A Survey on Fooling Sets as Effective Tools for Lower Bounds on Nondeterministic Complexity
Abstract
A fooling set for a regular language is a special set of pairs of strings whose size provides a lower bound on the number of states in any nondeterministic finite automaton accepting this language. We show that, in spite of the fact that the difference between the size of the largest fooling set and the nondeterministic state complexity may be arbitrarily large, the fooling set lower bound methods work in many cases. We modify the method in the case when multiple initial states may save one state. We also state some useful properties that allow us to avoid describing particular fooling sets which may often be difficult and tedious.
1 Introduction
The nondeterministic state complexity of a regular language is the smallest number of states in any nondeterministic finite automaton (NFA) accepting this language. To get lower bounds on the nondeterministic state complexity, usually a fooling set technique is used. A fooling set is a special set of pairs of strings whose size provides a lower bound on the number of states in any NFA for a given language.
The lower bound method based on fooling sets, as a version of the crossing sequence argument, has been used for proving lower bounds on VLSI computations [4, 14, 15, 19, 22]. The fooling set method as a method providing lower bounds on communication complexity has been formulated by Aho, Ullman, and Yannakakis [1]. In the settings of formal languages, the method has been first described by Birget [2, 3], and examined by Glaister and Shallit [6].
Although the gap between the size of a largest fooling set and the nondeterministic state complexity of a regular language may be arbitrarily large [7], in many cases, the fooling set method provides lower bounds that are tight.
In [12, 13, 18, 20], the nondeterministic state complexity of basic regular operations in the subclasses of convex languages has been investigated. The authors considered operations of union, intersection, concatenation, star, reversal, and complementation in the classes of prefix, suffix, factor, and subwordfree, closed, and convex languages, and right, left, twosided, and allsided ideals. For each operation and each class, except for complementation on factor and subwordconvex languages, tight upper bounds have been obtained, and to get lower bounds, a fooling set lower bound method has been used in each case.
Here we present the fooling set method as an effective tool for getting lower bounds on the nondeterministic state complexity. We state some sufficient properties on NFAs that guarantee the existence of a sufficiently large fooling set for the accepted language. As a result, we can avoid the description of a fooling set which may sometimes be rather difficult and tedious. Moreover, the size of a fooling set provides a lower bound on the size of NFAs even with multiple initial states. This means that, for example, in the case of union or reversal, where NFAs with multiple initial states may save one state in the resulting automaton, the method cannot yield matching bounds. We describe a modification of the method consisting in a possibility to divide a fooling set into two parts such that adding a pair with left component equal to the empty string results again in a fooling set. Since after reading the empty string, the NFA is in its unique initial state, this state must be different from all states given by the fooling set.
We start by restating the result from [7] that the gap between the size of a largest fooling set and the nondeterministic state complexity of a regular language may be arbitrarily large. Then we formulate two lemmas with sufficient conditions on NFAs that guarantee their minimality. We continue with a modification of the fooling set method providing lower bounds on the size of NFAs with a unique initial state. Finally, we give a sufficient condition for getting large lower bounds for the complementation operation.
2 Preliminaries
We assume that the reader is familiar with basic notions in formal languages and automata theory. For details and all the unexplained notions, the reader may refer to [11, 23, 24].
Let \(\varSigma \) be a finite nonempty alphabet of symbols. Then \(\varSigma ^*\) denotes the set of strings over the alphabet \(\varSigma \) including the empty string \(\varepsilon \). The length of a string w is denoted by w, and the number of occurrences of a symbol a in a string w by \(w_a\). A language is any subset of \(\varSigma ^*\). For a finite set X, the cardinality of X is denoted by X, and its power set by \(2^X\).
A nondeterministic finite automaton (with a nondeterministic choice of initial states; cf. [24]) (NNFA) is a quintuple \(A=(Q,\varSigma ,\cdot ,I,F)\), where Q is a finite nonempty set of states, \(\varSigma \) is a finite nonempty alphabet, \(I\subseteq Q\) is the set of initial states, \(F\subseteq Q\) is the set of final (or accepting) states, and the function \(\cdot :Q\times \varSigma \rightarrow 2^Q\) is the transition function which is naturally extended to the domain \(2^Q\times \varSigma ^*\). The language accepted by A is \(L(A)=\{w\in \varSigma ^{*}\mid I \cdot w\cap F \ne \emptyset \}\).
