these results, we conclude that model checking of the qualitative fragment of the logic Individually Randomised ATL – introduced in the recent work of Brice et al. – is $\mathsf{P}^{\mathsf{NP}}$-complete.

We classify prime accessible deterministic automata over finite alphabets, both among finite automata and among automata with arbitrary state sets. Every one-state automaton is prime. Over a one-letter alphabet, a finite nontrivial automaton is prime exactly when it is a cycle of prime-power length or a finite chain ending in a fixed state. The unique infinite accessible unary automaton, up to isomorphism, is also prime. Every nontrivial automaton over an alphabet with at least two letters is not prime. Shen proved the finite case; we give a shorter proof. We also characterize a decomposition of the automaton whose states are all finite input words. The reachable product of the quotients by the least right congruences identifying the empty word with nonempty words $p$ and $q$, respectively, is isomorphic to this automaton exactly when $p^r\ne q^s$ for all positive integers $r,s$.

Many network protocols and distributed systems combine recursion, unbounded local state, and quantitative behaviour such as cost, latency, or reliability. Existing decidability results for networks of pushdown systems are largely qualitative, and do not extend to quantitative analyses, which must jointly track traces and weights. We present a framework for the quantitative verification of acyclic networks of weighted pushdown systems. Finitely summarising a recursive network component requires collapsing the infinite family of runs obtained by pumping its nested loops, and doing so exactly, rather than by over-approximation, is a standing difficulty. We give a class of weight domains where this is possible: pumping semirings, those in which the weights accumulated by such a family collapse to a closed form; informally, the domain must be unable to count iterations. The class admits domains with infinite ascending chains, such as the arctic semiring and downward-closed languages. Over these, we give a terminating saturation algorithm computing quantitative reachability exactly, resting on two ideas: segment tree algebras, a compositional representation of runs, and thermal extensions, which symbolically separate accelerated weights so that further acceleration remains exact. We use this to compute upward and downward closures of context-free languages uniformly, and to lift the algorithm to acyclic networks over "thin" pumping semirings, yielding the first quantitative safety and reachability analyses for networks of pushdown systems.

If the machines of some type A have enough resources to (i) solve problems that no machine of type B can solve, and (ii) simulate any unary two-way deterministic finite automaton that has access to a linearly-bounded counter, then typically no recursive function can upper bound the increase in the size of description when a machine of type A is replaced by an equivalent machine of type B.

We prove that no recursive function can upper bound the increase in the size of description when a two-way deterministic finite automaton with k+1 heads is replaced by an equivalent two-way deterministic finite automaton with k heads. This is true for all k, and remains true if the automata are unary and/or nondeterministic.

Thompson's classical construction turns every regular expression into an equivalent epsilon-NFA whose transition graph is series-parallel. The construction uses epsilon-transitions, and eliminating them by the usual shortcut construction need not preserve planarity or the bound of two on the treewidth of the transition graph. Book and Chandra proved that every regular language nevertheless has an epsilon-free NFA with a planar transition graph. We show that the transition graph can in fact be chosen to be a bipartite cactus: every cycle is even, and any two cycles have at most one vertex in common. In particular, it is outerplanar and has treewidth at most two. This is a considerably more simple structure than the general series-parallel guarantee supplied by Thompson's construction. The proof is a direct induction on parity-typed regular expressions and uses only elementary closure properties of cactus graphs.

We introduce and study pattern-based tree transformations. As an illustrating example, consider a source pattern $(x \cdot y) + (x \cdot z)$ and a target pattern $x \cdot (y + z)$ as a pair. This source pattern matches any expression $e$ of the form $(e_1 \cdot e_2) + (e_1 \cdot e_3)$ (by substituting $x$ with $e_1$, $y$ with $e_2$, and $z$ with $e_3$) and the pair transforms it into the expression $e_1 \cdot (e_2 + e_3)$ as dictated by the target pattern. Note that in this example, the set of expressions that match the source pattern is not a regular tree language. We propose a model of tree transformations given by a finite representation of a (possibly infinite) set of such (source pattern, target pattern) pairs. The expressive power of this model comes at the cost of undecidability of checking equivalence. Nevertheless, we show that the type-checking problem is decidable for our model of pattern-based tree transformations. The type-checking problem asks whether applying a given transformation to trees having a given regular property (type) preserves the property. Our decision procedure is by a reduction to the emptiness problem of alternating tree automata.

