Nondeterministic Syntactic Complexity

We introduce a new measure on regular languages: their nondeterministic syntactic complexity. It is the least degree of any extension of the ‘canonical boolean representation’ of the syntactic monoid. Equivalently, it is the least number of states of any subatomic nondeterministic acceptor. It turns out that essentially all previous structural work on nondeterministic state-minimality computes this measure. Our approach rests on an algebraic interpretation of nondeterministic finite automata as deterministic finite automata endowed with semilattice structure. Crucially, the latter form a self-dual category.


Introduction
Regular languages admit a plethora of equivalent representations: finite automata, finite monoids, regular expressions, formulas of monadic second-order logic, and numerous others. In many cases, the most succinct representation is given by a nondeterministic finite automaton (nfa). Therefore, the investigation of stateminimal nfas is of both computational and mathematical interest. However, this turns out to be surprisingly intricate; in fact, the task of minimizing an nfa, or even of deciding whether a given nfa is minimal, is known to be PSPACE-complete [ ]. One intuitive reason is that minimal nfas lack structure: a language may have many non-isomorphic minimal nondeterministic acceptors, and there are no clearly identified and easily verifiable mathematical properties distinguishing them from non-minimal ones. As a consequence, all known algorithms for nfa minimization (and related problems such as inclusion or universality testing) require some form of exhaustive search [ , , ]. This sharply contrasts the situation for minimal deterministic finite automata (dfa): they can be characterized by a universal property making them unique up to isomorphism, which immediately leads to efficient minimization.
In the present paper, we work towards the goal of bringing more structure into the theory of nondeterministic state-minimality. To this end, we propose a novel algebraic perspective on nfas resting on boolean representations of monoids, i.e. morphisms M → JSL(S, S) from a monoid M into the endomorphism monoid Similarly, a morphism f : S → 2 corresponds uniquely to a prime filter F = f −1 [1] ⊆ S, i.e. an upwards closed subset such that X ∈ F implies X ∩ F = ∅ for every finite subset X ⊆ S. If S is finite, prime filters are precisely the sets F = {s ∈ S : s ≤ s 0 } for s 0 ∈ S. If S is a subsemilattice of a semilattice T , every prime filter F of S can be extended to the prime filter T \ (↓(S \ F )) of T , where ↓ X = { t ∈ T : t ≤ x for some x ∈ X } denotes the down-closure of a subset X ⊆ T . Equivalently, every morphism f : S → 2 can be extended to a morphism g : T → 2. In category-theoretic terminology, this means that the semilattice 2 forms an injective object of JSL.
The category JSL f of finite semilattices is self-dual [ ]. The equivalence functor JSL f − → JSL op f sends a semilattice S to its dual semilattice S op obtained by reversing the order, and a morphism f : S → T to the morphism f * : T op → S op mapping t ∈ T to the ≤ S -largest element s ∈ S with f (s) ≤ T t. Note that f is adjoint to f * : for s ∈ S and t ∈ T we have f (s) ≤ T t iff s ≤ S f * (t).

Languages.
A language is a subset L of Σ * , the set of finite words over an alphabet Σ. We let L = Σ * \ L denote the complement and L r = {w r : w ∈ L} the reverse, where w r = a n . . . a 1 for w = a 1 . . . a n . The left derivatives, right derivatives and two-sided derivatives of L are, respectively, given by u −1 L = {w ∈ Σ * : uw ∈ L}, Lv −1 = {w ∈ Σ * : wv ∈ L} and u −1 Lv −1 = {w ∈ Σ * : uwv ∈ L} for u, v ∈ Σ * . More generally, for U ⊆ Σ * the language U −1 L = u∈U u −1 L is called the left quotient of L w.r.t. U . We define the following sets of languages generated by L: In other words, SLD(L) is the ∪-semilattice of all left quotients of L, or equivalently, the ∪-subsemilattice of P(Σ * ) generated by all left derivatives. Moreover, BLD(L) and BLRD(L) form the boolean subalgebras of P(Σ * ) generated by all left derivatives and all two-sided derivatives, respectively.

