Formal Languages and Automata Theory

arXiv:cs.FL

Covers automata theory, formal language theory, grammars, and combinatorics on words.

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Theory & Algorithms

Formal Languages and Automata Theory

arXiv:cs.FL

Covers automata theory, formal language theory, grammars, and combinatorics on words.

Looking for a broader view? This category is part of:

Theory & Algorithms

Trending in Formal Languages and Automata Theory

Bounded treewidth, multiple context-free grammars, and downward closures

The reachability problem in multi-pushdown automata (MPDA) has many applications in static analysis of recursive programs. An example is safety verification of multi-threaded recursive programs with shared memory. Since these problems are undecidable, the literature contains many decidable (and efficient) underapproximations of MPDA. A uniform framework that captures many of these underapproximations is that of bounded treewidth (tw): To each execution of the MPDA, we associate a graph; then we consider the subset of all graphs that have a wt at most $k$, for some constant $k$. In fact, bounding tw is a generic approach to obtain classes of systems with decidable reachability, even beyond MPDA underapproximations. The resulting systems are also called MSO-definable bounded-tw systems. While bounded tw is a powerful tool for reachability and similar types of analysis, the word languages (i.e. action sequences corresponding to executions) of these systems remain far from understood. For the slight restriction of bounded special tw, or "bounded-stw" (which is equivalent to bounded tw on MPDA, and even includes all bounded-tw systems studied in the literature), this work reveals a connection with multiple context-free languages (MCFL), a concept from computational linguistics. We show that the word languages of MSO-definable bounded-stw systems are exactly the MCFL. We exploit this connection to provide an optimal algorithm for computing downward closures (dcl) for MSO-definable bounded-stw systems. Computing dcl is a notoriously difficult task that has many applications in the verification of complex systems: As an example application, we show that in programs with dynamic spawning of MSO-definable bounded-stw processes, safety verification has the same complexity as in the case of processes with sequential recursive processes.

2512.019731

Dec 2025Formal Languages and Automata Theory

Related categories:

Artificial Intelligence Hardware Architecture Computational Complexity Computational Engineering, Finance, and Science Computational Geometry

561 papers

(PAC-)Learning state machines from data streams: A generic strategy and an improved heuristic (Extended version)

This is an extended version of our publication Learning state machines from data streams: A generic strategy and an improved heuristic, International Conference on Grammatical Inference (ICGI) 2023, Rabat, Morocco. It has been extended with a formal proof on PAC-bounds, and the discussion and analysis of a similar approach has been moved from the appendix and is now a full Section. State machine models are models that simulate the behavior of discrete event systems, capable of representing systems such as software systems, network interactions, and control systems, and have been researched extensively. The nature of most learning algorithms however is the assumption that all data be available at the beginning of the algorithm, and little research has been done in learning state machines from streaming data. In this paper, we want to close this gap further by presenting a generic method for learning state machines from data streams, as well as a merge heuristic that uses sketches to account for incomplete prefix trees. We implement our approach in an open-source state merging library and compare it with existing methods. We show the effectiveness of our approach with respect to run-time, memory consumption, and quality of results on a well known open dataset. Additionally, we provide a formal analysis of our algorithm, showing that it is capable of learning within the PAC framework, and show a theoretical improvement to increase run-time, without sacrificing correctness of the algorithm in larger sample sizes.

2604.02244Apr 2026

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2603.29411

Exact Separation of Words via Trace Geometry

A basic question in the theory of two-state measure-once quantum finite automata (MO-QFAs) is whether two distinct input words can be separated with certainty. In the setting considered here, this exact separation problem reduces to a trace-vanishing question in $SU(2)$: given distinct positive words $u$ and $v$, find matrices $A,B\in SU(2)$ such that the evaluated trace of $u^{-1}v$ is zero. The central difficulty lies in the genuinely nonabelian regime where $u$ and $v$ have the same abelianization, so the obvious commutative information disappears and the fine structure of the word must be connected to the geometry of representations. This paper develops a slice-driven framework for that task and proves exact separation for every hard positive-word difference covered by four explicit certified conditions, thereby reducing the problem to a sharply delimited residual super-degenerate class. The method extracts algebraic data from the positive-word difference and uses them to select explicit low-dimensional families in $SU(2)^2$ on which the trace becomes computable. On the algebraic side, the metabelian polynomial is decomposed into explicit interval blocks determined by prefix statistics, and a suitable slope specialization preserves nontrivial information. On the analytic side, the paper derives a computable quadratic trace identity on a visible one-parameter family and complements it with a Laurent-matrix sum-of-squares identity in a parallel algebraic model. These certified criteria are already strong in numerical experiments. This paper also shows that no method based only on finitely many finite-image tests can be universal.

