HomDFAs

HomDFAs, short for Homomorphic Deterministic Finite Automata, are an extension of classical deterministic finite automata that incorporate a homomorphism as part of their transition structure. Formally a HomDFA is a quintuple (Q, Σ, δ, q₀, F) together with a monoid homomorphism h: Σ* → M, where M is a target monoid. Each transition δ(q, a) = p is labeled by a symbol a ∈ Σ, and the homomorphism assigns to each traced word a product in M that captures additional algebraic information about the run. The acceptance condition depends not only on reaching a final state in F but also on the homomorphic image of the input word satisfying a specified property in M, such as evaluating to a particular group element.

The incorporation of h allows HomDFAs to recognize languages that are homomorphic images of regular languages, thereby enabling representation of constraints that would otherwise require non‑regular constructs. For instance, a HomDFA can accept the language of all words over {a,b} whose number of a’s minus the number of b’s is congruent to 0 modulo 3 by mapping a to the generator 1 and b to –1 in the additive group ℤ₃.

Minimization of a HomDFA follows a refined equivalence relation: two states q and p are equivalent if, for every suffix w, the homomorphic images of the paths from q and p on w belong to the same residue class in M and both lead to acceptance or rejection. This leads to a canonical minimal representation analogous to the classical DFA minimization algorithm, but with an additional check on the homomorphism values.

Applications of HomDFAs appear in formal verification, pattern matching with constraints, and the synthesis of controllers where output signals must satisfy algebraic conditions. They also provide a unifying framework for studying closure properties of regular languages under homomorphic transformations, and they are central to proofs of the Myhill–Nerode theorem in settings that involve output alphabets or secondary monoids.