We prove that topological conjugacy of elements of the Brin–Thompson group $nV$ is undecidable for every $n\geq2$. From a Turing machine $T$, the reduction produces two elements of $nV$, which are conjugate by an involution in $nV$ if $T$ halts on the empty tape. If $T$ does not halt on the empty tape, the topological dynamics of the two group elements are distinguished by their visits to minimal subsystems of the set of (germ-)aperiodic points. In these groups, and any finitely-generated supergroups inside the homeomorphism group of Cantor space, we also obtain undecidability of the Whitehead problem (the problem of being in the same automorphism orbit), and undecidability of group-theoretic conjugacy.

We prove that some pairs of binary words of every sufficiently large prescribed length $n$ require $\Omega(\log n\log\log n)$ states to separate by a deterministic finite automaton whose letters act as permutations. We obtain this bound by constructing distinct equal-length positive binary words that form an identity of $\mathfrak{S}_k$ of length $\exp(O(k/\log k))$. This improves the previous lower bound $(3/2-o(1))\log n$ of Bulatov, Karpova, Shur, and Startsev (arXiv:1609.03199). The proof combines a positive-word construction with a cover of permutation orders. Consecutive segments of prime powers reduce the covering problem to enumerating sets of positive integers with bounded sum. The required estimates use the central binomial coefficient and the classical asymptotic formula for partitions into distinct parts. An appendix gives sharper constants and lower-order terms using a greedy cover and a quantitative prime number theorem.

String attractors provide a compact way of representing the complete factor structure of a word: a set of positions is an attractor if every distinct factor has at least one occurrence crossing one of the selected positions. Although the minimum attractor size is known for several classical families of words, describing all optimal attractors is typically much more difficult, since it requires understanding the geometry of all factor occurrences rather than constructing a single optimal solution. We give a complete description for the finite Thue–Morse words. Earlier work proved that four positions are necessary and sufficient for every order $n\geq 4$, but the collection of all smallest attractors remained unknown. For every $n\geq 6$, writing $h=2^{n-3}$, we prove that the smallest attractors are exactly the two reflected families where the four offsets are chosen independently from $\{0,1\}$. Hence there are exactly $32$ smallest attractors for every $n\geq6$. The initial cases are genuinely exceptional: $t_5$ has $40$ smallest attractors and $t_4$ has $87$. We also study the attractor condition independently of optimality. For every $n\geq5$, we characterize the complete antichain of inclusion-minimal factor coverages of $t_n$. It consists precisely of the coverages of $aa$, $bb$, and the eight minimal unique substrings of every generation $t_m$, $4\leq m\leq n$. Thus there are exactly $8n-22$ canonical constraints, forming an irredundant exact certificate for attractors of arbitrary cardinality. When attention is restricted to four-position sets, this linear-size system collapses to a constant one: it is enough to test the $24$ minimal unique substrings coming from three consecutive generations, and sixteen of these already force the two optimal families. We also show that three generations are necessary within this natural consecutive-generation hierarchy.

We show that the Turing complete rule 110 in elementary cellular automata can be emulated by alternating three simple symmetric rules in space. As the constituent rules themselves are trivial, this shows that diversity alone can generate emergent phenomena. The rules cannot distinguish left from right, and the chirality of rule 110 is created by their arrangement. Random initial conditions lock on to rule 110 over 90% of the time, explained through the motion of domain walls between desynchronized regions. Adding random noise creates errors that seed domain walls, whose fraction scales with the square root of noise strength, explained through a simple rate equation. This points to a route to complexity through repeated, perturbed units, common in biology, and shows that in cellular automaton models of quantum field theory, broken symmetries such as parity need not be present in the local rules.

