Papers

We introduce communication-complexity lifting techniques into the study of recognizable picture languages. As an application, we resolve a long-standing open problem of Anselmo et al. (2006) by constructing a language in UREC whose complement does not belong to REC. Our lower-bound argument is inspired by the communication-complexity approach to unambiguous automata of Göös et al. (2022), although its implementation in the setting of picture languages requires substantially different technical ingredients.

Symbolic finite state machines (SFSMs) describe input/output behaviour using guards and output assignments with possibly infinite data domains. We study deterministic and completely specified SFSMs whose guards and output assignments depend only on the current input. We define finite representative input sets that contain witnesses for relevant guard overlaps and separating witnesses for output assignments that differ on those overlaps. Our main theorem shows that language equivalence of the finite instantiations implies language equivalence over the full input domain. This result transfers complete testing methods for deterministic finite state machines (DFSMs) to SFSMs, provided finite sets of admissible guards and output assignments and an upper bound on the number of distinguishable reachable states are known. Under these assumptions, a DFSM learner with complete testing can learn a finite instantiation, which is then lifted to an equivalent SFSM. We establish a bound on the size of representative input sets and give an SMT construction whose correctness and termination hold under stated solver assumptions.

We introduce causal graph rewriting, a model of computation in which local rules are applied on directed acyclic graphs in an asynchronous manner. The non-determinism arising from asynchrony is disciplined by the oriented edges, which must be understood as both computational dependencies and locality constraints—and are themselves subject to the rewriting. We illustrate the model through two examples: a particle system, and a time-dilation example—reminiscent of general relativity. We study the well-definedness and properties of induced subgraphs and graph composition, which isolate and recombine the region affected by a rewrite. We then study locality with respect to these constructions, showing how a local rewrite preserves positions, borders, and context. Our main result concerns sequential composition: locality extends from single rule applications to arbitrary valid sequences, as any local rule is automatically $*$-local and $*$-extensive. We also formalise and prove the simulation of any one-dimensional cellular automaton.

We study complete-information debate systems in which a probabilistic finite-state verifier reads the alternating messages of a prover and a refuter. Demirci, Say, and Yakary{\i}lmaz showed that every language in $\mathsf{P}$ has such debates checkable with a constant number of random bits and arbitrarily small weak error. Their strong-error construction, which also counts nontermination as failure, did not permit arbitrary error reduction. We close this gap: for every $L\in\mathsf{P}$ and every $\varepsilon>0$, there is a constant-space verifier using a constant number of private coin tosses that has perfect completeness and strong error at most $\varepsilon$. The verifier simulates a polynomial-time alternating multihead finite automaton, privately spot-checking one of its input heads. The key observation is that, on a nonmember, the refuter may concede any round in which the prover first misreports a head reading. This ensures termination against every prover when the refuter follows the specified strategy, and permits strong-error reduction by repetition.

We reconsider the theory of probabilistic formal languages generated by n-gram models and by probabilistic context-free grammars (PCFGs). The expected hierarchy of probabilistic grammars is established by proving that every probabilistic language generated by an n-gram model is also generated by some PCFG, while some probabilistic languages generated by PCFGs cannot be generated by any $n$-gram model. We introduce the notion of fully connected PCFGs, namely PCFGs in Chomsky normal form where every production rule only involving non-terminals has non-zero probability. Our main result shows that any probabilistic language generated by an $n$-gram model differs from any probabilistic language generated by a fully connected PCFG. Therefore, the class of probabilistic languages generated by $n$-gram models is not a subset of the class generated by fully connected PCFGs.

The Lyndon array stores, at each position of a word, the length of the longest Lyndon factor starting at that position and plays an important role in combinatorics on words, for example, in the construction of fundamental data structures such as the suffix array. In this paper, we introduce the Inverse Lyndon array, the analogous structure for inverse Lyndon words, namely words that are lexicographically greater than all their proper nonempty suffixes. Unlike standard Lyndon words, inverse Lyndon words may have non-trivial borders, which introduces a genuine theoretical difficulty. We show that the Inverse Lyndon array can be characterized in terms of the next greater suffix array together with a border-correction term, and we prove that this correction coincides with a longest common extension (LCE) value. Building on this characterization, we adapt the nearest-suffix framework underlying Ellert's linear-time construction of the Lyndon array to the inverse setting, obtaining an O(n)-time algorithm for general ordered alphabets. Finally, we show that the Inverse Lyndon array can also be used to reconstruct the canonical inverse Lyndon factorization in linear time.

