Forbidding Sets and Normal Forms
for Language Forbidding-Enforcing Systems
Daniela Genova
Department of Mathematics and Statistics
University of North Florida Jacksonville, FL 32224, USA
[email protected]
Abstract. This paper investigates ways to reduce redundancy in forbidding sets for language forbidding-enforcing systems. A language forbidding set disallows combinations of subwords in a word, while permitting
the presence of some parts of these combinations. Since a forbidding set
is a potentially infinite set of finite sets of words, finding normal forms for
forbidding sets is interesting from a combinatorics on words perspective
and important for the theoretical investigation of language fe-systems,
the connection between variants of fe-systems, and their applications to
molecular computation. This paper shows that the minimal normal forms
for forbidding sets defining classes of languages (fe-families) are also normal forms for forbidding sets defining single languages (fe-languages),
but not necessarily minimal. Thus, an investigation of minimality and
sufficient conditions for fe-languages are presented and it is shown that
in special cases they coincide with a minimal normal form for fe-families.
Keywords: fe-systems, forbidden words, biomolecular computing, normal forms, formal languages.
1
Introduction
Forbidding-enforcing systems (fe-systems) can be viewed, in general, as boundary
restrictions imposed on classes of structures that can be defined over any category
of objects and morphisms [9]. Abstracting from the non-deterministic behavior
of molecules in molecular reactions, A. Ehrenfeucht and G. Rozenberg introduced the forbidding and enforcing paradigm ([2,3,4,16]) as fe-systems that define classes of languages (fe-families) capable of providing means for information
processing. These classes of languages were shown to be different than Chomsky’s hierarchy [8]. Fe-systems have been proposed in the framework of membrane computing [1], used to model DNA self-assembly [5], and defined on graphs
[6]. Detailed discussion of DNA computing models, splicing systems, membrane
systems, and DNA self-assembly, can be found in [10,11,12,13,14,15,17]).
This paper investigates ways to simplify forbidding sets for a variant of fesystems introduced in [7], in which one fe-system is used to define a single language (fe-language fe-system), as opposed to a family of languages (fe-family
fe-system) as defined in [4]. Unlike a grammar or an automaton, which generates or accepts a word symbol by symbol, a language fe-system defines a language
A.-H. Dediu and C. Martı́n-Vide (Eds.): LATA 2012, LNCS 7183, pp. 289–300, 2012.
c Springer-Verlag Berlin Heidelberg 2012
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based on forbidden and enforced subwords. Characterizations of local and factorial languages by fe-systems were presented in [7] and it was shown that such
systems can define the solutions to the k-colorability problem and model splicing
rules. Using one system to define one set of words is motivated by molecular computation, where the result of a computation is a set of molecules (a set of words
over a DNA alphabet), restriction enzymes require specific sequences (subwords)
to perform a cut, and from DNA involution codes, where subwords are essential
to the word structure.
The focus here is on theoretical properties of forbidding sets and the sets of
words that they define and shies away from applications. This paper investigates
how redundancy of forbidding sets can be avoided, i.e. how by shrinking certain
forbidden combinations of subwords or deleting many combinations from the
forbidding set, one can find an equivalent reduced set of forbidden combinations
of subwords that defines the same set of consistent words (fe-language). From
a combinatorics on words perspective, the combinations of forbidden words are
non-strict, i.e., some parts of them are allowed, and the number of forbidden
combinations may be infinite.
Following the related definitions and examples in Section 2, Section 3 investigates the similarities and differences of the normal forms for forbidding sets
for both the fe-family and fe-language fe-systems models and shows that the
subword free and subword incomparable minimal normal form proved in [2,16]
for fe-families is not necessarily minimal for the single language forbidding set.
Minimal language forbidding sets are investigated in Section 4 and Section 5
provides normal forms for strict forbidding sets.