Duality Theory of Semilattice Automata
In this section, we set up the algebraic framework in which nondeterministic automata can be studied. Since it involves considering several different types of automata, it is convenient to view them all as instances of a general categorical concept. For the rest of this paper, let Σ denote a fixed finite input alphabet.
Definition . . Let C be a category and let X, Y ∈ C be two fixed objects. An automaton in C is a quadruple (S, δ, i, f ) consisting of an object S ∈ C of states, a family δ = (δ a : S → S) a∈Σ of morphisms representing transitions, and two morphisms i : X → S and f : S → Y representing initial and final states (see the left-hand diagram below). A morphism between automata (S, δ, i, f ) and (S , δ , i , f ) is given by a morphism h : S → S in C preserving transitions, initial Let Aut(C ) denote the category of automata in C and their morphisms.
Example . . ( ) An automaton D = (S, δ, i, f ) in Set, the category of sets and functions, with X = 1 and Y = 2, is precisely a classical deterministic automaton. It is called a dfa if S is finite. We identify the map i : 1 → S with an initial state s 0 = i( * ) ∈ S, and the map f : S → 2 with a set F = f −1 [1] ⊆ S of final states. The language L(D, s) accepted by a state s ∈ S is the set of all words w ∈ Σ * such that δ w (s) ∈ F . The language L(D) accepted by D is the language accepted by the state s 0 .
( ) An automaton N = (S, δ, i, f ) in Rel, the category of sets and relations, with X = Y = 1, is precisely a classical nondeterministic automaton. It is called an nfa if S is finite. We identify i ⊆ 1 × S with a set I ⊆ S of initial states and f ⊆ S × 1 with a set F ⊆ S of final states. Thus, in our view an nfa may have multiple initial states. The language L(N, R) accepted by a subset R ⊆ S consists of all w ∈ Σ * such that (r, s) ∈ δ w for some r ∈ R and s ∈ F . The language L(N ) accepted by N is the language accepted by the set I.
( ) An automaton A = (S, δ, i, f ) in JSL with X = Y = 2, shortly a JSLautomaton, is given by a semilattice S of states, a family δ = (δ a : S → S) a∈Σ of semilattice morphisms specifying transitions, an initial state s 0 ∈ S (corresponding to i : 2 → S), and a prime filter F ⊆ S of final states (corresponding to f : S → 2). It is called a JSL-dfa if S is finite. The language accepted by a state s ∈ S or by the automaton A, resp., is defined as for deterministic automata.
Remark . (JSL-dfas vs. nfas). Dfas, nfas and JSL-dfas are expressively equivalent; they all accept precisely the regular languages. The interest of JSLdfas is that they constitute an algebraic representation of nfas: ; the initial states are those s ∈ J(S) with s ≤ s 0 , and the final states form the set J(S) ∩ F .
( ) Conversely, for every nfa N = (Q, δ, I, F ), the subset construction yields an equivalent JSL-dfa P(N ) with states P(Q) (the ∪-semilattice of subsets of Q), transitions Pδ a : P(Q) → P(Q), X → δ a [X], initial state I ∈ P(Q), and final states those subsets of Q containing some state from F . Note that J(P(Q)) ∼ = Q. It follows that the task of finding a state-minimal nfa for a given language is equivalent to finding a JSL-dfa with a minimum number of join-irreducibles [ ]. This idea has recently been extended to a general coalgebraic framework [ , ].
Recall that the minimal dfa [ ] for a regular language L, denoted by dfa(L), has states LD(L) (the set of left derivatives of L), transitions K a − → a −1 K for K ∈ LD(L) and a ∈ Σ, initial state L = ε −1 L, and final states those K ∈ LD(L) containing ε. Up to isomorphism, it can be characterized as the unique dfa accepting L that is reachable (i.e. every state is reachable from the initial state via transitions) and simple (i.e. any two distinct states accept distinct languages). We now develop the analogous concepts for JSL-automata; they are instances of the categorical theory of minimality due to Arbib and Manes [ ] and Goguen [ ].
( ) The final JSL-automaton Fin(L) for L has states P(Σ * ) (the ∪-semilattice of all languages), initial state L, final states all languages K containing ε, and transitions K → a −1 K for K ∈ P(Σ * ) and a ∈ Σ.
As suggested by the terminology, these automata form the initial and the final object in the category of JSL-automata accepting L: Remark . . ( ) The category Aut(JSL) has a factorization system given by surjective and injective morphisms. Thus, for every JSL-automata morphism h : using that 2 op ∼ = 2. Thus, the initial state of A op is the ≤ S -largest non-final state of A, and its final states are those s ∈ S with s 0 ≤ S s. Given s, t ∈ S and a ∈ Σ, there is a transition s The dualization of JSL-dfas can be seen as an algebraic generalization of the reversal operation on nfas. Recall that the reverse of an nfa N is the nfa N r obtained by flipping all transitions and swapping initial and final states. If N accepts the language L, then N r accepts the reverse language L r .
Lemma . . For each nfa N = (Q, δ, I, F ), we have the JSL-dfa isomorphism The following lemma summarizes some important properties of A op : ( ) If A accepts the language L, then A op accepts the reverse language L r .
Our next goal is to give, for every regular language L, dual characterizations of SLD(L), BLD(L) and BLRD(L), the JSL-subautomata of Fin(L) carried by all finite unions of left derivatives, boolean combinations of left derivatives and boolean combinations of two-sided derivatives, respectively. These results form the core of our duality-based approach to (sub-)atomic nfas in the next section. The minimal JSL-dfa SLD(L) admits the following dual description: Proposition . . For every regular language L, the minimal JSL-dfas for L and L r are dual. More precisely, we have the JSL-dfa isomorphism Remark . . ( ) The isomorphism dr L induces a bijection between the left and right factors of L, i.e. the inclusion-maximal left/right solutions of X · Y ⊆ L.
Conway [ ] observed that the left and right factors are respectively {K r : K ∈ SLD(L r )} and {K : K ∈ SLD(L)} and that they biject. Backhouse [ ] observed that they are dually isomorphic posets. Proposition . provides an explicit automata-theoretic lattice isomorphism arising canonically via duality.
( ) The isomorphism dr L is tightly connected to the dependency relation [ , ] of a regular language L, i.e. the binary relation given by Its restriction DR j L := DR L ∩ J(SLD(L)) × J(SLD(L r )) to the ∪-irreducible left derivatives of L and L r is called the reduced dependency relation. The following theorem shows that the semilattice of left quotients and the dependency relation are essentially the same concepts. In part ( ), we use that the isomorphism dr L restricts to a bijection between the ∪-irreducible derivatives of L r and the meet-irreducible elements of the lattice SLD(L).