We say that (p, a, q) is a transition in A if \(q \in p\cdot a\). If (p, a, q) is a transition in A, then we say that the state q has an intransition, and the state p has an outtransition. We sometimes write \(p\xrightarrow {w} q\) if \(q\in p\cdot w\).
An NNFA A is a trim NNFA if each its state q is reachable and useful, that is, there are strings u and v in \(\varSigma ^*\) such that \(q\in I \cdot u\) and \(q\cdot v \cap F\ne \emptyset \).
If \(I=1\), we say that A is a nondeterministic finite automaton (NFA). In an \(\varepsilon \)NFA, we also allow transitions on the empty string. It is known that the \(\varepsilon \)transitions can be removed without increasing the number of states in the resulting NFA (cf. [11, Theorem 2.2] and [24, Theorem 2.3]).
An NFA A is a deterministic finite automaton (DFA) if \(q \cdot a=1\) for each q in Q and each a in \(\varSigma \). Next, A is a partial deterministic finite automaton if \(q \cdot a\le 1\) for each q in Q and each a in \(\varSigma \).
Every NNFA \(A=(Q,\varSigma ,\cdot ,I,F)\) can be converted to an equivalent DFA \(\mathcal {D}(A)=(2^Q,\varSigma ,\cdot \,,I,\{S\in 2^Q\mid S\cap F\ne \emptyset \})\). We call the DFA \(\mathcal {D}(A)\) the subset automaton of the NNFA A. The subset automaton might not be minimal since some of its states may be unreachable or equivalent to other states.
The reversal \(w^R\) of a string w is defined as \(\varepsilon ^R=\varepsilon \) and \((wa)^R=aw^R\) for each symbol a and string w. The reversal of a language L is \(L^R=\{w^R\mid w\in L\}\). If a language L is accepted by an NNFA \(A=(Q,\varSigma ,\cdot ,I,F)\), then the language \(L^R\) is accepted by the NNFA \(A^R\) obtained from A by reversing all the transitions, and by swapping the roles of the initial and final states. Formally, we have \(A^R=(Q,\varSigma ,\cdot ^R,F,I)\) where \(q \cdot ^R a =\{p\in Q\mid q \in p\cdot a\}\).
Let \(A=(Q,\varSigma ,\cdot ,I,F)\) be an NNFA and \(S,T \subseteq Q\). We say that S is reachable in A if there is a string w in \(\varSigma ^*\) such that \(S= I \cdot w\). Next, we say that T is coreachable in A if T is reachable in \(A^R\). Notice that if T is coreachable in A, then there is a string w in \(\varSigma ^*\) such that w is accepted by A from each state in T and rejected from each state in \(T^c\).
If \(u,v,w,x\in \varSigma ^*\) and \(w=uxv\), then u is a prefix of w, x is a factor of w, and v is a suffix of w. If \(w=u_0v_1u_1\cdots v_nu_n\), where \(u_i,v_i \in \varSigma ^*\), then \(v_1v_2\cdots v_n\) is a subword of w. A prefix v (suffix, factor, subword) of w is proper if \(v\ne w\).
A language L is prefixfree if \(w\in L\) implies that no proper prefix of w is in L; it is prefixclosed if \(w\in L\) implies that each prefix of w is in L; and it is prefixconvex if \(u,w\in L\) and u is a prefix of w imply that each string v such that u is a prefix of v and v is a prefix of w is in L. Suffix, factor, and subwordfree, closed, and convex languages are defined analogously.
A language L is a right (respectively, left, twosided, allsided) ideal if \(L=L\varSigma ^*\) (respectively, Open image in new window ), where Open image in new window denotes the shuffle operation [5]. Notice that the classes of free, closed, and ideal languages are subclasses of convex languages.
3 Fooling Set Lower Bound Method
To get lower bounds on the number of states in an NNFA accepting a regular language, the fooling set technique has been successfully used in the literature. We start with the definition of a fooling set, and with a lemma showing that the size of a fooling set for a regular language provides a lower bound on the nondeterministic state complexity of this language.