2
Language Forbidding-Enforcing Systems
A finite set of symbols (alphabet) is denoted by A and the free monoid consisting
of all words over A is denoted by A∗ . A subset of A∗ is called a language. The
length of a word w ∈ A∗ is denoted by |w| and Am is the set of all words of
length m, whereas Am is the set of all words of length at most m. The empty
word, denoted by λ has length 0. The language A+ consists of all words over A
with positive length.
The word y ∈ A∗ is a subword (factor) of x ∈ A∗ , if there exist s, t ∈ A∗ , such
that x = syt. The set of subwords of a word x is denoted by sub (x) and the set
of subwords of a language L by sub (L), where sub (L) = ∪x∈L sub (x).
When referring to fe-families, this paper uses the definitions and notation
from [4]. For more details about properties of fe-systems defining fe-families of
languages, the reader is referred to [2,3,4,6,8,16].
The normal forms in this paper relate to the fe-systems model introduced in
[7], in which one forbidding-enforcing system defines a single language as opposed
to a family of languages. The related definitions are recalled below. Assume that
the alphabet A is given.
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Definition 1. A forbidding set F is a family of finite nonempty subsets of A+ ;
each element of a forbidding set is called a forbidder. A word w is consistent
with a forbidder F , denoted by w con F , if and only if, F ⊆ sub (w). A word w is
consistent with a forbidding set F, denoted by w con F, if and only if, w con F for
all F ∈ F. If w is not consistent with F (resp. F), the notation is w ncon F (resp.
w ncon F). The language L(F) = {w | w con F}. A language L is a forbidding
language or f-language, if there is a forbidding set F such that L = L(F). Two
forbidding sets F and F are equivalent, denoted by F ∼ F , if and only if,
L(F) = L(F ). A forbidding set F is called strict, if and only if, |F | = 1 for every
F ∈ F.
The following forbidding set was discussed in [2,4,6,8,16], where it was used to
define a family of languages. The example below is from [7], where the same
forbidding set was used to define a single set of words (f-language).
Example 2. Assume that A = {a, b}. Let F = {{ab, ba}, {aa, bb}}. Then L(F) =
{an , bn , abn , an b, ban , bn a | n ≥ 0}.
Other examples of f-languages over the same alphabet A include: L(F1 ) = a∗
for F1 = {{b}} and if F2 = {{bb}}, then L(F2 ) contains words where any two b’s
are separated by at least one a. Note that a∗ ⊂ L(F2 ). Also note that if nothing
is forbidden, then everything is allowed, i.e. L(F) = A∗ if and only if F is empty.
Theorem 1 in [7] establishes a connection between the fe-family defined by a
forbidding set and the fe-language defined by the same forbidding set. It states
that the union of maximal languages in the fe-family gives the fe-language. The
same statement holds, if we replace “maximal languages” by “languages”. The
following result is used in the proof of Theorem 8.
Theorem 3. Let F be a forbidding set. Then, L(F) = ∪L∈L(F) L.
Proof. Let F be given. Assume w ∈ L(F). Then, the language {w} ∈ L(F),
otherwise there is a forbidder F ∈ F, such that F ⊆ sub ({w}) and we have F ⊆
sub (w), which contradicts the assumption that w con F . Hence, w ∈ ∪L∈L(F) L.
Therefore, L(F) ⊆ ∪L∈L(F) L. Conversely, assume w ∈ ∪L∈L(F) L. Then, there
exists a language K ∈ L(F) such that w ∈ K. Let F ∈ F. Since K con F, it
follows that F ⊆ sub (K). Then, F ⊆ sub (w), otherwise F ⊆ sub (w) and
sub(w) ⊆ sub (K) imply F ⊆ sub (K), a contradiction. Since w con F for an
arbitrary F ∈ F, we have that w con F, i.e., w ∈ L(F). Thus, ∪L∈L(F) L ⊆ L(F).
Consequently, L(F) = ∪L∈L(F) L.
The other boundary condition used in the fe-language fe-systems model proposed
in [7] is an enforcing set.