Theorem . (Dependency theorem). ( ) We have the JSL-isomorphism
Note that its codomain forms a subsemilattice of P(LD(L r )).
( ) The following diagram in Rel commutes: y y

Nondeterministic Syntactic Complexity
Let us now turn to a dual characterization of the JSL-dfa BLD(L): Proposition . . For every regular language L, the JSL-dfa BLD(L) is dual to the subset construction of the minimal dfa for L r : To state the dual characterization of BLRD(L), we recall two standard concepts from algebraic language theory [ ]. The transition monoid of a deterministic Thus, tm(M ) is carried by the set of extended transition maps δ w (w ∈ Σ * ) with multiplication given by δ v • δ w = δ vw and unit id S = δ ε : S → S. We may view tm(D) as a deterministic automaton with initial state id S , final states all δ w such that w is accepted by D, and transitions δ w a − → δ wa for w ∈ Σ * and a ∈ Σ. This automaton accepts the same language as D. The syntactic monoid syn(L) of a regular language L ⊆ Σ * is the transition monoid of its minimal dfa: Equivalently, syn(L) is the quotient monoid of the free monoid Σ * modulo the syntactic congruence of L, i.e the monoid congruence on Σ * given by v ≡ L w iff ∀x, y ∈ Σ * : xvy ∈ L ⇐⇒ xwy ∈ L.
Proposition . . For every regular language L, the JSL-dfa BLRD(L) is dual to the subset construction of syn(L r ), viewed as a dfa: Our final duality result in this section concerns the transition semiring [ ], a generalization of the transition monoid to JSL-automata. Note that the monoid JSL(S, S) of endomorphisms of a semilattice S forms an idempotent semiring with join defined pointwise: for any f, g : Here P f (Σ * ) is the free idempotent semiring on Σ, with composition given by concatenation of languages and join given by union. Thus, ts(A) is the semiring carried by all morphisms n i=1 δ wi for w 1 , . . . , w n ∈ Σ * , with join given as above and multiplication j δ vj • i δ wi = i,j δ vj wi . We view ts(A) as a JSL-automaton with initial state id S = δ ε , final states all i δ wi such that some w i is accepted by A, and transitions . . , w n ∈ Σ * and a ∈ Σ. This JSL-automaton is reachable and accepts the same language as A. It has the following dual characterization: Notation . . Given a simple JSL-automaton A = (S, δ, i, f ), the subautomaton of Fin(L) obtained by closing S (viewed as a set of languages) under right derivatives is called the right-derivative closure of A and denoted rdc(A).
Proposition . . Let A be a reachable JSL-dfa. Then the transition semiring of A, viewed as a JSL-dfa, is dual to the right-derivative closure of A op : Note that both [ts(A)] op and rdc(A op ) are simple, hence subautomata of Fin(L). Thus, the isomorphism just expresses that their states accept the same languages.