We present a fast numerical method for safe continuous-time motion planning under Temporal Logic (TL) specifications. The method generates smooth continuous trajectories that remain collision-free while robustly satisfying temporal and logical task requirements. A central component of our method is the formulation of nonconvex safety and logic constraints as unions of convex sets where associated discrete decisions are encoded in a joint feasibility graph. This graph representation allows Euclidean projection onto the feasible set and proximal robustness maximization to be reformulated as shortest- and widest-path problems, respectively. Building on this structure, we develop a nonconvex splitting method based on the Alternating Direction Method of Multipliers (ADMM), which decouples smooth spatio-temporal trajectory optimization from nonsmooth discrete constraint handling within the optimization. The resulting algorithm exhibits reliable convergence across benchmarks and scales to large-scale motion-planning problems, providing a 4.7x average speedup over the state of the art on discrete and continuous-time logic problems.

Which cellular-automaton diagonal families form bases in every finite window? For canonical polynomial lifts of elementary rules, two truth-table bits determine triangularity, and units on the matrix diagonal determine invertibility. Exactly 24 rules give universal binary bases; all remain universal over every modulus. Among triangular binary coordinate maps, the Pascal transform is uniquely characterized by converting OR convolution into pointwise multiplication, while increment becomes strict prefix summation. Explicit inverses and coordinate comparisons distinguish sparsity from evaluation cost. A Rule 30 polynomial construction gives Fibonacci bounds on interpolation order and prime-modulus periods. Exact additive periods anchor a finite census modulo two and three. These results separate all-window basis classification from optimization of a representation and from period patterns observed in finite windows.

Let $\Lambda=KQ/I$ be a finite-dimensional zero-relation algebra. We encode Crawley–Boevey tree modules over $\Lambda$ by finite rooted trees labelled by arrows of $Q$ and their formal inverses, and construct a deterministic finite bottom-up tree automaton recognizing exactly these encodings. We define an accepted tree to be an automata-induced tree brick when it has no non-trivial factor–image self-overlap, and use Crawley–Boevey's graph-map basis to prove that this is equivalent to brickness of the associated tree module. We also introduce local colourings of $Q_1$ and show that the arrow alphabet can be compressed without changing the tree data, graph maps, or brick property. The optimal number of colours for such a compression is the maximum of the in-degree and out-degree of $Q$. We conclude by asking whether the tree language consisting only of bricks is regular.

Interval Markov decision processes (IMDPs) provide a natural framework for modeling stochastic systems with uncertain transition probabilities, represented by probability intervals and resolved adversarially. Such uncertainty arises naturally, for example, when the transition model is learned from finite data or obtained through model-based reinforcement learning. In this paper, we study the automata-theoretic verification of IMDPs against rich temporal specifications, including all LTL specifications, by considering the broader class of {\omega}-regular objectives. We show that classical automata-theoretic verification techniques extend to IMDPs, but with a sharp distinction determined by the structure of the transition intervals. For stable IMDPs, where either the upper bound is zero or the lower bound is strictly positive, verification reduces to ordinary MDP analysis and can be carried out using the standard automata used in that setting (good-for-MDP automata). For unstable IMDPs, where intervals may include zero while the upper bound is strictly positive, verification becomes game-like and requires automata whose nondeterminism can be resolved on the fly (good-for-games automata). Building on these insights, we develop algorithms for verifying {\omega}-regular specifications over IMDPs and derive probabilistic guarantees when the interval model is learned from sampled data. The resulting framework enables principled verification of stochastic systems under probabilistic model uncertainty, connecting automata-based verification with data-driven stochastic modeling.

The satisfiability problem for word equations asks whether variables can be replaced by words so that the two sides become equal. For regular word equations, in which each variable occurs at most once on each side, satisfiability is NP-complete. For general quadratic word equations, in which each variable occurs at most twice in total, satisfiability is NP-hard, but its membership in NP remains open. We consider an intermediate class: quadratic word equations with a linear side, where each variable occurs at most once on one designated side. We show that the Nielsen graph of an equation $U=V$ in this class, with total length $N=|U|+|V|$, has diameter $O(N^{12})$, measured over reachable pairs of vertices. Together with the known NP-hardness for regular word equations, this result establishes NP-completeness of satisfiability for this class. We also show that each strongly connected component is isomorphic to the length-preserving reachability graph of a regular equation, and that the condensation graph has depth at most $|U|+2|V|$.For arbitrary quadratic word equations, we further show that Nielsen paths can be shortened to a length polynomial in the equation size and the number of prefix operations between variables repeated on opposite sides. This yields NP membership for subclasses defined by variable occurrence arrangements.