2603.29411Mar 2026

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Evolution-Based Timed Opacity under a Universal Observation Model

Existing literature on timed opacity uses specific definitions for restricted subclasses of timed automata or limited observation models. This lack of a unified definition makes it difficult to establish formal relationships and compare the expressiveness of different opacity variants. This paper establishes a unified framework for timed opacity by introducing a universal observation model for timed automata. First, we introduce an observation model with full observation of time delay and partial observation of locations, clocks, and events. Second, based on this model, we define the notion of evolution-based timed opacity. Third, we mathematically prove that evolution-based timed opacity strictly implies language-based timed opacity and establish a formal equivalence with execution-time opacity under constrained observations. This framework establishes a unified semantic hierarchy for characterizing the landscape of timed opacity.

2603.26573Mar 2026

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Synchronous Signal Temporal Logic for Decidable Verification of Cyber-Physical Systems

Many Cyber Physical System (CPS) work in a safety-critical environment, where correct execution, reliability and trustworthiness are essential. Signal Temporal Logic (STL) provides a formal framework for checking safety-critical CPS. However, static verification of STL is undecidable in general, except when we want to verify using run-time-based methods, which have limitations. We propose Synchronous Signal Temporal Logic (SSTL), a decidable fragment of STL, which admits static safety and liveness property verification. In SSTL, we assume that a signal is sampled at fixed discrete steps, called ticks, and then propose a hypothesis, called the Signal Invariance Hypothesis (SIH), which is inspired by a similar hypothesis for synchronous programs. We define the syntax and semantics of SSTL and show that SIH is a necessary and sufficient condition for equivalence between an STL formula and its SSTL counterpart. By translating SSTL to LTL_P (LTL defined over predicates), we enable decidable model checking using the SPIN model checker. We demonstrate the approach on a 33-node human heart model and other case studies.

2603.25531Mar 2026

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Nominal Automata with Name Deallocation

Data words with binders formalize concurrently allocated memory. Most name-binding mechanisms in formal languages, such as the $λ$-calculus, adhere to properly nested scoping. In contrast, stateful programming languages with explicit memory allocation and deallocation, such as C, commonly interleave the scopes of allocated memory regions. This phenomenon is captured in dedicated formalisms such as dynamic sequences and bracket algebra, which similarly feature explicit allocation and deallocation of letters. One of the classical formalisms for data languages are register automata, which have been shown to be equivalent to automata models over nominal sets. In the present work, we introduce a nominal automaton model for languages of data words with explicit allocation and deallocation that strongly resemble dynamic sequences, extending existing nominal automata models by adding deallocating transitions. Using a finite NFA-type representation of the model, we establish a Kleene theorem that shows equivalence with a natural expression language. Moreover, we show that our non-deterministic model allows for determinization, a quite unusual phenomenon in the realm of nominal and register automata.

2603.24468Mar 2026

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The SCAN Statistical Model Checker

This paper lays out the formal foundations upon which the SCAN statistical model checker is built.

2603.28794Mar 2026

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Interpreted Higher-Dimensional Automata for Concurrent Discrete-Event Control

In recent years the theory of Higher Dimensional Automata (HDA) has seen significant advances from a theoretical point of view, reflecting standard automata theory. There have also been first attempts to use the mathematical framework provided by HDAs to known problems, in particular Petri Net analysis. However real-world applications are still lacking and issues from real-world system, as concurrency, is still opened in the context of controller generation. In this work we show how the framework of HDAs can be adapted to help transforming controllers given as interpreted Petri nets (IPN) into an actual closed loop controller and how the HDA helps in identifying ambiguous or even contradictory specifications that remain hidden in the IPN. We demonstrate the feasibility by connecting the obtain controller to a virtual environment for closed loop control, exemplified by an industrial example.

