arXiv subjects

Karoliina Lehtinen

Publications and source records attributed to Karoliina Lehtinen.

At least 19 recordsLinked to original sources

arXiv2025

Explorability in Pushdown Automata

We study explorability, a measure of nondeterminism in pushdown automata, which generalises history-determinism. An automaton is k-explorable if, while reading the input, it suffices to follow k concurrent runs, built step-by-step based only on the input seen so far, to construct an accepting one, if it exists. We show that the class of explorable PDAs lies strictly between history-deterministic and fully nondeterministic PDAs in terms of both expressiveness and succinctness. In fact increasing explorability induces an infinite hierarchy: each level k defines a strictly more expressive class than level k-1, yet the entire class remains less expressive than general nondeterministic PDAs. We then introduce a parameterized notion of explorability, where the number of runs may depend on input length, and show that exponential explorability precisely captures the context-free languages. Finally, we prove that explorable PDAs can be doubly exponentially more succinct than history-deterministic ones, and that the succinctness gap between deterministic and 2-explorable PDAs is not recursively enumerable. These results position explorability as a robust and operationally meaningful measure of nondeterminism for pushdown systems.

cs.FL↗

arXiv2025

Monitorability for the Modal mu-Calculus over Systems with Data: From Practice to Theory

Runtime verification, also known as runtime monitoring, consists of checking whether a system satisfies a given specification by observing the trace it produces during its execution. It is used as a lightweight verification technique to complement or substitute costlier methods such as model-checking. In the regular setting, Hennessy-Milner logic with recursion, a variant of the modal mu-calculus, provides a versatile formalism for expressing linear- and branching-time specifications of the control flow of the system. In this paper, we shift the focus from control to data and study the monitorability of an extension of this logic that allows one to express properties of the data flow. Data values are modelled as values from an infinite domain. They are stored using data variables and manipulated using predicates and first-order quantification. The resulting logic is closely related to register automata with guessing. This correspondence yields a monitor synthesis algorithm, and allows us to derive a strict monitorability hierarchy between the different fragments of the logic, in stark contrast to the regular setting. In particular, restricting to deterministic monitors strictly reduces the set of monitorable properties. Last, we exhibit a fragment of the logic that can express all monitorable formulae in the logic without greatest fixed-points but not in the full logic. We finally show that this is unavoidable because, in fact, there is no decidable fragment of the logic that captures all monitorable properties.

cs.LO↗

arXiv2025

History-deterministic Parikh Automata

Parikh automata extend finite automata by counters that can be tested for membership in a semilinear set, but only at the end of a run. Thereby, they preserve many of the desirable properties of finite automata. Deterministic Parikh automata are strictly weaker than nondeterministic ones, but enjoy better closure and algorithmic properties. This state of affairs motivates the study of intermediate forms of nondeterminism. Here, we investigate history-deterministic Parikh automata, i.e., automata whose nondeterminism can be resolved on the fly. This restricted form of nondeterminism is well-suited for applications which classically call for determinism, e.g., solving games and composition. We show that history-deterministic Parikh automata are strictly more expressive than deterministic ones, incomparable to unambiguous ones, and enjoy almost all of the closure properties of deterministic automata. Finally, we investigate the complexity of resolving nondeterminism in history-deterministic Parikh automata.

cs.FL↗

arXiv2025

Using games and universal trees to characterise the nondeterministic index of tree languages

The parity index problem of tree automata asks, given a regular tree language $L$ and a set of priorities $J$, is $L$ $J$-feasible, that is, recognised by a nondeterministic parity automaton with priorities $J$? This is a long-standing open problem, of which only a few sub-cases and variations are known to be decidable. In a significant but technically difficult step, Colcombet and Löding reduced the problem to the uniform universality of distance-parity automata. In this article, we revisit the index problem using tools from the parity game literature. We add some counters to Lehtinen's register game, originally used to solve parity games in quasipolynomial time, and use this novel game to characterise $J$-feasibility. This provides a alternative proof to Colcombet and Löding's reduction. We then provide a second characterisation, based on the notion of attractor decompositions and the complexity of their structure, as measured by a parameterised version of their Strahler number, which we call $n$-Strahler number. Finally, we rephrase this result using the notion of universal tree extended to automata: a guidable automaton recognises a $[1,2j]$-feasible language if and only if it admits a universal tree with $n$-Strahler number $j$, for some $n$. In particular, a language recognised by a guidable automaton $A$ is Büchi-feasible if and only if there is a uniform bound $n\in \mathbb{N}$ such that all trees in the language admit an accepting run with an attractor decomposition of width bounded by $n$, or, equivalently, if and only $A$ admits a \textit{finite} universal tree. While we do not solve the decidability of the index problem, our work makes the state-of-the-art more accessible and brings to light the deep relationships between the $J$-feasibility of a language and attractor decompositions, universal trees and Lehtinen's register game.