Boolean Representations and Subatomic NFAs
Based upon the duality results of the previous section, we will now introduce our algebraic approach to nondeterministic state minimality. It rests on the concept of a representation of a monoid on a finite semilattice.

( ) Given boolean representations ρ
If f is injective, we say that the representation ρ 2 extends ρ 1 .
( ) The category of boolean representations of M coincides with the functor category JSL M f , viewing M as a one object category.

Definition . (Canonical representation).
For every regular language L, the canonical boolean representation of the syntactic monoid syn(L) is given by It induces the canonical boolean presentation of the free monoid Σ * given by The representation κ L • µ L amounts to constructing the transition semiring of the minimal JSL-automaton SLD(L), i.e. the syntactic semiring [ ] of L.
The next theorem links minimal nfas and representations. S. One can equip S with a JSL-dfa structure making h an automata morphism. Since morphisms preserve accepted languages, it follows that S accepts L. Then the nfa of join-irreducibles of S, see Remark . , is a k-state nfa accepting L.
As an application, let us return to the dependency relation DR L introduced in Remark . ( ). Recall that a biclique of a relation R ⊆ X × Y (viewed as a bipartite graph) is a subset of the form X × Y ⊆ R, where X ⊆ X and Y ⊆ Y . A biclique cover of R is a set C of bicliques with R = C . The bipartite dimension dim(R) is the least cardinality of any biclique cover of R.
We give a new algebraic proof of this result based on boolean representations.
Proof. ( ) The task of computing biclique covers is well-known to be equivalent to the set basis problem. Given a family C ⊆ P(Y ) of subsets of a finite set Y , a set basis for C is a family B ⊆ P(Y ) such that each element of C can be expressed as a union of elements of B. A relation R ⊆ X × Y has a biclique cover of size k iff the family C R = {R[x] : x ∈ X} ⊆ P(Y ) of neighborhoods of nodes in X has a set basis of size k.
( ) Given an instance C ⊆ P(Y ) of the set basis problem, consider the ∪subsemilattice C ⊆ P(Y ) generated by C, i.e. the semilattice of all unions of sets in C. We claim that C has a set basis of size at most k iff there exists an extension of C of degree at most k, i.e. a monomorphism C S into some finite semilattice S with |J(S)| ≤ k.
For the "only if" direction, suppose that B ⊆ P(Y ) is a set basis of C of size at most k. The the embedding C B gives an extension of C with the desired property: since the semilattice B has a set of generators with at most k elements, it has at most k join-irreducibles. For the "if" direction, suppose that m : C S with |J(S)| ≤ k is given. Since the free semilattice P(Y ) is an injective object of JSL [ , Corollary . ], there exists a morphism f : S → P(Y ) extending the embedding C P(Y ). Consider the image S ⊆ P(Y ) of f , leading to the commutative diagram below: We thus have C ⊆ S ⊆ P(Y ). Every set of generators of the semilattice S is a basis of C. Since the morphism e is surjective, we have |J(S )| ≤ |J(S)| ≤ k, i.e. S has a set of generators with at most k elements.
( ) Let C DR L ⊆ P(LD(L r )) be the instance of the set basis problem corresponding to the dependency relation DR L ⊆ LD(L) × LD(L r ). Note that C DR L consists of all DR L [X] for X ⊆ LD(L). Thus, Theorem . ( ) shows that C DR L ∼ = SLD(L). In particular, every extension of the canonical boolean representation of Σ * yields an extension of the semilattice C DR L of the same degree. Therefore, by part ( ) and ( ) and Theorem . , we have dim(DR L ) ≤ ns(L), as required.
Theorem . motivates the following definition, which can be considered the key concept of our paper: Definition . . The nondeterministic syntactic complexity nµ(L) of a regular language L is the least degree of any boolean representation of syn(L) extending the canonical boolean representation κ L : syn(L) → JSL(SLD(L), SLD(L)).
Just like the degrees of boolean representations of Σ * determine the state complexity of nfas, we will provide an automata-theoretic characterization of nµ(L) in terms of subatomic nfas in Theorem . below.
Definition . . An nfa accepting the language L is called ( ) atomic if each state accepts a language from BLD(L), and ( ) subatomic if each state accepts a language from BLRD(L).
The notion of an atomic nfa goes back to Brzozowski and Tamm [ ], as does the following characterization.
Notation . . For any nfa N , let rsc(N ) denote the dfa obtained via the reachable subset construction, i.e. the dfa-reachable part of P(N ).