Definition 1
 (1)
\(x_i y_i \in L\), and
 (2)
if \(i\ne j\), then \(x_i y_j \notin L\) or \(x_j y_i \notin L\).
Example 1
Let \(L=\{a^{n}\}\) where \(n\ge 0\). Consider the set of pairs of strings \(\mathcal {F}=\{(a^i, a^{ni}) \mid i=0,1,\ldots ,n \}\). The string \(a^n\) is in L, while for each k with \(k\ne n\), the string \(a^k\) is not in L. It follows that \(\mathcal {F}\) is a fooling set for L. \(\square \)
Lemma 1
(Birget [2, Lemma 1]). Let \(\mathcal {F}\) be a fooling set for a regular language L. Then every NNFA for L has at least \(\mathcal {F}\) states.
Proof
Our first aim is to show that a gap between the maximal size of the fooling set and the nondeterministic state complexity of a language can be arbitrarily large. To this end, we introduce the notion of a fooling set for an automaton.
Definition 2
 (1)
\(X_i\) is reachable and \(Y_i\) is coreachable in A,
 (2)
\(X_i \cap Y_i\ne \emptyset \), and
 (3)
if \(i\ne j\), then \(X_i\cap Y_j=\emptyset \) or \(X_j\cap Y_i=\emptyset \).
Example 2
Proposition 1
Let A be an NNFA. Then a fooling set of size n for the language L(A) exists if and only if a fooling set of size n for the automaton A exists.
Proof
Conversely, since \(X_i\) is reachable, there is a string \(x_i\) such that \(X_i = I \cdot x_i\). Since \(Y_i\) is coreachable, there is a string \(y_i\) such that \(Y_i=F\cdot ^R y_i\). Then the set \(\{(x_i,y_i^R)\mid i=1,2,\ldots ,n\}\) is a fooling set for L(A). \(\square \)
Theorem 1
(cf. Gruber and Holzer [7, Theorem 10]). Let \(n\ge 4\). There exists a language L such that every fooling set for L is of size at most 3, while every NFA for L has at least \(\log _2 n\) states.
Proof
Let L be the unary language accepted by the DFA A with the state set \(Q=\{1,2,\ldots ,n\}\) shown in Fig. 2. The DFA A has n states and it is a minimal DFA for L. It follows that every NFA for L must have at least \(\log _2 n\) states.

\(\mathcal {R}=\{S\subseteq Q\mid S \text { is reachable in A}\}=\{ \{i\} \mid i=1,2,\ldots ,n\}\),

\(\mathcal {C}=\{T\subseteq Q\mid T \text { is coreachable in A}\} = \{ Q\setminus \{i\} \mid 2\le i \le n\}\cup \{Q\setminus \{1,n\}\}\),

\((S,T)\in E\) iff \(S\cap T \ne \emptyset \).
Notice that each \(\{i\}\) in \(\mathcal {R}\), except for \(\{n\}\), is connected to each set in \(\mathcal {C}\), except for the set \(Q\setminus \{i\}\). The set \(\{n\}\) is connected to each set in \(\mathcal {C}\), except for \(Q\setminus \{n\}\) and \(Q\setminus \{1,n\}\).
Now the set \(\mathcal {S}=\{(X_i,Y_i) \mid i=1,2,3,4\}\) is a fooling set for A by assumption. Consider the subgraph G of \((\mathcal {R},\mathcal {C},E)\) induced by the set \(\{X_i \mid i=1,2,3,4\}\cup \{Y_i \mid i=1,2,3,4\}\). If \(X_i\ne \{n\}\), then \(X_i\) is not connected to at most one of \(Y_i, \ i=1,2,3,4\). If \(X_i=\{n\}\), then \(X_i\) is not connected to at most two of \(Y_i,\ i=1,2,3,4\). This means that there are at least 11 edges in G. However, since \(\mathcal {S}\) is a fooling set, for every two distinct pairs \((X_i,Y_i)\) and \((X_j,Y_j)\), at least one edge must be missing in G. In total, at least 6 edges must be missing in G – one for every two distinct pairs. However, this only gives 10 possible edges in G, a contradiction. \(\square \)
Although the previous theorem shows that the gap between the size of a fooling set for a regular language and its nondeterministic state complexity may be arbitrarily large, our next aim is to show that in most cases, the fooling set technique provides lower bounds that are tight.