Definition 4. An enforcing set E is a family of ordered pairs called enforcers
(x, Y ), such that x ∈ A∗ and Y = {y1 , . . . , yn } where yi ∈ A+ for i = 1, . . . , n,
x ∈ sub (yi ) and x = yi for every yi ∈ Y . A word w satisfies an enforcer (x, Y )
(w sat (x, Y )), if and only if, w = uxv for some u, v ∈ A∗ implies that there
exists yi ∈ Y and u1 , u2 , v1 , v2 ∈ A∗ such that yi = u2 xv2 and w = u1 u2 xv2 v1 .
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In the case that x ∈ sub (w), w is said to satisfy the enforcer trivially. A word
w satisfies an enforcing set E (w sat E), if and only if, w satisfies every enforcer
in that set. For an enforcing set E the set of all words that satisfy it is denoted
by L(E). A language L is called an e-language, if there exists an enforcing set E
such that L = L(E).
Enforcers in which x = λ are called brute. In this case, a word from Y has to be
a subword of w in order for w to satisfy the enforcer. Note that if y ∈ Y , then
y sat (x, Y ). Also, L(E) = A∗ if and only if E = ∅. An enforcer (x, Y ) is called
strict, if |Y | = 1. Consider the enforcing set E = {(λ, {a})}∪{(ai , {ai+1 }) | i ≥ 1}
over an alphabet A that contains a. It consists of strict enforcers only, one of
which is brute and requires that ai ∈ sub (w) for any i ≥ 1. Since L(E) can only
contain finite words, L(E) = ∅.
The idea of a forbidding-enforcing system for families of languages from [4] is
preserved for a set of words (fe-language) in [7] and the definition is stated next.
Definition 5. A forbidding-enforcing system is an ordered pair (F, E), such that
F is a forbidding set and E is an enforcing set. The language L(F, E) defined by
this system consists of all words that are consistent with F and satisfy E, i.e.,
L(F, E) = L(F) ∩ L(E). A language L is called an fe-language, if there exists an
fe-system (F, E), such that L = L(F, E).
Basic properties of fe-language fe-systems are stated in [7] and are reminiscent
of fe-family fe-systems properties from [4,16]. For example, Property 7 from
Proposition 1 in [7] states that if F and F are two forbidding sets and E and E
are two enforcing sets, then L(F ∪ F , E ∪ E ) = L(F, E) ∩ L(F , E ). This property
is used in the following example from [7] to define the language in Item 3 as the
intersection of the languages in Items 1 and 2.
Example 6. Let A = {a, b}.
1. Let F = {{ba}} and E1 = {(λ, {a})} ∪ {(ai , {ai+1 , ai bi }) | i ≥ 1}. Then,
L1 = L(F, E1 ) = {an bm | n ≤ m and n, m ≥ 1}.
2. Let F = {{ba}} and E2 = {(λ, {b})} ∪ {(bi , {bi+1 , ai bi }) | i ≥ 1}. Then,
L2 = L(F, E2 ) = {an bm | n ≥ m and n, m ≥ 1}.
3. Then, L = L1 ∩ L2 = {an bn | n ≥ 1} = L(F, E1 ∪ E2 ).
Since a forbidding (enforcing) set can be empty, an fe-system can be defined using
only one of the boundary conditions as constraints, i.e., L(∅, E) = A∗ ∩ L(E) =
L(E) and L(F, ∅) = L(F) ∩ A∗ = L(F). In this sense, forbidding languages
(enforcing languages) are fe-languages.
3
Subword-Free and Subword-Incomparable Normal
Forms
A forbidding set may be be redundant and in that case it may be reduced by
removing some parts of its forbidders or entire forbidders without changing the
language that it defines and without changing the family of languages that it
defines.
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Definition 7. A forbidding set F is called subword free if all of its forbidders
are subword free and subword incomparable if for any two forbidders F1 , F2 ∈ F
with F1 = F2 , it holds that sub (F1 ) ⊆ sub (F2 ) and sub (F2 ) ⊆ sub (F1 ).