This volume contains the proceedings of GandALF 2026, the Seventeenth International Symposium on Games, Automata, Logics, and Formal Verification. The symposium was held in Aalborg, Denmark, on September 15-17, 2026. The aim of GandALF 2026 is to bring together researchers from academia and industry who are actively working in the fields of Games, Automata, Logics, and Formal Verification. The idea is to cover an ample spectrum of themes, ranging from theory to applications, and stimulate cross-fertilisation.

Ambiguity measures the number of accepting runs in nondeterministic finite automata (NFA). We consider finitely ambiguous NFA, where there exists a constant $N$ such that over every word $w$ there are at most $N$ accepting runs. In such a case we also say that the NFA is $N$-ambiguous. Importantly $N$ depends only on the NFA, it does not depend on the length of the word. Weber and Seidl showed that every NFA is $N$-ambiguous for $N = 2^{O(n \log n)}$, where $n$ is the number of states. We improve this to $N = 2^{O(n)}$, which is asymptotically tight.

We investigate the parameterized reachability problem for concurrent register machines over infinite data domains. In this framework, each machine is a program equipped with a set of local registers, the communication across machines is mediated through a set of shared registers. Both local and shared registers can take values from an infinite data domain. The program's primitive operations include copying values between registers, assigning constants, comparing registers for (dis-)equality, and nondeterministic assignments that store an arbitrary domain value into a local register. The parameterized reachability problem considers a program and a target location, asking whether there exists some n in Naturals such that an execution of n identical machines (referred to as instances) results in at least one instance reaching the specified location. We show that this problem is Pspace-complete in the general case and it becomes undecidable if a freshness assumption (i.e., each assignment must produce a unique value distinct from all constants) is applied to nondeterministic assignments. This undecidability persists even for systems restricted to two shared and two local registers. Finally, we establish optimal decidability results for two restricted settings: when each thread is limited to a single local register, or when the system utilizes only one shared register.

Experiments on binary run-length differentiability lead to sharp computer-assisted criteria for shuffle squares. Every $C^3$-word of length greater than $16$ is a shuffle square exactly when both letter multiplicities are even. All $34$ nonempty even-Parikh exceptions are smooth and persist in every higher differentiability class. For $C^2$ the sharp threshold is $48$, with $212$ nonempty exceptions. At level $C^1$ no global parity threshold exists, but every even-Parikh non-shuffle-square of length at least $36$ has proper nonempty shuffle-square prefixes and suffixes. Exactly $230$ nonempty even-Parikh $C^1$-words have no nonempty shuffle-square prefix. The full tree avoiding such prefixes eventually consists of $422$ periodic rays. Consequently, a nonempty Kolakoski prefix is a shuffle square exactly when both multiplicities are even, apart from lengths $4$ and $8$. We also characterize classes of morphisms reflecting shuffle squares. For doubly binary words, we determine the exact deletion distance and largest twins, and prove sharp bounds for single local repairs. Exact recurrences, residual-state checks, and separate Python programs make the finite computations reproducible.

We introduce and investigate the irreducibility of endomorphisms of finitely generated free semigroups, i.e., we investigate when an endomorphism $\phi: \Sigma^+ -> \Sigma^+$, where $\Sigma$ is any alphabet, can be nontrivially expressed as a composition $\phi = \psi_2 \circ \psi_1$ of endomorphisms $\psi_1, \psi_2: \Sigma^+ -> \Sigma^+$. We, hence, study a notion of primality in the endomorphism monoid of the free semigroup—a natural and fundamental concept in this algebraic structure. We establish that irreducibility is a nontrivial property for the class of so-called rank-preserving endomorphisms, and we provide a characteristic condition separating the reducible and irreducible endomorphisms. We also characterise when an endomorphism is a factor of another endomorphism, analyse the non-uniqueness of factorisations of a rank-preserving endomorphism into its irreducible components, and investigate the use of incidence matrices to give insights into the (ir-)reducibility of rank-preserving endomorphisms.