The next example illustrates how both types of fooling sets can be used to obtain the nondeterministic state complexity of the star operation. The upper bound on the nondeterministic state complexity of the star operation is \(n+1\) since we can construct an \(\varepsilon \)NFA for the star of a language given by an NFA by adding a new initial and final state connected through \(\varepsilon \)transitions to the original initial state, and by adding \(\varepsilon \)transitions from every final state to the original initial state. The next example provides a unary witness language.
Example 3
Then for each \(i=0,1,\ldots ,n\), the set \(X_i\) is reachable and the set \(Y_i\) is coreachable in \(A^*\). Next, if \((i,j)\in \{(0,n),(n2,n1),(n1,n)\}\), then \(X_j\cap Y_i=\emptyset \), otherwise, \(X_i \cap Y_j =\emptyset \) if \(i<j\). It follows that the set \(\{(X_i, Y_i) \mid i=0,1,\ldots ,n\}\) is a fooling set for \(A^*\), so every NNFA for \(L^*\) has at least \(n+1\) states. \(\square \)
4 Simplifications of the Fooling Set Method
In this section, we state some sufficient conditions on an NNFA that guarantee its minimality. Having such an NNFA, there is no need to describe a fooling set for the accepted language since we know that every equivalent NNFA has at least as many states as the given NNFA.
Lemma 2
Let \(A=(Q,\varSigma ,\cdot ,I,F)\) be an NNFA. Suppose that, for each state q in Q, the oneelement set \(\{q\}\) be reachable as well as coreachable in A. Then every NNFA for L(A) has at least Q states.
Proof
Since \(\{q\}\) is reachable in A, there is a string \(x_q\) such that \(I\cdot x_q=\{q\}\). Since \(\{q\}\) is coreachable in A, there is a string \(y_q\) accepted by A from and only from the state q. Then \(\{(x_q,y_q) \mid q\in Q\}\) is a fooling set for the language L(A). By Lemma 1, every NNFA for L(A) has at least Q states. \(\square \)
Notice that if A is a trim partial DFA, then for each state q of A, the singleton set \(\{q\}\) is reachable. If moreover \(A^R\) is a partial DFA, then \(\{q\}\) is coreachable in A. So we get the following result.
Lemma 3
Let A be an nstate trim NFA. If both A and \(A^R\) are partial DFAs, then every NNFA for L(A) has at least n states. \(\square \)
Let us show how Lemma 2 can be used to get the nondeterministic state complexity of the union operation on suffixfree languages. Recall that if two NFAs A and B accept suffixfree languages, then we may assume that their initial states do not have any intransitions [8, 21] This means that we can merge the initial states to get an NFA for \(L(A)\cup L(B)\). This gives an upper bound of \(m+n1\). In the next example, we use Lemma 2 to prove the tightness of this upper bound.
Example 4
(Union on suffixfree languages; cf. [13, Theorem 9]). Consider the NFAs A and B shown in Fig. 5; notice that the languages L(A) and L(B) are suffixfree. Construct an NFA for \(L(A) \cup L(B) \) by merging the initial states of A and B; see Fig. 6. For each state q of the resulting NFA, the set \(\{q\}\) is reachable, as well as coreachable. By Lemma 2, every NNFA for \(L(A)\cup L(B)\) has at least \(m+n1\) states. \(\square \)
Now, we use Lemma 3 to get the nondeterministic complexity of intersection on regular languages. The upper bound is mn since the product automaton \(A\times B = (Q_A \times Q_B,\varSigma ,\cdot ,(s_A,s_B), F_A\times F_B)\), where \((p,q) \cdot a = (p\cdot _A a )\times (q\cdot _B a)\), recognizes \(L(A)\cap L(B)\). The next example provides binary witness languages.