cs.FL↗

arXiv2025

The 2-Token Theorem: Recognising History-Deterministic Parity Automata Efficiently

History-determinism is a restricted notion of nondeterminism in automata, where the nondeterminism can be successfully resolved based solely on the prefix read so far. History-deterministic automata still allow for exponential succinctness in automata over infinite words compared to deterministic automata (Kuperberg and Skrzypczak, 2015), allow for canonical forms unlike deterministic automata (Abu Radi and Kupferman, 2019 and 2020; Ehlers and Schewe, 2022), and retain some of the algorithmic properties of deterministic automata, for example for reactive synthesis (Henzinger and Piterman, 2006; Ehlers and Khalimov, 2024). Despite the topic of history-determinism having received a lot of attention over the last decade, the complexity of deciding whether a parity automaton is history-deterministic has, up till now, remained open. We show that history-determinism for a parity automaton with a fixed parity index can be checked in PTIME, thus improving upon the naive EXPTIME upper bound of Henzinger and Piterman that has stood since 2006. More precisely, we show that the so-called 2-token game, which can be solved in PTIME for parity automata with a fixed parity index, characterises history-determinism for parity automata. This game was introduced by Bagnol and Kuperberg in 2018, who showed that to decide if a Büchi automaton is history-determinism, it suffices to find the winner of the 2-token game on it. They conjectured that this 2-token game based characterisation extends to parity automata. Boker, Kuperberg, Lehtinen, and Skrzypcak showed in 2020 that this conjecture holds for coBüchi automata as well. We prove Bagnol and Kuperberg's conjecture that the winner of the 2-token game characterises history-determinism on parity automata.

cs.FL↗

arXiv2024

Mostowski Index via extended register games

The parity index problem of tree automata asks, given a regular tree language L, what is the least number of priorities of a nondeterministic parity tree automaton that recognises L. This is a long-standing open problem, also known as the Mostowski or Rabin-Mostowski index problem, of which only a few sub-cases and variations are known to be decidable. In a significant step, Colcombet and Löding reduced the problem to the uniform universality of distance-parity automata. In this brief note, we present a similar result, with a simplified proof, based on on the games in Lehtinen's quasipolynomial algorithm for parity games. We define an extended version of these games, which we call parity transduction games, which take as parameters a parity index J and an integer bound N. We show that the language of a guidable automaton A is recognised by a nondeterministic automaton of index J if and only if there is a bound N such that the parity transduction game with parameters J and N captures membership of the language, that is, for all trees t, Eve wins the parity transduction game on the acceptance parity game of t in A if and only in t is in L(A).

cs.FL↗

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

History-Determinism vs Fair Simulation

An automaton is history-deterministic if its nondeterminism can be resolved on the fly, only using the prefix of the word read so far. This mild form of nondeterminism has attracted particular attention for its applications in synthesis problems. An automaton $A$ is guidable with respect to a class $C$ of automata if it can fairly simulate every automaton in $C$ whose language is contained in that of $A$. In other words, guidable automata are those for which inclusion and simulation coincide, making them particularly interesting for model-checking. We study the connection between these two notions, and specifically the question of when they coincide. For classes of automata on which they do, deciding guidability, an otherwise challenging decision problem, reduces to deciding history-determinism, a problem that is starting to be well-understood for many classes. We provide a selection of sufficient criteria for a class of automata to guarantee the coincidence of the notions, and use them to show that the notions coincide for the most common automata classes, among which are $ω$-regular automata and many infinite-state automata with safety and reachability acceptance conditions, including vector addition systems with states, one-counter nets, pushdown-, Parikh-, and timed-automata. We also demonstrate that history-determinism and guidability do not always coincide, for example, for the classes of timed automata with a fixed number of clocks.

cs.FL↗

arXiv2024

From Muller to Parity and Rabin Automata: Optimal Transformations Preserving (History) Determinism

We study transformations of automata and games using Muller conditions into equivalent ones using parity or Rabin conditions. We present two transformations, one that turns a deterministic Muller automaton into an equivalent deterministic parity automaton, and another that provides an equivalent history-deterministic Rabin automaton. We show a strong optimality result: the obtained automata are minimal amongst those that can be derived from the original automaton by duplication of states. We introduce the notions of locally bijective morphisms and history-deterministic mappings to formalise the correctness and optimality of these transformations. The proposed transformations are based on a novel structure, called the alternating cycle decomposition, inspired by and extending Zielonka trees. In addition to providing optimal transformations of automata, the alternating cycle decomposition offers fundamental information on their structure. We use this information to give crisp characterisations on the possibility of relabelling automata with different acceptance conditions and to perform a systematic study of a normal form for parity automata.