Theorem . . An nfa N is atomic iff rsc(N r ) is a minimal dfa.
We present a new conceptual proof, interpreting this theorem as an instance of the self-duality of JSL-dfas.
Proof (Sketch). Let L be the language accepted by N . We establish the theorem by showing each of the following statements to be equivalent to the next one: ( ) N is atomic.
( ) There exists a JSL-automata morphism from P(N ) to BLD(L).
( ) There exists a JSL-automata morphism from P(dfa(L r )) to P(N r ).
( ) There exists a dfa morphism from dfa(L r ) to P(N r ).
( ) There exists a dfa morphism from dfa(L r ) to rsc(N r ).
The key step is ( )⇔( ), which follows via duality from Lemmas . and . , and Proposition . . All remaining equivalences follow from the definitions.
The next theorem gives an analogous characterization of subatomic nfas. Again, the proof is based on duality.
Theorem . . An nfa N accepting the language L is subatomic iff the transition monoid of rsc(N r ) is isomorphic to the syntactic monoid syn(L r ).

Proof (Sketch).
Each of the following statements is equivalent to the next one: ( ) N is subatomic.
( ) There exists a dfa morphism from syn(L r ) to ts(reach(P(N r ))).
( ) There exists a dfa morphism from syn(L r ) to tm(rsc(N r )).
The equivalence ( )⇔( ) follows via duality from Lemma . , Proposition . and Proposition . . All remaining equivalences follow from the definitions.
We are prepared to state the main result of our paper, an automata-theoretic characterization of the nondeterministic syntactic complexity: Theorem . . For every regular language L, the nondeterministic syntactic complexity nµ(L) is the least number of states of any subatomic nfa accepting L.

Proof (Sketch).
( ) Let N be a k-state subatomic nfa accepting the language L. As in the proof of Theorem . , we consider the semilattice langs(N ) = simple(P(N )). ( ) Conversely, let ρ : syn(L) → JSL(S, S) be a boolean representation extending κ L , and let h : SLD(Q) S be the embedding. As in the proof of Theorem . , we can equip S with the structure of a JSL-dfa making h an automata morphism. Its nfa of join-irreducibles, see Remark . , is a subatomic nfa accepting L with deg(ρ) states.
We conclude this section with the observation that the state complexity of unrestricted nfas, subatomic nfas and atomic nfas generally differs: Example .
(Subatomic more succinct than atomic). Consider the language L accepted by the nfa N shown below, along with the minimal dfas for L and L r . Each automaton has exactly one initial state, namely 0.
Brzozowski and Tamm [ ] showed that there is no atomic nfa with four states accepting L. However, N is subatomic: one can verify that the transition monoids of dfa(L r ) and rsc(N r ) both have 22 elements. Since the former is the syntactic monoid of L r , they are isomorphic, and so Theorem . applies.

Example . (Subatomic less succinct than general nfas).
There is a regular language for which no state-minimal nfa is subatomic: It is accepted by the following nfa: An exhaustive search shows that no subatomic nfa with five states accepts L. In fact, L is the unique (!) unary language with ns(L) ≤ 5 and ns(L) < nµ(L). Moreover, the above nfa and its reverse are the only state-minimal nfas for L.