Example 5
(Intersection on regular languages; cf. [10, Theorem 3]). Consider the partial DFAs A and B shown in Fig. 7. The product automaton \(A\times B\) for \(L(A)\cap L(B)\) is a trim partial DFA, and its reverse is a partial DFA as well; see Fig. 8 for \(m=3\) and \(n=4\). By Lemma 3, every NNFA for \(L(A) \cap L(B)\) has at least mn states. \(\square \)
5 Modification of the Fooling Set Method
The fooling set method provides a lower bound on the number of states in any NNFA, that is, in any nondeterministic finite automaton with, possibly, multiple initial states. However, sometimes the NFA with a unique initial state must have one additional state. This is true for the case of union, where for every pair of languages K and L accepted by an mstate and nstate NFA, respectively, there exists an NNFA of size \(m+n\) accepting \(K\cup L\), hence no fooling set for \(K\cup L\) can be of size more than \(m+n\). Similarly, no fooling set for \(L^R\) can be of size more than n.
The idea for getting lower bounds of \(m+n+1\) for union and of \(n+1\) for reversal, is to divide a fooling set into two parts \(\mathcal {A}\) and \(\mathcal {B}\) and to find pairs \((\varepsilon ,u)\) and \((\varepsilon ,v)\) such that \(\mathcal {A}\cup \{(\varepsilon ,u)\}\) and \(\mathcal {B}\cup \{(\varepsilon ,v)\}\) are fooling sets. This implies that a unique initial state, reached after reading the empty string, must be different from all states given by fooling set \(\mathcal {A}\cup \mathcal {B}\).
Lemma 4
(Jirásková and Masopust [17, Lemma 4]). Let \(\mathcal {A}\) and \(\mathcal {B}\) be disjoint sets of pairs of strings and let u and v be two strings such that \(\mathcal {A}\cup \mathcal {B}\), \(\mathcal {A}\cup \{(\varepsilon ,u)\}\), and \(\mathcal {B}\cup \{(\varepsilon ,v)\}\) are fooling sets for a language L. Then every NFA for L has at least \(\mathcal {A}+\mathcal {B}+1\) states.
Proof
It is shown in [16, Theorem 2] that there is a binary regular language L accepted by an nstate NFA such that every NFA for \(L^R\) has at least \(n+1\) states. An NFA for the language is shown in Fig. 9, and the proof in [16] is by a counting argument. In the next example we use Lemma 4 to get the lower bound.
Example 6
The following two examples use Lemma 4 to get the nondeterministic state complexity of the union operation on regular and prefixfree languages.
Example 7
Example 8
(Union on prefixfree languages; cf. [13, Theorem 9]). A minimal NFA for a prefixfree language has exactly one final state with no outtransitions. We can construct an NNFA for the union of prefixfree languages given by minimal NFAs A and B (with disjoint states sets) just by merging their final states. This gives \(m+n1\) states in the resulting NNFA. Therefore \(m+n\) is an upper bound on the nondeterministic complexity of the union of prefixfree languages.
In [9] it is claimed that the upper bound \(m+n\) is met by the union of the prefixfree languages \(K=(a^{m1})^*b\) and \(L=(c^{n1})^*d\), and a set P of pairs of strings of size \(m+n\) is described in [9, Proof of Theorem 3.2]. The authors claim that the set P is a fooling set for \(K\cup L\). However, as shown above, the language \(K\cup L\) is accepted by an NNFA of \(m+n1\) states. Therefore P cannot be a fooling set for \(K\cup L\); indeed, the pairs \((\varepsilon ,a^{m1}b)\) and \((a^{m1},b)\) do not satisfy the second condition in Definition 1.
Here we prove the tightness of the upper bound \(m+n\) for the union of prefixfree languages using a binary alphabet and Lemma 4. Let \(m,n\ge 3\) and consider binary languages K and L accepted by the mstate and nstate NFA A and B, respectively, shown in Fig. 11. Notice that K is prefixfree since every string in K ends with b while every proper prefix of every string in K is in \(a^*\). Now, using Lemma 4, we show that every NFA for \(K\cup L\) has at least \(m+n\) states.
Finally, we use Lemma 4 to show that the upper bound \(m+n+1\) for the union of regular languages can be met by binary subwordclosed languages.
Example 9
6 Fooling Sets for Complementation
The next very useful observation allows us to significantly simplify the proofs of lower bounds on the nondeterministic state complexity of complementation.
Lemma 5
Let A be an nstate NNFA in which each subset of the state set is reachable as well as coreachable. Then every NNFA for the complement of the language L(A) has at least \(2^n\) states.