2603.21742Mar 2026

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Complexity of Linear Subsequences of Fibonacci-Automatic Sequences

We construct automata with input(s) in Fibonacci representation (also known as Zeckendorf representation) recognizing some basic arithmetic relations and study their number of states. We also consider some basic operations on Fibonacci-automatic sequences and discuss their state complexity. Furthermore, as a consequence of our results, we improve a bound in a recent paper of Bosma and Don. We also discuss the state complexity and runtime complexity of using a reasonable interpretation of Büchi arithmetic to actually construct some of the studied automata recognizing relations.

2603.21645Mar 2026

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Logic-Gated Time-Shared Feedforward Networks for Alternating Finite Automata: Exact Simulation and Learnability

We present a formal and constructive framework for simulating Alternating Finite Automata (AFAs) using Logic-Gated Time-Shared Feedforward Networks (LG-TS-FFNs). Unlike prior neural automata models limited to Nondeterministic Finite Automata (NFAs) and existential reachability, our architecture integrates learnable, state-dependent biases that function as differentiable logic gates, enabling the representation of both Existential \textsc{\textsc{OR}} and Universal \textsc{\textsc{AND}} aggregation within a shared-parameter linear recurrence. We prove that this architectural modification upgrades the network's computational class to be structurally isomorphic to AFAs, thereby inheriting their exponential succinctness: the network can represent regular languages requiring $2^n$ states in an NFA with only $n$ neurons. We rigorously establish that the forward pass of an LG-TS-FFN exactly simulates the reachability dynamics of an AFA, including instantaneous $\varepsilon$-closures. Furthermore, we demonstrate empirical learnability: a continuous relaxation of the logic gates allows the network to simultaneously recover the automaton's topology and logical semantics from binary labels via standard gradient descent. Extensive experiments confirm that our model achieves perfect recovery of ground-truth automata, bridging the gap between statistical learning and succinct, universal logical reasoning.

2604.01228Mar 2026

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State Complexity of Shifts of the Fibonacci Word

The Fibonacci infinite word ${\bf f} = (f_i)_{i \geq 0} = 01001010\cdots$ is one of the most celebrated objects in combinatorics on words. There is a simple $5$-state automaton that, given $i$ in lsd-first Zeckendorf representation, computes its $i$'th term $f_i$, and a $2$-state automaton for msd-first. In this paper we consider the state complexity of the automaton generating the shifted sequence $(f_{i+c})_{i \geq 0}$, and show that it is $O(\log c)$ for both msd-first and lsd-first input. This is close to the information-theoretic minimum for an aperiodic sequence. The techniques involve a mixture of state complexity techniques and Diophantine approximation.

2603.18858Mar 2026

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An automata-based test for bricks over string algebras

Motivated by the recent work of Deaconu, Mousavand and Paquette on the connection between infinite string bricks for certain gentle algebras and Sturmian words, we develop a decorated version of a deterministic automaton, called a multi-entry inverse automaton (MIA, for short) that accepts pointed words. We then associate an MIA $\mathsf M_{Λδ}$ over $\{0,1\}$ to a string algebra $Λ$, and show that strings over $Λ$ can be viewed as certain equivalence classes of the pointed words accepted by $\mathsf M_{Λδ}$. By defining (weak) brick words over this MIA, we show that a finite/infinite string module (resp. band module) is a brick if and only if every word in the associated equivalence class of pointed binary words is a brick word (resp. a weak brick word) over $\mathsf M_{Λδ}$. The result of Deaconu et al. follows as an immediate consequence.

2603.18820Mar 2026

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2603.20274

Solomonoff induction

This chapter discusses the Solomonoff approach to universal prediction. The crucial ingredient in the approach is the notion of computability, and I present the main idea as an attempt to meet two plausible computability desiderata for a universal predictor. This attempt is unsuccessful, which is shown by a generalization of a diagonalization argument due to Putnam. I then critically discuss purported gains of the approach, in particular it providing a foundation for the methodological principle of Occam's razor, and it serving as a theoretical ideal for the development of machine learning methods.

