NAAs: Nondeterministic Asynchronous Finite Automata

• NAAs are a conceptual extension of nondeterministic finite automata that incorporate asynchronous actions and semilattice structures to explore derivative-based behaviors.
• The framework leverages boolean representations and syntactic monoid actions to draw parallels with classical NFAs while suggesting new avenues for asynchronous modeling.
• Although hypothetical, this approach outlines practical research directions for defining transition semantics, acceptance conditions, and minimization theorems in NAAs.

Searching arXiv for papers directly about “Nondeterministic Asynchronous Finite Automata” and closely related asynchronous automata models. Search results for query: Nondeterministic Asynchronous Finite Automata arXiv Nondeterministic Asynchronous Finite Automata (NAAs), as a named automaton model, are not explicitly discussed in "Nondeterministic Syntactic Complexity" (Myers et al., 2021). Within that evidentiary scope, they can therefore be addressed only indirectly. The established theory concerns ordinary nondeterministic finite automata (NFAs), reinterpreted algebraically as deterministic finite automata endowed with semilattice structure, together with boolean representations of syntactic monoids, the canonical semilattice $\mathrm{SLD}(L)$ of finite unions of left derivatives, and a self-duality principle for finite semilattice automata. These results do not constitute a formal theory of NAAs, but they suggest a template for asynchronous variants in which derivative structure, semilattice actions, and monoid-based minimization remain central (Myers et al., 2021).

1. Scope of the term and status of the model

The principal limitation is explicit: the paper does not explicitly discuss Nondeterministic Asynchronous Finite Automata (NAAs) by name (Myers et al., 2021). Accordingly, no direct formal definition, transition semantics, acceptance condition, or minimization theorem for NAAs is established there. The results apply directly to ordinary NFAs and to semilattice automata.

This point is methodologically important. A common misconception would be to read the semilattice framework as already covering asynchronous nondeterministic models. The paper does not make that claim. Instead, it identifies several concepts that are potentially relevant to NAAs or other asynchronous nondeterministic models: the categorical treatment of automata, the semilattice representation of state spaces, the duality between forward and reverse behavior, the use of derivative-closed language families, and the idea of minimizing a structured subclass of nondeterministic automata by counting join-irreducibles (Myers et al., 2021).

A plausible implication is that NAAs, if they admit an appropriate algebraic semantics, could be studied by adapting this framework rather than by importing the results unchanged. In that sense, the paper provides a possible research program, not an already completed theory of NAAs.

2. Boolean representations and syntactic-monoid actions

The paper introduces nondeterministic syntactic complexity as a measure on regular languages, defined through boolean representations of the syntactic monoid (Myers et al., 2021). For a finite join-semilattice $S$, the relevant endomorphism monoid is $JSL(S,S)$, the monoid of join-preserving self-maps of $S$. A boolean representation of a monoid $M$ is a morphism

$\rho \colon M \to JSL(S,S),$

and its degree is

$\deg(\rho) := |J(S)|,$

the number of join-irreducible elements of $S$.

The extension relation between representations is expressed through equivariant semilattice morphisms. If

$\rho_i \colon M \to JSL(S_i,S_i),$

then an equivariant map is a semilattice morphism $f \colon S_1 \to S_2$ satisfying

$S$0

If $S$1 is injective, then $S$2 is said to extend $S$3 (Myers et al., 2021).

For a regular language $S$4, the canonical boolean representation of the syntactic monoid is

$S$5

Here $S$6 is the syntactic monoid, $S$7 is the semilattice of finite unions of left derivatives of $S$8, and $S$9 is the syntactic congruence class of $JSL(S,S)$0. Via the syntactic morphism $JSL(S,S)$1, this yields the induced boolean presentation

$JSL(S,S)$2

This construction is the paper’s central algebraic object. It reframes a regular language as a canonical action of $JSL(S,S)$3, and of $JSL(S,S)$4, on a finite semilattice built from derivatives of the language. For any future treatment of NAAs, this suggests that the decisive question would be whether asynchronous behavior admits an analogous canonical action.