Let $\mathcal A$ be a complete deterministic finite automaton on a state set $Q$ of size $n$ with $k$ letters, and for a proper nonempty subset $S$ of $Q$ let $\mathrm{minext}(S)$ be the length of a shortest word $u$ with $|Su^{-1}|>|S|$, where $Su^{-1}=\{q: q\cdot u\in S\}$. To each state $q$ attach the integer $\beta^{\ast}_q=\sum_{t=1}^{n-1}k^{\,n-1-t}(\mathrm{indeg}_t(q)-k^{t})$, where $\mathrm{indeg}_t(q)$ counts the pairs $(p,u)$ with $|u|=t$ and $p\cdot u=q$, and let $B(S)=\sum_{q\in S}\beta^{\ast}_q$. On every synchronizing automaton, $B(S)\ge0$ implies $\mathrm{minext}(S)\le n-1$, so, as $B(Q)=0$, one of $S$ and $Q\setminus S$ extends within $n-1$; when $B(S)>0$ no hypothesis is needed. Kari's Eulerian extension lemma is the case $\beta^{\ast}=0$, and $\beta^{\ast}$, like every member of the family $\sum_{t=1}^{n-1}c_t\sigma_t$, $c_t>0$, vanishes identically if and only if the automaton is Eulerian, where $\sigma_t(S)=\sum_{q\in S}(\mathrm{indeg}_t(q)-k^{t})$. On strongly connected automata $\sigma_t(S)/k^{t}$ has Cesàro limit $n\,e(S)/e(Q)-|S|$ for Friedman's weight $e$; that limit certifies singletons but no larger subset in general. The hypothesis $B(S)\ge0$ cannot be relaxed by one integer unit, nor can the constant $n-1$ be improved. A second-moment test on the sizes $|Su^{-1}|$ certifies 60 to 95 percent of the subsets with $B(S)<0$ at $n\le7$. Along non-Eulerian automata whose words of length $n-1$ merge a fraction of the state pairs bounded below, with $\max_q\mathrm{indeg}_{n-1}(q)=o(nk^{n-1})$, it certifies all but a vanishing share of them. The functional $B$ certifies half of the subsets outside $\{B=0\}$. At each subset size coprime to $n$ ($n\ge4$) some synchronizing Eulerian binary automaton attains the constant $n-1$; whether only there is open. No reset bound follows: Černý's automata have subsets not extending within $n-1$.

We develop a new framework for proving the undecidability of first-order theories of structures of the form $\langle \mathbb{N}; +, P \rangle$, $\langle \mathbb{N}; <, f \rangle$, and $\langle \mathbb{N}; +, f\rangle$, where $P \subseteq \mathbb{N}$ and $f \colon \mathbb{N} \to \mathbb{N}$. It is based on the recent proof of Hieronymi and Schulz that the first-order theory of $\langle \mathbb{N}; +, \{2^n \colon n \in \mathbb{N}\}, \{3^n \colon n \in \mathbb{N}\}\rangle$ is undecidable, and capable of transforming various randomness results about integer sequences into undecidability proofs. We apply our method to a large class of integer linear recurrence sequences, as well as various special functions, in particular showing that the first-order theories of $\langle \mathbb{N}; +, \{u_n \colon n \in \mathbb{N}\} \cap \mathbb{N}\rangle$, $\langle\mathbb{N}; <, n \mapsto \max\{0,u_n\}\rangle$, and $\langle \mathbb{N}; <, \phi\rangle$ are undecidable, where $(u_n)_{n\in\mathbb{N}}$ is any integer LRS with exactly two non-repeated dominant roots satisfying a non-degeneracy assumption, and $\phi$ is Euler's totient function.

In previous papers, we began the study of sequence prediction algorithms adapted to stringological word complexity measures. In particular, we defined a complexity measure called Arithmetic Repetition Complexity (ARC) which admits a polynomial-time prediction algorithm with a mistake bound quasilinear in the complexity. Here, we show a weaker complexity measure related to ARC that admits an especially efficient prediction algorithm: an algorithm that runs in quasilinear time and polylog space for appropriate highly-structured sequences. The complexity measure is defined via a restricted class of "zipline programs" (a variant of straight-line programs), which we call layered. We thus get a less expressive measure with a more efficient algorithm (compared to our results for ARC), demonstrating a possible tradeoff.