The subword free and subword incomparable normal forms for families of languages were introduced in [2] and discussed in detail in [16]. The authors proved
that these two conditions combined (called minimal normal form) define a normal form that is indeed minimal and unique for fe-families. In this section, two
questions are investigated: whether the normal forms proved for fe-families forbidding sets are also normal forms for fe-language forbidding sets and if so,
whether the minimal and unique normal form for fe-families is minimal and
unique for fe-languages, as well. The answer to the first question is affirmative.
However, as shown in this section, the analogous normal form for fe-languages
is neither minimal nor unique.
Example 5 in [2] shows that since the minimal normal form of the forbidding
set F = {{ai , bi , ai bi } | i ≥ 1} is F = {{ab}}, they both define the same family
of languages and so, an infinite forbidding set in this case can be reduced to a
finite one. Observe that, the same is true for the fe-language model, as well, since
{ai , bi , ai bi } ⊆ sub (w) for all i ≥ 1 iff {a, b, ab} ⊆ sub (w) iff {ab} ⊆ sub (w). In
fact, consider the following.
Theorem 8. Every normal form for forbidding sets for fe-families is also a
normal form for forbidding sets for fe-languages.
Proof. Let F be a forbidding set and F be a forbidding set equivalent to it
in some normal form for fe-families. Then, L(F) = L(F ). From Theorem 3,
L(F) = ∪L∈L(F) L. By the same theorem, we have that L(F ) = ∪L∈L(F ) L.
Thus, L(F) = ∪L∈L(F) L = ∪L∈L(F ) L = L(F ). Hence, L(F) = L(F ). Therefore,
F ∼ F for fe-language forbidding sets, as well.
It follows from the above theorem that the minimal normal form for fe-family
forbidding sets is a normal form for fe-language forbidding sets, as well.
Corollary 9. For every language forbidding set there is an equivalent subword
free and subword incomparable forbidding set.
So, given a forbidding set, we can find a fe-language equivalent forbidding set that
is subword free and subword incomparable. However, the next example shows
that a subword free and subword incomparable forbidding set is not necessarily
fe-language minimal.
Example 10. Let A = {a, b}, F = {{aabb}, {bbaa}, {bbabaa}, {aa, bb, abab}}, and
F = {{aabb}, {bbaa}, {bbabaa}, {aa, bb}}. Clearly, F is subword free and subword incomparable. In the fe-families model, this forbidding set is minimal. If
we remove a forbidder, the removed forbidder is a language that is not in the old
family but it is in the new one, so the obtained forbidding set is not going to be
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equivalent to F. If we remove a word from a forbidder, say abab, the remaining
words from that forbidder, i.e., {aa, bb} form a language that is consistent with
F but not consistent with the newly obtained forbidding set. In the fe-language
model F is not minimal, since it can be reduced to a smaller forbidding set which
defines the same language, i.e., is equivalent to F . Observe that every word that
contains both aa and bb as subwords and does not contain abab as a subword
contains either aabb, or bbaa, or bbabaa as a subword. Hence, w con F implies
w con F . The converse is obvious. Therefore, F ∼ F . Further more, since F is
not subword incomparable, i.e., sub ({aa, bb}) ⊆ sub (G) for every G ∈ F with
G = {aa, bb}, F can be reduced to F where F = {{aa, bb}}. Thus, F ∼ F .
4
Connecting Words and Minimal Normal Form
In [7] connecting words were used to prove that for every forbidding set there
exists an equivalent enforcing set that defines the same set of words, i.e. language.
In this paper, connecting words are used to investigate the relationship between
general forbidding sets and strict ones and to prove some normal forms.
Definition 11. Given a finite set of words (a forbidder) F , a word x such that
F ⊆ sub (x) is called a connecting word of F . The set of all connecting words
of F is called the connect of F and denoted by C(F ). If s ∈ C(F ) and for no
t ∈ C(F ), t = s it holds that t ∈ sub (s), then s is a minimal connecting word
of F . The set of minimal connecting words of F is called the minimal connect
of F and denoted by Cmin (F ).