3. $JSL(S,S)$5-automata and the two complexity measures

To connect NFAs with algebra, the paper works in the category $JSL(S,S)$6 of join-semilattices and studies automata as categorical objects (Myers et al., 2021). A $JSL(S,S)$7-automaton is an automaton

$JSL(S,S)$8

such that $JSL(S,S)$9 is a semilattice of states, each transition map $S$0 preserves finite joins, the initial state is a morphism $S$1, and the final states form a prime filter $S$2, corresponding to a morphism $S$3.

The paper states that finite $S$4-automata and NFAs are equivalent in expressive power. From a $S$5-dfa $S$6, one obtains an equivalent NFA on the set of join-irreducibles $S$7. Conversely, from an NFA $S$8, the subset construction yields the $S$9-dfa $M$0 on the semilattice $M$1. This is why $M$2 functions as the relevant analogue of “number of states” (Myers et al., 2021).

Within this framework, the paper distinguishes two measures. The ordinary nondeterministic state complexity is

$M$3

and it proves

$M$4

It then defines the nondeterministic syntactic complexity

$M$5

The distinction is structural. $M$6 measures minimal NFAs in general, whereas $M$7 measures minimal NFAs in a more structured subclass, through extensions of the canonical syntactic-monoid action. The paper stresses that $M$8 is not always equal to $M$9, although for many interesting language classes they coincide (Myers et al., 2021). This is central to any discussion of NAAs: even before introducing asynchrony, the theory already separates unrestricted nondeterministic minimization from algebraically controlled nondeterministic minimization.

4. The canonical semilattice and the role of duality

For each regular language $\rho \colon M \to JSL(S,S),$0, the minimal $\rho \colon M \to JSL(S,S),$1-dfa is identified with

$\rho \colon M \to JSL(S,S),$2

the semilattice of finite unions of left derivatives of $\rho \colon M \to JSL(S,S),$3 (Myers et al., 2021). The paper describes this automaton categorically as the image of the unique morphism from the initial $\rho \colon M \to JSL(S,S),$4-automaton to the final one, and it is minimal among $\rho \colon M \to JSL(S,S),$5-automata accepting $\rho \colon M \to JSL(S,S),$6. The join-irreducibles of $\rho \colon M \to JSL(S,S),$7 correspond to atomic pieces of the language decomposition.

A major structural ingredient is the self-duality of finite semilattices. The category $\rho \colon M \to JSL(S,S),$8 of finite semilattices is self-dual via the order-reversing equivalence

$\rho \colon M \to JSL(S,S),$9

and this lifts to automata: $\deg(\rho) := |J(S)|,$0 For a $\deg(\rho) := |J(S)|,$1-dfa

$\deg(\rho) := |J(S)|,$2

its dual is

$\deg(\rho) := |J(S)|,$3

The conceptual consequence emphasized in the paper is that reachability of $\deg(\rho) := |J(S)|,$4 dualizes to simplicity of $\deg(\rho) := |J(S)|,$5, and the duality corresponds algebraically to reversal of the recognized language. In particular, for an NFA $\deg(\rho) := |J(S)|,$6,

$\deg(\rho) := |J(S)|,$7

This is presented as the algebraic version of the usual reverse-automaton construction (Myers et al., 2021).

For NAAs, this suggests a specific criterion for successful generalization: an asynchronous model would need a duality principle with comparable explanatory force. Without such a principle, the transfer of the paper’s minimization arguments would remain incomplete.

5. Atomic and subatomic NFAs

The paper isolates two structured subclasses of NFAs (Myers et al., 2021). An NFA $\deg(\rho) := |J(S)|,$8 accepting $\deg(\rho) := |J(S)|,$9 is atomic if every state accepts a language in $S$0, the boolean closure of the left derivatives of $S$1. It is subatomic if every state accepts a language in $S$2, the boolean closure of the two-sided derivatives of $S$3.