cs.FL↗

arXiv2024

A Bit of Nondeterminism Makes Pushdown Automata Expressive and Succinct

We study the expressiveness and succinctness of history-deterministic pushdown automata (HD-PDA) over finite words, that is, pushdown automata whose nondeterminism can be resolved based on the run constructed so far, but independently of the remainder of the input word. These are also known as good-for-games pushdown automata. We prove that HD-PDA recognise more languages than deterministic PDA (DPDA) but not all context-free languages (CFL). This class is orthogonal to unambiguous CFL. We further show that HD-PDA can be exponentially more succinct than DPDA, while PDA can be double-exponentially more succinct than HD-PDA. We also study HDness in visibly pushdown automata (VPA), which enjoy better closure properties than PDA, and for which we show that deciding HDness is ExpTime-complete. HD-VPA can be exponentially more succinct than deterministic VPA, while VPA can be exponentially more succinct than HD-VPA. Both of these lower bounds are tight. We then compare HD-PDA with PDA for which composition with games is well-behaved, i.e. good-for-games automata. We show that these two notions coincide, but only if we consider potentially infinitely branching games. Finally, we study the complexity of resolving nondeterminism in HD-PDA. Every HDPDA has a positional resolver, a function that resolves nondeterminism and that is only dependant on the current configuration. Pushdown transducers are sufficient to implement the resolvers of HD-VPA, but not those of HD-PDA. HD-PDA with finite-state resolvers are determinisable.

cs.FL↗

arXiv2023

Token Games and History-Deterministic Quantitative-Automata

A nondeterministic automaton is history-deterministic if its nondeterminism can be resolved by only considering the prefix of the word read so far. Due to their good compositional properties, history-deterministic automata are useful in solving games and synthesis problems. Deciding whether a given nondeterministic automaton is history-deterministic (the HDness problem) is generally a difficult task, which can involve an exponential procedure, or even be undecidable, as is the case for example with pushdown automata. Token games provide a PTime solution to the HDness problem of Büchi and coBüchi automata, and it is conjectured that 2-token games characterise HDness for all $ω$-regular automata. We extend token games to the quantitative setting and analyse their potential to help deciding HDness of quantitative automata. In particular, we show that 1-token games characterise HDness for all quantitative (and Boolean) automata on finite words, as well as discounted-sum (DSum), Inf and Reachability automata on infinite words, and that 2-token games characterise HDness of LimInf and LimSup automata, as well as Sup automata on infinite words. Using these characterisations, we provide solutions to the HDness problem of Safety, Reachability, Inf and Sup automata on finite and infinite words in PTime, of DSum automata on finite and infinite words in NP$\cap$co-NP, of LimSup automata in quasipolynomial time, and of LimInf automata in exponential time, where the latter two are only polynomial for automata with a logarithmic number of weights.

cs.FL↗

arXiv2023

Good-for-games $ω$-Pushdown Automata

We introduce good-for-games $ω$-pushdown automata ($ω$-GFG-PDA). These are automata whose nondeterminism can be resolved based on the input processed so far. Good-for-gameness enables automata to be composed with games, trees, and other automata, applications which otherwise require deterministic automata. Our main results are that $ω$-GFG-PDA are more expressive than deterministic $ω$- pushdown automata and that solving infinite games with winning conditions specified by $ω$-GFG-PDA is EXPTIME-complete. Thus, we have identified a new class of $ω$-contextfree winning conditions for which solving games is decidable. It follows that the universality problem for $ω$-GFG-PDA is in EXPTIME as well. Moreover, we study closure properties of the class of languages recognized by $ω$-GFG- PDA and decidability of good-for-gameness of $ω$-pushdown automata and languages. Finally, we compare $ω$-GFG-PDA to $ω$-visibly PDA, study the resources necessary to resolve the nondeterminism in $ω$-GFG-PDA, and prove that the parity index hierarchy for $ω$-GFG-PDA is infinite. This is a corrected version of the paper arXiv:2001.04392v6 published originally on January 7, 2022.