We study the circuit complexity of regular languages in terms of unbounded fan-in Boolean circuit families. We characterize the regular languages of constant circuit complexity in terms of the one-variable fragment of first-order logic with regular predicates, in terms of the pseudovariety of stamps $\mathbf{QEJ}_\mathbf{1}$, suitable word congruences and regular expressions. We analogously characterize the neutral letter regular languages of constant circuit complexity. Our lower bound result implies that the class of regular languages of sublogarithmic circuit complexity coincides with the one of constant circuit complexity. In addition we show that deciding whether a regular language, given as a nondeterministic finite automaton, has constant circuit complexity is $\mathbf{PSPACE}$-complete. We introduce a strong notion of reduction, called rational truth-table reduction, that is tailored towards algebraically defined classes of languages. We show that, for a class of functions we call mild, rational truth-table reductions preserve both upper and lower bounds on circuit complexity. We show that the class of regular languages, whose circuit complexity is bounded by a mild function, is in fact a length-multiplying variety of languages. Slightly extending the class of regular languages of constant circuit complexity, we analogously characterize the class of regular languages that are in the pseudovariety $\mathbf{QEACom}$. For these we derive logarithmic circuit complexity upper bounds.

We study the language-theoretic structure of fixed points and finite Picard iterates for guarded q-power language operators. For the general operator, we prove that a context-free seeded language deter- mines a unique context-free fixed point and give an effective construction of a context-free grammar generating this fixed point, independently of the initial language.We then consider a marked single-guard special case in which two new symbols separate the recursive contribution from the the contribution of the seed. In this setting, the initial language can be traced and recovered exactly from every finite Picard iterate by means of a regular slice and fixed-word quotients. This yields injectivity of the finite-time maps and an abstract finite-time class-preservation prin- ciple. As a consequence, we obtain a classification of the finite Picard iterates according to the exact position of the initial language in the Chomsky hierarchy. Thus the exact language-theoretic complexity may persist at every finite stage, while all Picard orbits converge to the same context-free fixed point.

In Model-Based Testing (MBT), test suites are generated automatically from a formal specification. The theory of testing real-time systems is rich, but often underused in practice, partly because applying the timed machinery demands expertise practitioners should not need. In prior work we addressed this for timed testing with a canonic lifting operator, which lets a modeller specify behaviour as plain labelled transition systems while implicitly obtaining the timed automata that express their quiescent behaviour (the explicit absence of outputs) with timers. This paper takes the next step: we show that our lifting still works when each component carries its own time-out on its own channel. This way, we introduce a multi-channel lifting. We show that it commutes with parallel composition, i.e. composing and lifting is the same as lifting first and then composing. We show that the MBT apparatus survives: conformance, test generation and verdicts are preserved by the lifting, on the testable traces that a time-out based tester can observe.

Timed automata, which extend finite state automata by introducing clock variables, serve as a popular formalism for specifying and analyzing the timed behaviors of real-time systems. Extracting the timed behaviors of a black-box, safety-critical system is crucial for designing and analyzing its real-time requirements, yet it remains challenging. In this paper, we address this problem by generating a deterministic timed automaton (DTA) consistent with a given set of system behaviors, comprising both positive and negative examples. To this end, we adapt the formalism of simple elementary languages (sEL) and introduce the timed augmented prefix tree acceptor (tAPTA). Our approach proceeds as follows: First, we preprocess samples by translating them into sEL, which discards redundancy and detects conflicts; then, we rewrite the resulting sELs in an incremental form and construct a tAPTA to further simplify the samples; finally, we encode the search for a DTA that accepts the simplified tAPTA as an SMT formula. We evaluate our approach on randomly generated benchmarks and a scheduling case study. The results demonstrate the effectiveness of our simplification method in reducing the size of the encoded SMT formula and the efficiency of our approach in mining a DTA.

For every $n\ge9$, we construct a strongly connected synchronizing automaton with two input letters and $n$ states whose shortest word merging some three distinct states has length $\lfloor4n/3\rfloor$. One letter is a permutation; the other has an image of size $n-1$. We give complete transition maps and an explicit word attaining the bound. For the lower bound, we assign an integer to each unordered pair of states and prove that applying either input letter can decrease it by at most one. For $n\not\equiv0\pmod3$, the same construction contains a specified state whose shortest merging word, with its partner chosen freely, has length $\lfloor4n/3\rfloor-1$.

