automata-based equivalence

This work addresses the problem of explaining why a finite automaton makes a specific decision on a given input word and how its output can be altered through minimal modifications. We formally define the notion of a minimal feature explanation for automaton decisions as the smallest set of critical input symbols responsible for the current acceptance or rejection outcome. To compute all such minimal explanations exactly, we propose an efficient algorithm that integrates formal methods, automata theory, and combinatorial optimization. Experimental evaluation demonstrates that our approach scales well in complex scenarios and consistently produces unbiased, accurate minimal explanations, thereby providing rigorous interpretability guarantees for automaton-based decisions.

Students commonly struggle to grasp the operational semantics of nondeterministic finite automata (NFAs) and pushdown automata (PDAs), particularly regarding multi-path computation, stack-state dependencies, and distinguishing configurations with identical control states but differing stack contents. To address this, we design and implement FSM—a domain-specific language for automata theory education—that uniquely supports full visualization of nondeterministic execution paths and dynamic stack evolution in NFAs and PDAs. We introduce a state-semantic verification mechanism to help users validate transition semantics, integrating dynamic rendering, path-traversal algorithms, and interactive state tracking. Empirical evaluation demonstrates that FSM significantly improves students’ understanding of nondeterminism and stack-dependent behavior, especially in discerning stack-sensitive state equivalence.

This paper addresses the problem of black-box implementation and specification equivalence verification for automata in monoidal closed categories. Methodologically, it generalizes the classical W-method to the abstract level of monoidal closed categories, enabling a unified conformance testing framework for diverse automaton models—including deterministic finite automata (DFAs), Moore/Mealy machines, weighted automata, and deterministic nominal automata. Key contributions include: (i) the first categorical foundation for constructing provably complete test suites, establishing universal construction principles grounded in category theory; (ii) the derivation of the first provably complete test suites for weighted automata and deterministic nominal automata; and (iii) the rigorous reconstruction and categorical generalization of the W-method for DFAs and classical finite-state machines. The framework unifies theoretical rigor with broad model applicability, providing a principled categorical semantics for formal verification of automata-based systems.

Byte Pair Encoding (BPE) tokenization yields subword sequences that lack direct support for formal language operations, hindering rigorous pattern matching and compositional verification in open-vocabulary NLP systems. Method: We propose the first deterministic finite automaton (DFA) construction algorithm tailored to BPE output—treating tokenized sequences as constrained symbol strings without reconstructing original bytes or characters. Our approach introduces a novel equivalence-class partitioning scheme and transition function synthesis mechanism grounded in BPE merge rules, enabling linear-time O(n) DFA construction while preserving semantic fidelity. Contribution/Results: The resulting DFA supports efficient subword-level regular expression matching, lexicon equivalence checking, and formal language composition operations. It significantly improves both efficiency and composability of pattern recognition and formal verification in open-vocabulary NLP, establishing foundational automata infrastructure for verifiable, scalable, tokenization-aware language processing.

To address the low efficiency of graph-structured pattern recognition and the lack of a unified automaton framework for graphs, this paper introduces the Deterministic Finite Graph Automaton (DFGA)—the first extension of classical finite automata to the graph domain. Based on hyperedge replacement graph grammars, we formalize DFGA as a rigorous computational model; devise an enhanced powerset construction algorithm enabling backtrack-free graph recognition; and establish a sufficient condition for efficient DFGA recognition, proving theoretically that DFGAs recognize corresponding graph languages in polynomial time. Our main contributions are: (1) the first formal construction and semantic definition of a deterministic graph automaton; (2) an efficient, linear graph grammar–driven recognition mechanism; and (3) foundational theoretical results on decidability and computational complexity for graph automata.

This paper addresses the problem of universal program equivalence verification. It proposes an algebraic approach based on Kleene Algebra (KA), modeling program behavior as regular expressions and reducing program equivalence to equation derivation within the KA framework. The main contributions are: (1) a formal proof of logical completeness of KA for regular expression equivalence—i.e., two expressions are equivalent iff their equality is derivable from KA axioms; and (2) the first systematic integration of coalgebraic methods into automata theory reconstruction, yielding a unified, abstract algebraic foundation for state-machine semantics. This framework bridges automata, regular languages, and program semantics cohesively, while substantially enhancing both the reliability and mechanizability of equivalence verification. A complementary exercise suite further strengthens conceptual intuition and formal reasoning skills.

Extending ω-automata to infinite alphabets poses significant challenges due to the complexity of managing unbounded memory. Method: This paper introduces obligation-based symbolic ω-automata (OSωA), a novel model that replaces explicit register-based memory with symbolic guards and obligation assignments triggered at transitions—thereby eliminating traditional register mechanisms and yielding simpler semantics and improved decidability. OSωA integrates existential/universal branching with Emerson–Lei acceptance conditions. Contribution/Results: The language class of OSωA strictly subsumes ω-regular languages, as formally proven. A complete toolchain supports core operations—including automaton product and emptiness checking—and demonstrates practical feasibility and effectiveness in verification tasks.

This work addresses the limited expressiveness of traditional visibly pushdown automata by proposing a novel model called Visibly Recursive Automata (VRA), which extends procedural automata systems through mutually recursive classical automata. Introducing the new notion of “co-determinism,” the paper replaces conventional determinism constraints while preserving expressive power, thereby enabling efficient algorithms for crucial operations such as complementation. By integrating formal language theory, automata composition, and complexity analysis, the study rigorously demonstrates that VRA strictly subsumes existing models in expressiveness. Furthermore, it establishes that co-determinism enjoys favorable algorithmic properties, providing a solid foundation for efficient practical implementations.

This paper establishes logical and expression-based characterizations for the class of languages recognized by nondeterministic register automata with guessing (NRA) over infinite alphabets. We introduce Scoped MSO, a logic featuring a novel segment modality and syntactic restrictions on data comparisons. We prove this logic is expressively equivalent to NRA over data domains where ``strong guessing''can be eliminated. Furthermore, we define Data-Regular Expressions, a minimalist regular-expression calculus built from quantifier-free regions and equipped with $k$-contracting concatenation, and demonstrate its equivalence to NRA over arbitrary relational structures. Together, these formalisms provide a robust descriptive theory for register automata, bridging the gap between automata, logic, and expressions.

To address the double-exponential blowup in monitoring complexity arising from deterministic finite automaton (DFA) construction for Linear Temporal Logic (LTL) formulas in runtime verification, this paper proposes a direct trace-evaluation method that bypasses automata construction entirely. The core innovation lies in the first formal characterization and exploitation of semantic properties of safety and co-safety formulas, enabling complete avoidance of DFA generation over the safety and co-safety fragments of LTLf and LTL. By combining syntactic fragment identification with semantics-driven online evaluation, the method reduces monitoring time complexity to polynomial in both trace length and formula size: LTLf monitoring achieves PTIME, while key decision problems for LTL are optimized to PSPACE. This approach significantly enhances monitoring efficiency and establishes a new paradigm for lightweight, scalable runtime verification.