Applications
While subatomic nfas are generally less succinct then unrestricted ones, all structural results concerning nondeterministic state complexity we have encountered in the literature are actually about nondeterministic syntactic complexity: they implicitly identify classes of languages where the two measures coincide. In the present section, we illustrate this in a few selected applications. .

Unary languages
For unary languages L ⊆ {a} * , two-sided derivatives are left derivatives. Thus, a unary nfa is atomic iff it is subatomic.

Example . (Cyclic unary languages).
A unary language L is cyclic if its minimal dfa is a cycle [ ]. We claim that ns(L) = nµ(L). To see this, let d := |LD(L)| be the period (i.e. number of states) of the minimal dfa. By Fact of [ ] (originally from [ ]) every state-minimal nfa N accepting L is a disjoint union of cyclic dfas whose periods divide d. Then |rsc(N r )| = d: we have |rsc(N r )| ≥ d since rsc(N r ) is a dfa accepting L = L r and d is the size of the minimal dfa for L, and |rsc(N r )| ≤ d because after d steps, each cycle will be back in its initial state. Thus N is atomic by Theorem . and hence subatomic.
We deduce the following result for (not necessarily unary) regular languages:

Example . (nµ(L) no larger than Chrobak normal form).
A unary nfa is in Chrobak normal form [ , ] if it has a single initial state and at most one state with multiple successors, all of which lie in disjoint cycles. We claim that for any nfa N in Chrobak normal form accepting the language L, we have nµ(L) ≤ |N |, In [ ] nfas are restricted to have a single initial state and so are distinguished from unions of dfas; the latter are valid nfas from our perspective.
where |N | denotes the number of states of N . To see this, observe that each state of N up to and including the unique choice state accepts some left derivative of L. The successors of the choice state collectively accept a derivative u −1 L; this language is cyclic because it is a finite union of cyclic languages. Therefore, by Example . we may replace the cycles by an atomic nfa accepting u −1 L, without increasing the number of states. The resulting nfa is atomic.
Since every unary nfa on n states can be transformed into an nfa in Chrobak normal form with O(n 2 ) states [ , Lemma . ], we get: Corollary . . If L is a unary regular language, then nµ(L) = O(ns(L) 2 ). .

Languages with a canonical state-minimal nfa
There are several natural classes of regular languages for which canonical stateminimal nondeterministic acceptors have been identified. We show that these acceptors are actually subatomic. In our arguments, we frequently consider the length of a finite semilattice S, i.e. the maximum length n of any ascending chain s 0 < s 1 < . . . < s n in S. Note that since every element is uniquely determined by the set of join-irreducibles below it, the length of S is at most |J(S)|.

Example . (Bideterministic and biseparable languages).
( ) A language is called bideterministic if it is accepted by a dfa whose reverse is also a dfa. In this case, the minimal dfa is a minimal nfa [ , ]. Bideterministic languages have been studied in the context of automata learning [ ] and coding theory, where they are known as rectangular codes [ , ]. We show that for every bideterministic language L, ns(L) = nµ(L) = |LD(L)|.
To this end, we first note that by [ , Theorem . ] a language L ⊆ Σ * is bideterministic iff the left derivatives of L are pairwise disjoint. This implies that SLD(L) is a boolean algebra with atoms LD(L). Since the length of a boolean algebra equals the number of atoms (= join-irreducibles), we conclude that for every finite semilattice extension SLD(L) S, the semilattice S has length at least |LD(L)|. Thus, |LD(L)| ≤ |J(S)|, so any representation ρ extending κ L or κ L • µ L satisfies |LD(L)| ≤ deg(ρ). Hence, ns(L) = nµ(L) = |LD(L)| by Theorem . and . . In particular, the minimal dfa of L is a minimal nfa.
For every biseparable language L, the canonical residual automaton [ ], i.e. the nfa N L of join-irreducibles of the minimal JSL-dfa SLD(L), is a state-minimal nfa; it is subatomic because every state of N L accepts a derivative of L. This follows exactly as in ( ): our argument only used that SLD(L) is a boolean algebra.
Actually [ ] defines biseparability as a property of nfas, and characterizes biseparable nfas as those accepting a language L for which no ∪-irreducible left derivative is contained in the union of other ∪-irreducible left derivatives. This is equivalent to the lattice SLD(L) being boolean, i.e. to L being 'biseparable' in our sense.