The Skolem Problem asks whether a given linear recurrence sequence (LRS) has a zero term, and is equivalent to proving an effective version of the Skolem-Mahler-Lech theorem, which states that a non-degenerate LRS has finitely many zeros. Decidability of the Skolem Problem however, has remained open for many decades. Bilu et al. (2022) showed that the Skolem Problem for simple LRS is decidable subject to the weak $p$-adic Schanuel Conjecture and the Skolem Conjecture (also known as the exponential local-global principle), the latter of which states that an LRS has an integer zero if and only if it has a zero modulo every integer $m$. This paper works towards understanding the Skolem Conjecture. We provide an example showing that a natural strengthening of the Skolem Conjecture, where we restrict $m$ to be a prime power, is false. This failure is accounted for by the relationship between $p$-adic zeros of LRS (which arise naturally in the proof of the Skolem–Mahler–Lech theorem and have been studied algorithmically by Bacik et al. (2026)) and twisted rational zeros (introduced by Bilu et al. (2025)). By studying this relationship further, we are able to prove the main result of this paper: a local-global principle for simultaneous zeros of two coprime LRS, subject to the $p$-adic Schanuel Conjecture. A fundamental step in proving this result is characterising when twisted rational zeros are (or are not) $p$-adic zeros for infinitely many primes $p$, which we do unconditionally. This also answers two open questions of Bilu et al. (2025). Finally, we conjecture that the existence of twisted rational zeros is the only way that the aforementioned strengthened Skolem Conjecture may fail, which is supported by a heuristic argument.

Vectors addition systems with states (VASS), a model equivalent to Petri nets, are finite-state machines with finitely many counters ranging over the natural numbers. The decidable reachability problem for VASS has many applications in logic, automata, and verification. In this paper we study the reachability problem for BVASS, a branching generalization of VASS. We show that BVASS reachability sets are very similar to VASS reachability sets, namely that they are sections of VASS. Our proof relies on a new well-quasi-order (wqo) on BVASS runs that generalizes the well-known wqo on VASS runs. By leveraging an amalgamation property, we prove that every BVASS run can be transformed into an equivalent one of bounded branching complexity. This allows us to derive several results on the geometry of BVASS reachability sets. As an application we obtain that reachability sets of 5-dimensional BVAS are effectively semilinear, as is the case for 5-dimensional VAS.

It is shown that the (scattered) subsequence problem for two words represented by straight-line programs is $\mathsf{PSPACE}$-complete, even over a binary alphabet. The lower bound is obtained by a polynomial-time reduction from quantified subset sum. If the two words are given explicitly, i.e., uncompressed, then the subsequence problem is hard for uniform $\mathsf{NC}^1$ and in deterministic logspace.

We study the complexity of simple semi-conditional grammars (SSCGs) in terms of the number of their terminals and non-terminals. We show that SSCGs with three non-terminals can generate all RE languages; two non-terminals suffice for linear languages; and one suffices for unary regular languages. We also determine both upper bounds and fundamental limitations of SSCGs: while certain one-non-terminal SSCGs can generate non-context-free languages, every language generated by a one-non-terminal SSCG is in non-deterministic linear space. We also prove that there exists a regular language of alphabet size at least 3 which cannot be generated by any one-non-terminal SSCG. Finally, we prove that membership testing for SSCGs of two non-terminals is already NP-hard.

The language $\mathrm{MIX}$ consists of all words over a three-letter alphabet that have an equal number of occurrences of each letter. It is also the word problem of $\mathbb{Z}^2$ with respect to a suitable choice of generators. The Kanazawa–Salvati conjecture states that $\mathrm{MIX}$ is not a well-nested multiple context-free language. Every well-nested multiple context-free language is an indexed language. We reduce the conjecture to an explicit combinatorial problem about tuples of words, which is easier to state than the original formulation in terms of arbitrary well-nested multiple context-free grammars. More generally, for every surjective monoid homomorphism $\psi\colon \Sigma^* \to \mathbb{Z}^d$, we define a family of well-nested multiple context-free grammars $G_\psi[r]$ for $r \geq 1$, each of which generates a sublanguage of $\psi^{-1}(\mathbf{0})$. We prove that every well-nested multiple context-free sublanguage of $\psi^{-1}(\mathbf{0})$ is contained in $L(G_\psi[r])$ for some $r \geq 1$. Using this family, we prove that the four-letter analogue $\mathrm{MIX}_4$, which is a word problem of $\mathbb{Z}^3$, is not a well-nested multiple context-free language. The proof reduces this claim to a result of Bishop–Elder–Evetts–Gallot–Levine stating that the two-letter analogue $\mathrm{MIX}_2$ is not generated by any non-branching multiple context-free grammar. The Kanazawa–Salvati conjecture itself remains open.