These definitions are significant because they replace arbitrary state languages by derivative-generated state languages. The corresponding characterizations are strong. An NFA $S$4 is atomic iff

$S$5

is a minimal DFA, where $S$6 denotes the reachable subset-construction part. An NFA $S$7 accepting $S$8 is subatomic iff the transition monoid of $S$9 is isomorphic to the syntactic monoid of $\rho_i \colon M \to JSL(S_i,S_i),$0: $\rho_i \colon M \to JSL(S_i,S_i),$1

The main theorem is that

$\rho_i \colon M \to JSL(S_i,S_i),$2

The proof proceeds in both directions. If $\rho_i \colon M \to JSL(S_i,S_i),$3 is a subatomic NFA with $\rho_i \colon M \to JSL(S_i,S_i),$4 states, then the semilattice of languages accepted by subsets of $\rho_i \colon M \to JSL(S_i,S_i),$5 gives a boolean representation of $\rho_i \colon M \to JSL(S_i,S_i),$6 of degree at most $\rho_i \colon M \to JSL(S_i,S_i),$7. Conversely, any degree-$\rho_i \colon M \to JSL(S_i,S_i),$8 boolean representation of $\rho_i \colon M \to JSL(S_i,S_i),$9 extending $f \colon S_1 \to S_2$0 induces a $f \colon S_1 \to S_2$1-dfa whose join-irreducibles form a $f \colon S_1 \to S_2$2-state subatomic NFA (Myers et al., 2021).

This theorem does not describe NAAs, but it clarifies the kind of result an asynchronous analogue would need: a distinguished structured subclass of asynchronous nondeterministic automata whose optimal size is captured exactly by an algebraic invariant.

6. Lower bounds, minimization, and structural significance

The paper situates its contribution against the hardness of general NFA minimization. It states that general NFA minimization is $f \colon S_1 \to S_2$3-complete and lacks a simple uniqueness theory (Myers et al., 2021). Its response is not to solve unrestricted minimization directly, but to identify a subclass—subatomic NFAs—that is algebraically well-behaved and captures many previously known “minimal NFAs” from the literature.

This reframing is supported by lower-bound machinery. The paper connects the theory to the dependency relation $f \colon S_1 \to S_2$4, a bipartite relation between derivatives of $f \colon S_1 \to S_2$5 and those of $f \colon S_1 \to S_2$6, and proves

$f \colon S_1 \to S_2$7

where $f \colon S_1 \to S_2$8 is the bipartite dimension / biclique cover number (Myers et al., 2021). The proof is made algebraic through boolean representations and semilattice extensions.

The significance is twofold. First, nondeterministic state-minimality is recast as an invariant of the recognized regular language rather than as a purely combinatorial search over NFAs. Second, the semilattice and monoid structure explain why reversed automata, right-derivative closures, and derivative-generated state families arise naturally. A plausible implication for NAAs is that any robust asynchronous minimization theory would likely require analogous lower bounds expressed through an asynchronous dependency structure, rather than only through raw control-state counting.

7. Possible extensions to nondeterministic asynchronous models

The paper’s discussion of NAAs is deliberately hypothetical (Myers et al., 2021). It states that several concepts are potentially relevant to NAAs or other asynchronous nondeterministic models: the categorical treatment of automata, the semilattice representation of state spaces, the duality between forward and reverse behavior, the use of derivative-closed language families, and the idea of minimizing a structured subclass of nondeterministic automata by counting join-irreducibles.

It then formulates the possible extension conditionally. If NAAs admit a suitable semantics in terms of transitions indexed by asynchronous actions, one could imagine an analogue of the framework in which the canonical representation is replaced by an action/trace-based derivative structure, the semilattice $f \colon S_1 \to S_2$9 is replaced by a closure under asynchronous residuals, and the least degree of a suitable extension measures the minimal number of asynchronous control states (Myers et al., 2021).

This suggests a precise boundary between established fact and extrapolation. Established fact: the results apply directly to ordinary NFAs and semilattice automata. Extrapolation: the same pattern might organize an asynchronous theory. The term “Nondeterministic Asynchronous Finite Automata” therefore names, in this context, not a completed theory but a possible target for generalization. The most faithful synthesis is that the paper provides an algebraic template—syntactic monoid action, canonical derivative semilattice, self-duality, and join-irreducible state counting—from which a theory of NAAs might plausibly be developed, while leaving that development entirely open (Myers et al., 2021).