Remark 12. Note that for a forbidder F and a word w ∈ A∗ , either w con F or
w ∈ C(F ). More precisely, w ∈ C(F ) if and only if F ⊆ sub (w) if and only if
w ncon F .
Two forbidders F1 and F2 are equivalent if and only if w con F1 implies w con F2
and vice versa. Hence, the following remark.
Remark 13. Let F be a forbidding set and F1 , F2 ∈ F. Then, F1 and F2 are
equivalent if and only if C(F1 ) = C(F2 ).
Even for a very simple forbidder, such as the one in the next example, the set
of minimal connecting words and thus, the set of all connecting words for this
forbidder may be infinite.
Example 14. Let A = {a, b} and consider the forbidder F = {aa, bb}. Then
aabbabb ∈ C(F ), but aabbabb ∈ Cmin (F ), since aabb ∈ sub (aabbabb). However, aababb ∈ Cmin (F ), since none of its proper subwords is in C(F ). In fact,
aa(ba)i bb ∈ Cmin (F ) for any i ≥ 0. Moreover, Cmin (F ) = {aa(ba)i bb, bb(ab)i aa |
i ≥ 0}.
The following useful result can be proved directly by the above definition.
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Lemma 15. Let F and F be finite sets of words (forbidders) with F ⊆ F .
Then, C(F ) ⊆ C(F ).
Lemma 16. Let F be a forbidding set and F1 , F2 ∈ F such that C(F2 ) ⊆ C(F1 ).
Then F ∼ (F\{F2 }).
Proof. Let F, F1 , and F2 be as in the hypothesis of the lemma. Obviously, L(F) ⊆
L(F\{F2 }). Let w ∈ L(F\{F2 }). Since w con F1 , it follows that w ∈ C(F2 ).
Otherwise, since C(F2 ) ⊆ C(F1 ), we have w ∈ C(F1 ), a contradiction with
Remark 12 and so, w con F2 , in view of the same remark. Therefore, L(F\{F2 }) ⊆
L(F).
The above lemma is generalized below to allow removal of possibly infinitely
many forbidders.
Lemma 17. Let F and F be forbidding sets with F ⊆ F such that for each
F ∈ F there is F ∈ F such that C(F ) ⊆ C(F ). Then F ∼ F.
Proof. Obviously, L(F) ⊆ L(F ). Let w ∈ L(F ) and let F ∈ F . Then, there
exists F ∈ F such that C(F ) ⊆ C(F ). By Remark 12, w ∈ C(F ), which
implies that w ∈ C(F ). Hence, w con F and the lemma follows.
Example 10 shows that some words in a forbidder may be redundant depending
not only on other words in the forbidder, but also on the other forbidders.
Proposition 18. Let F be a forbidding set and F ∈ F be a forbidder. Let x ∈ F
be such that for every w ∈ C(F \ {x}) with x ∈ sub (w), there exists G ∈ F,
G = F with w ∈ C(G). Then, F ∼ (F \ {F }) ∪ {F }, where F = F \ {x}.
Proof. Assume F, F , and x are given as in the statement and let F = (F \
{F }) ∪ {F }, where F = F \ {x}. Clearly, L(F ) ⊆ L(F). Assume w ∈ L(F) and
w ncon F . By Remark 12, w ∈ C(F ). Since w con F , it follows that x ∈ sub (w).
Then, there exists G ∈ F, G = F such that w ∈ C(G), which contradicts the
assumption that w ∈ L(F). Hence, w con F and L(F) ⊆ L(F ). Consequently,
F ∼ F .
Note that the above process is transitive. It follows from the above proposition
that if F can be reduced to F = F \ {x} and F can be reduced to some
F = F \ {y}, then F can be reduced to F = F \ {y, x}. In fact, F may be
reduced by a subset X, if every element in X satisfies the conditions of the above
proposition.
Definition 19. A forbidding set is called connecting free if for every forbidder
F ∈ F with |F | ≥ 2 and every word x ∈ F there exists w ∈ C(F \{x}) with
x ∈ sub (w) such that w con G for every G ∈ F, G = F .