2603.20274Mar 2026

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Irreducibility of Semigroup Morphisms

We study the notion of irreducibility of semigroup morphisms. Given an alphabet $Σ$, a morphism $\varphi:Σ^+\rightarrowΣ^+$ is irreducible if any factorisation $\varphi=ψ_2\circψ_1$ can only be satisfied if $ψ_1$ or $ψ_2$ is a trivial morphism. Otherwise, $\varphi$ is reducible. We introduce the notion of irreducibility, characterise this property and study a number of fundamental questions on the concepts under consideration.

2603.15177Mar 2026

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Visibly Recursive Automata

As an alternative to visibly pushdown automata, we introduce visibly recursive automata (VRAs), composed of a set of classical automata that can call each other. VRAs are a strict extension of so-called systems of procedural automata, a model proposed by Frohme and Steffen. We study the complexity of standard language-theoretic operations and classical decision problems for VRAs. Since the class of deterministic VRAs forms a strict subclass in terms of expressiveness, we propose a (weaker) notion that does not restrict expressive power and which we call codeterminism. Codeterminism comes with many desirable algorithmic properties that we demonstrate by using it, e.g., as a stepping stone towards implementing complementation of VRAs.

2603.11648Mar 2026

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The complexity of finite smooth words over binary alphabets

Smooth words over an alphabet of non-negative integers $\{a,b\}$ are infinite words that are infinitely derivable, the most famous example being the Oldenburger-Kolakoski word over $\{1,2\}$. The main way to study their language is to consider a finite version of smooth words that we call f-smooth words. In this paper we prove that the f-smooth words are exactly the factors of smooth words, and we make progress towards the conjecture of Sing that the complexity of f-smooth words over $\{a,b\}$ grows like $Θ\left(n^{\log(a+b)/\log((a+b)/2)}\right)$: we prove it over even alphabets, we prove the lower bound over any binary alphabet and we improve the known upper bound over odd alphabets.

2603.10733Mar 2026

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Context-Free Trees

Muller and Schupp introduced the concept of context-free graphs (originating from Cayley graphs of context-free groups). These graphs are always tree-like (i.e. quasi-isometric to a tree) and in this paper we investigate the subclass of bona fide context-free trees. We show that they have a finite-state description using multi-edge NFAs and that this specializes to certain partial DFAs in the case of deterministic graphs. We investigate this form of encoding algorithmically and show that the isomorphism problem for deterministic context-free trees is NL-complete in the rooted and the non-rooted case.

2603.08624Mar 2026

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2603.08331

Turn Complexity of Context-free Languages, Pushdown Automata and One-Counter Automata

A turn in a computation of a pushdown automaton is a switch from a phase in which the height of the pushdown store increases to a phase in which it decreases. Given a pushdown or one-counter automaton, we consider, for each string in its language, the minimum number of turns made in accepting computations. We prove that it cannot be decided if this number is bounded by any constants. Furthermore, we obtain a non-recursive trade-off between pushdown and one-counter automata accepting in a finite number of turns and finite-turn pushdown automata, that are defined requiring that the constant bound is satisfied by each accepting computation. We prove that there are languages accepted in a sublinear but not constant number of turns, with respect to the input length. Furthermore, there exists an infinite proper hierarchy of complexity classes, with the number of turns bounded by different sublinear functions. In addition, there is a language requiring a number of turns which is not constant but grows slower than each of the functions defining the above hierarchy.

2603.08331Mar 2026

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Forgetting Event Order in Higher-Dimensional Automata

Higher dimensional automata (HDAs) provide a geometric model of true concurrency, yet their standard formulation encodes an artificial total order on events. This representational artifact causes a fundamental mismatch between the combinatorial structure of HDAs and their observable behavior, leading to logical asymmetries and complicating the application of categorical tools. In this paper, we resolve this tension by developing a semantics for HDAs that is independent of event order, based on interval ipomsets (partially ordered multisets with interfaces) that preserve only precedence and concurrency. We prove that for any HDA, the traditional ST trace of an execution path corresponds precisely to its associated interval ipomset. On the structural side, we show that the presheaf theoretic presentation with an unordered base and the combinatorial presentation of symmetric HDAs are categorically isomorphic. Finally, by characterizing ST and hereditary history preserving (hhp) bisimulation via ipomset isomorphism, we provide a unified, order free foundation for HDA semantics. Our results resolve several critical ambiguities in the literature: they provide the necessary path category structure to canonically apply the Open Maps framework, eliminate representational artifacts in temporal and modal logics, and bridge systematic mismatches between HDAs and other models of concurrency such as Petri nets.