Example . (Maximal reachability).
A folklore result asserts that if N is an nfa whose accepted language L satisfies |LD(L)| = 2 |N | , then N is stateminimal. Since LD(L) forms the set of states of the minimal dfa for L and rsc(N ) accepts L, we have rsc(N ) = P(N ). It follows the JSL-dfa P(N ) is reachable and simple, hence isomorphic to the minimal JSL-dfa SLD(L). This proves that SLD(L) is a boolean algebra, i.e. L is a biseparable language. We conclude from Example . ( ) that ns(L) = nµ(L) = |N | and N L is a subatomic minimal nfa.
Example . (BiRFSA and topological languages). So far SLD(L) has been a boolean algebra. But the argument in Example . also applies when SLD(L) is a distributive lattice, noting that the length of a finite distributive lattice is equal to the number of its join-irreducibles [ , Corollary . ]. Languages with this property are called topological [ ]. It thus follows as in Example . ( ) that for any topological language L, the canonical residual automaton N L is subatomic and a state-minimal nfa. Thus, ns(L) = nµ(L) = |J(SLD(L))|.
There is another class of languages where N L is known to be a state-minimal nfa, the biRFSA . Surprisingly, these languages are exactly the topological ones: ( ) Suppose that L is topological. Recall that N L is the nfa of join-irreducibles of the minimal JSL-dfa. Thus, it has states J(SLD(L)) and transitions given by Moreover, a join-irreducible j is initial iff j ⊆ L and final iff ε ∈ j. Since the lattice SLD(L) is distributive, we have a canonical bijection between its join-and meet-irreducibles: Let θ be the unique map making the following diagram commute, where dr L is the restriction of the isomorphism of Proposition . : One can show θ to be an nfa isomorphism from N L to (N L r ) r . Thus, L is biRFSA.
( ) Suppose that L is biRFSA. Then we have a surjective JSL-morphism where the first isomorphism follows from N L ∼ = (N L r ) r and Lemma . , the second isomorphism is given by Proposition . , and e L r sends X ⊆ J(SLD(L r )) to X. The dual of this morphism is the injective JSL-morphism ( ) L is topological iff DR j L is essentially an order relation ≤ P ⊆ P × P of a finite poset [ , Example . . ]. Theorem ]. The latter means the adjacency matrix of the bipartite graph DR j L can be put in upper triangular form with ones along the diagonal, by permuting rows and columns. An order relation is upper unitriangularizable because it may be extended to a linear order.

Conclusion and Future Work
Motivated by the duality theory of deterministic finite automata over semilattices, we introduced a natural class of nondeterministic finite automata called subatomic nfas and studied their state complexity in terms of boolean representations of syntactic monoids. Furthermore, we demonstrated that a large body of previous work on state minimization of general nfas actually constructs minimal subatomic ones. There are several directions for future work.
As illustrated by Theorem . , the dependency relation DR L forms a useful tool for proving lower bounds on nfas. It is also a key element of the Kameda-Weiner algorithm [ , ] for minimizing nfas, which rests on computing biclique covers of DR L . We aim to give an algebraic interpretation of dependency relations based on the representation of finite semilattices by contexts [ ], which can be augmented to a categorical equivalence between JSL f and a suitable category of bipartite graphs [ ]. Under this equivalence, JSL-dfas correspond to dependency automata; in particular, the minimal JSL-dfa SLD(L) corresponds to a dependency automaton whose underlying bipartite graph is precisely the dependency relation DR L . We expect that this observation can lead to a fresh algebraic perspective on the Kameda-Weiner algorithm, as well as a generalization of it computing minimal (sub-)atomic nfas.
On a related note, we also intend to investigate the complexity of the minimization problem for (sub-)atomic nfas. While minimizing general nfas is PSPACEcomplete, even if the input automaton is a dfa, we conjecture that the additional structure present in (sub-)atomic acceptors will simplify their minimization to an NP-complete task. First evidence in this direction is provided by Geldenhuys, van der Merve, and van Zijl [ ] whose work implies that minimal atomic nfas can be efficiently computed in practice using SAT solvers. Open Access This chapter is licensed under the terms of the Creative Commons Attribution . International License (http://creativecommons.org/licenses/by/ . /), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made. The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.