Proof
Let \(A=(Q,\varSigma ,\cdot ,I,F)\) be an NNFA. Let \(S\subseteq Q\). Since S is reachable, there exists a string \(x_S\) in \(\varSigma ^*\) such that \(S = I\cdot x_S\). Since the set \(S^c\) is coreachable, there is a string \(y_S\) which is accepted by A from each state in \(S^c\), but rejected from each state in S. It follows that the set \(\{(x_S,y_S) \mid S\subseteq Q\}\) is a fooling set for \((L(A))^c\). Hence every NNFA for \((L(A))^c\) has at least \(2^n\) states. \(\square \)
Recall that to get an NFA (even DFA) for the complement of a language represented by an NFA A, we first apply the subset construction to the NFA A, and in the resulting DFA, we interchange the accepting and rejecting states. This gives an upper bound of \(2^n\) on the nondeterministic state complexity of complementation. In the next example we use Lemma 5 to show that the complement of the binary language from [16, Proof of Theorem 5] meets this upper bound. Notice that a rather complicated fooling set is described in [16], and the proof is almost three pages long. Moreover, the NFA for this binary witness and its reverse are isomorphic, so we only need to show reachability of all subsets.
Example 10
First, notice that the automata A and \(A^R\) are isomorphic. Hence to prove that every NFA for \((L(A))^c\) has at least \(2^n\) states, we only need to show that every subset of \( \{0,1,\ldots ,n1\}\) is reachable in A. Since every subset S with \(0\notin S\) can be reached by \(a^{\min S}\) from the set \(\{s\min S\mid s\in S\}\) which contains state 0, we only need to show the reachability of subsets containing 0. The proof is by induction on the size of subsets. The set \(\{0\}\) is the initial subset, and every subset \(\{0,i_2,\ldots ,i_k\}\) of size k, where \(1 \le i_2< \cdots < i_k \le n1\), is reachable from the set \(\{0,i_3i_2,\ldots ,i_ki_2\}\) of size \(k1\) by the string \(a b^{i_21}\). \(\square \)
Finally, we prove that the upper bound \(2^n\) can be met by the complement of a suffixconvex language. Let us emphasize that such a language must be socalled proper suffixconvex, that is, it can be neither suffixfree, nor suffixclosed, nor left ideal since it is proved in [18, Lemma 4, Theorem 2], [12, Theorem 10], and [20, Theorem 26], respectively, that the nondeterministic complexity of complementation is less than \(2^n\) in these three subclasses of convex languages.
Example 11
Moreover, we can shift each subset of \(\{1,2,\ldots ,n1\}\) cyclically by one using the symbol a, that is, we have \(S \cdot a = \{(s+1)\bmod n \mid s \in S\}\). Next, we can eliminate the state 1 from each subset containing 1 by reading the symbol b. It follows that each subset is reachable. To prove coreachability, notice that the initial subset of \(A^R\) is \(\{1,2,\ldots ,n1\}\) and it goes to \(\{0,1,\ldots ,n1\}\) on e. We again use symbol a to shift the subsets of \(\{1,2,\ldots ,n1\}\) and symbol b to eliminate the state 1. It follows that every subset is coreachable. By Lemma 5, every NNFA for \(L^c\) has at least \(2^n\) states. \(\square \)
7 Conclusions
The fooling set method provides lower bounds on the number of states in nondeterministic finite automata that are tight in many cases despite the fact that the gap between the size of a fooling set and the nondeterministic state complexity may be arbitrarily large. We illustrated this on a number of examples.
We also provided sufficient conditions on nondeterministic finite automata that guarantee the existence of appropriate fooling sets. This allowed us to avoid the tedious description of such fooling sets.
Since fooling sets provide lower bounds on the number of states in nondeterministic finite automata with multiple initial states, in the case of union or reversal, where such automata may save one state, the fooling set method cannot be used. However, if a fooling set can be divided into two parts such that adding a pair with its left component equal to the empty string results in another fooling set, we get tight lower bounds also for automata with a unique initial state.
Finally, we provided a very useful observation from [13, Proof of Theorem 13] claiming that if all subsets of the state set of a nondeterministic finite automaton are reachable and coreachable, then the accepted language is a witness for the complementation operation
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