2603.08134Mar 2026

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History-Deterministic Büchi Automata are Succinct

We describe a history-deterministic Büchi automaton that has strictly less states than every language-equivalent deterministic Büchi automaton. This solves a problem that had been open since the introduction of history-determinism and actively investigated for over a decade. Our example automaton has 65 states, and proving its succinctness requires the combination of theoretical insights together with the aid of computers.

2603.05380Mar 2026

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Computational Complexity of Alignments

In process mining, alignments quantify the degree of deviation between an observed event trace and a business process model and constitute the most important conformance checking technique. We study the algorithmic complexity of computing alignments over important classes of Petri nets. First, we show that the alignment problem is PSPACE-complete on the class of safe Petri nets and also on the class of safe and sound workflow nets. For live, bounded, free-choice systems, we prove the existence of optimal alignments of polynomial length which positions the alignment problem in NP for this class. We further show that computing alignments is NP-complete even on basic subclasses such as process trees and T-systems. We establish NP-completeness on several related classes as well, including acyclic systems. Finally, we demonstrate that on live, safe S-systems the alignment problem is solvable in P and that both assumptions (liveness and safeness) are crucial for this result.

2603.05331Mar 2026

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Formal Languages and Automata Theory Papers (cs.FL) - ScienceStack

Trending in Formal Languages and Automata Theory

Bounded treewidth, multiple context-free grammars, and downward closures

2512.019731

Dec 2025Formal Languages and Automata Theory

Related categories:

Artificial Intelligence Hardware Architecture Computational Complexity Computational Engineering, Finance, and Science Computational Geometry

561 papers

(PAC-)Learning state machines from data streams: A generic strategy and an improved heuristic (Extended version)

2604.02244Apr 2026

View

2603.29411

Exact Separation of Words via Trace Geometry

2603.29411Mar 2026

View

Evolution-Based Timed Opacity under a Universal Observation Model

2603.26573Mar 2026

View

Synchronous Signal Temporal Logic for Decidable Verification of Cyber-Physical Systems

2603.25531Mar 2026

View

Nominal Automata with Name Deallocation

2603.24468Mar 2026

View

The SCAN Statistical Model Checker

This paper lays out the formal foundations upon which the SCAN statistical model checker is built.

2603.28794Mar 2026

View

Interpreted Higher-Dimensional Automata for Concurrent Discrete-Event Control

2603.21742Mar 2026

View

Complexity of Linear Subsequences of Fibonacci-Automatic Sequences

2603.21645Mar 2026

View

Logic-Gated Time-Shared Feedforward Networks for Alternating Finite Automata: Exact Simulation and Learnability

2604.01228Mar 2026

View

State Complexity of Shifts of the Fibonacci Word

2603.18858Mar 2026

View

An automata-based test for bricks over string algebras

2603.18820Mar 2026

View

2603.20274

Solomonoff induction

2603.20274Mar 2026

View

Irreducibility of Semigroup Morphisms

2603.15177Mar 2026

View

Visibly Recursive Automata

2603.11648Mar 2026

View

The complexity of finite smooth words over binary alphabets

2603.10733Mar 2026

View

Context-Free Trees

2603.08624Mar 2026

View

2603.08331

Turn Complexity of Context-free Languages, Pushdown Automata and One-Counter Automata

2603.08331Mar 2026

View

Forgetting Event Order in Higher-Dimensional Automata

2603.08134Mar 2026

View

History-Deterministic Büchi Automata are Succinct

2603.05380Mar 2026

View

Computational Complexity of Alignments

2603.05331Mar 2026

View

Page 1 of 29