A recent result by the present authors established a quadratic lower bound, in the worst case, for the increase in the number of states when a one-way nondeterministic finite automaton is converted to a two-way deterministic finite automaton. Although this simply matched a well-known pre-existing quadratic lower bound by Chrobak, it used a distinct proof method. We show that, much like Chrobak's, this new method is also unable to deliver any lower bound strictly greater than quadratic.

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.

Power profiles are widely used to describe power demand and production in energy systems. Yet, real-world usage often involves uncertainty, making it challenging to determine, e.g., whether a battery can reliably meet a given profile. Formal modeling enables the computation of probabilities that a battery will satisfy such demands. Given the combination of discrete and continuous dynamics, along with uncertain usage patterns, stochastic hybrid automata (SHA) offer a suitable framework for such modeling and analysis. In this paper, we formalize two orthogonal approaches to specify the (uncertain) power profiles that a specific battery can serve as well as a storage model. For both types of power profiles, we provide formal construction rules for SHA reflecting the behavior of the storage model given a specific power profile. We analyze how design choices, such as the power profile, storage model, and representation of uncertainty, affect the complexity of the resulting SHA. Finally, we demonstrate how these models quantify feasibility using the tools Modest and realyst, and assess how the analysis scales with profile size and the number of stochastic components.

We prove a lower bound of {\Omega}(log n/ log log n) on the approximation ratio of the global grammar-based compression algorithm Greedy. To our knowledge, the previously best lower bound was a constant, and the existence of a nonconstant lower bound had remained open for more than twenty years. Our bound holds on an infinite family of words of length n, over alphabets of growing size, for every execution using left-to-right occurrence replacement and arbitrary tie-breaking. The lower bound is also formally verified in Lean 4.

Input metrics evaluate the progress of testing in terms of features of inputs present in a test suite. Previous works, as early as the 1950s, established a number of such metrics, but few endeavored to compare them. This paper does so by utilizing existing methods proposed for other metric classes in partition testing literature. After defining and reviewing common input metrics, we begin with a short case study revealing that typical empirical comparison strategies are fundamentally insufficient for comparing metrics. Then, we demonstrate how one rigorously improves a standard metric by defining and implementing $k$-alt-path, a new metric which reduces redundancy while improving sensitivity over $k$-path. Each of the other common input metrics are then systematically compared before discussing the implications of our findings. With these contributions, we bring forward partition testing analysis methods that justify and form a strategy for future research in refining input metrics.

Transient reachability analysis of rare events in Continuous-Time Stochastic Vector Addition Systems (CTSVAS) such as Chemical Reaction Networks (CRNs) has proven a formidable challenge to cutting-edge tools. Underlying a CTSVAS is a continuous-time Markov chain (CTMC), and CTMC transient reachability analysis calls for Probabilistic Model Checking (PMC). This analysis requires the explicit representation of a model's entire state space. Rare events occur with extremely low probability, compounding the challenge of probabilistic analysis. In CRNs, it is imperative to verify the probability of rare events; even a low concentration of a species can have pathological consequences. This paper presents the RAGTIMER 1.0 tool, which efficiently builds a partial state space for a CTSVAS by enumerating traces to a rare event of interest and expanding them to exploit concurrency and cycles, providing a guaranteed lower bound on the probability of a rare event. Guaranteed lower bounds are particularly useful in synthetic biological applications because they indicate how and when a rare event can be experimentally observed. RAGTIMER is an attractive alternative to existing rare event analysis methods for CTSVAS models. It outperforms existing PMC tools and refutes multiple probability estimates from rare-event stochastic simulation on multiple challenging CRN models. RAGTIMER uses optimized data structures, a simple input format, and memory-safe Rust code to improve the scalability and accessibility of PMC for industry professionals.