cs.FL↗

arXiv2022

Parikh Automata over Infinite Words

Parikh automata extend finite automata by counters that can be tested for membership in a semilinear set, but only at the end of a run, thereby preserving many of the desirable algorithmic properties of finite automata. Here, we study the extension of the classical framework onto infinite inputs: We introduce reachability, safety, Büchi, and co-Büchi Parikh automata on infinite words and study expressiveness, closure properties, and the complexity of verification problems. We show that almost all classes of automata have pairwise incomparable expressiveness, both in the deterministic and the nondeterministic case; a result that sharply contrasts with the well-known hierarchy in the $ω$-regular setting. Furthermore, emptiness is shown decidable for Parikh automata with reachability or Büchi acceptance, but undecidable for safety and co-Büchi acceptance. Most importantly, we show decidability of model checking with specifications given by deterministic Parikh automata with safety or co-Büchi acceptance, but also undecidability for all other types of automata. Finally, solving games is undecidable for all types.

cs.FL↗

arXiv2022

On the Translation of Automata to Linear Temporal Logic

While the complexity of translating future linear temporal logic (LTL) into automata on infinite words is well-understood, the size increase involved in turning automata back to LTL is not. In particular, there is no known elementary bound on the complexity of translating deterministic $ω$-regular automata to LTL. Our first contribution consists of tight bounds for LTL over a unary alphabet: alternating, nondeterministic and deterministic automata can be exactly exponentially, quadratically and linearly more succinct, respectively, than any equivalent LTL formula. Our main contribution consists of a translation of general counter-free deterministic $ω$-regular automata into LTL formulas of double exponential temporal-nesting depth and triple exponential length, using an intermediate Krohn-Rhodes cascade decomposition of the automaton. To our knowledge, this is the first elementary bound on this translation. Furthermore, our translation preserves the acceptance condition of the automaton in the sense that it turns a looping, weak, Büchi, coBüchi or Muller automaton into a formula that belongs to the matching class of the syntactic future hierarchy. In particular, it can be used to translate an LTL formula recognising a safety language to a formula belonging to the safety fragment of LTL (over both finite and infinite words).

cs.FL↗

arXiv2022

On the size of good-for-games Rabin automata and its link with the memory in Muller games

In this paper, we look at good-for-games Rabin automata that recognise a Muller language (a language that is entirely characterised by the set of letters that appear infinitely often in each word). We establish that minimal such automata are exactly of the same size as the minimal memory required for winning Muller games that have this language as their winning condition. We show how to effectively construct such minimal automata. Finally, we establish that these automata can be exponentially more succinct than equivalent deterministic ones, thus proving as a consequence that chromatic memory for winning a Muller game can be exponentially larger than unconstrained memory.

cs.FL↗

arXiv2022

A Recursive Approach to Solving Parity Games in Quasipolynomial Time

Zielonka's classic recursive algorithm for solving parity games is perhaps the simplest among the many existing parity game algorithms. However, its complexity is exponential, while currently the state-of-the-art algorithms have quasipolynomial complexity. Here, we present a modification of Zielonka's classic algorithm that brings its complexity down to $n^{O\left(\log\left(1+\frac{d}{\log n}\right)\right)}$, for parity games of size $n$ with $d$ priorities, in line with previous quasipolynomial-time solutions.

cs.GT↗

arXiv2021

History Determinism vs. Good for Gameness in Quantitative Automata

Automata models between determinism and nondeterminism/alternations can retain some of the algorithmic properties of deterministic automata while enjoying some of the expressiveness and succinctness of nondeterminism. We study three closely related such models -- history determinism, good for gameness and determinisability by pruning -- on quantitative automata. While in the Boolean setting, history determinism and good for gameness coincide, we show that this is no longer the case in the quantitative setting: good for gameness is broader than history determinism, and coincides with a relaxed version of it, defined with respect to thresholds. We further identify criteria in which history determinism, which is generally broader than determinisability by pruning, coincides with it, which we then apply to typical quantitative automata types. As a key application of good for games and history deterministic automata is synthesis, we clarify the relationship between the two notions and various quantitative synthesis problems. We show that good-for-games automata are central for "global" (classical) synthesis, while "local" (good-enough) synthesis reduces to deciding whether a nondeterministic automaton is history deterministic.

cs.FL↗

arXiv2020

On the Succinctness of Alternating Parity Good-for-Games Automata

We study alternating parity good-for-games (GFG) automata, i.e., alternating parity automata where both conjunctive and disjunctive choices can be resolved in an online manner, without knowledge of the suffix of the input word still to be read. We show that they can be exponentially more succinct than both their nondeterministic and universal counterparts. Furthermore, we present a single exponential determinisation procedure and an Exptime upper bound to the problem of recognising whether an alternating automaton is GFG. We also study the complexity of deciding "half-GFGness", a property specific to alternating automata that only requires nondeterministic choices to be resolved in an online manner. We show that this problem is PSpace-hard already for alternating automata on finite words.

cs.FL↗