The forbidding set from Example 2 is connecting free. Observe that if F =
{ab, ba}, and x = ab, then there exists w = ba such that w ∈ C(F \ {x}) and
w con {aa, bb}. Similarly, for x = ba we can take w = ab, for x = aa let w = bb,
and for x = bb take w = aa.
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Proposition 20. Every connecting free forbidding set is subword free.
Proof. Suppose a connecting free forbidding set F is not subword free. This
implies that there exists a forbidder F and x, y ∈ F such that x ∈ sub (y).
Then, there is no w ∈ C(F \ {x}) for which x ∈ sub (w), since if y ∈ sub (w),
so is x. Hence, there doesn’t exist w ∈ C(F \ {x}) with x ∈ sub (w) such that
w con G for every G ∈ F, G = F . This contradicts the assumption that F is
connecting free. Hence, the proposition follows.
Example 10 shows that, in general, the converse of Proposition 20 does not hold,
i.e. a subword free forbidding set is not necessarily connecting free. The next
proposition states that for subword free and subword incomparable forbidding
sets with forbidders consisting of no more than two elements, the converse of
Proposition 20 holds.
Proposition 21. Let F be a subword free and subword incomparable forbidding
set with |F | ≤ 2 for every F ∈ F. Then, F is connecting free.
Proof. Let F satisfy the hypothesis of the statement and let F ∈ F with |F | = 2.
Since F is subword free, F = {x, y} for some x, y ∈ A+ such that x ∈ sub (y)
and y ∈ sub (x). Then, y ∈ C(F \ {x}) is such that y con G for every G ∈
F, otherwise there exists G ∈ F such that G ⊆ sub (y), which implies that
sub (G) ⊆ sub (y) ⊆ sub (F ) and contradicts the assumption that F is subword
incomparable. Similarly, x ∈ C(F \{y}) is such that x con G for every G ∈ F. The following result states that if in addition to subword free and subword
incomparable, a forbidding set is also connecting free, it is minimal in the sense
that a removal of only one word from one forbidder changes the forbidding
language.
Lemma 22. Let F be subword incomparable and connecting free. Then, for every
F ∈ F and every x ∈ F , L(F ) ⊂ L(F), where F = (F\{F }) ∪ {F } such that
F = F \{x}.
Proof. Let F be subword incomparable and connecting free. Let F ∈ F and
x ∈ F . Consider F = F \{x} and F = (F\{F }) ∪ {F }. It is obvious that
L(F ) ⊆ L(F). Since F is connecting free, there exists w ∈ C(F ) with x ∈ sub(w)
such that w con G for every G ∈ F, G = F . Then, w is such that w ∈ L(F), but
w ∈ L(F ). Hence, L(F ) ⊂ L(F).
The next example shows that even if a forbidding set contains minimal forbidders, it may still contain redundant forbidders.
Example 23. Let A = {a, b} and consider F = {{ab}, {ba}, {aa, bb}} and let
F = {aa, bb}. From Example 14, we have that Cmin (F ) = {aa(ba)i bb, bb(ab)i aa |
i ≥ 0}. Since w ncon F implies that one of the words in Cmin (F ) is a subword of
w, and for every word in Cmin (F ) either ab or ba is a subword of that word, it
follows that w ncon {ab} or w ncon {ba}. This implies that w con F if and only if
w con (F\{F }). Hence, F is redundant and can be removed without changing the
forbidding language. The forbidding language L(F) = a∗ ∪b∗ = L(F\{{aa, bb}}).
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The next proposition is a generalization of Example 23.
Proposition 24. Let F be a forbidding set and let F ∈ F be such that for every
w ∈ C(F ) there exists G ∈ F, G = F such that w ∈ C(G). Then, F ∼ (F \ {F }).
Proof. Let F be a forbidding set and let F ∈ F be such that for every w ∈ C(F )
there exists G ∈ F, G = F such that w ∈ C(G). Obviously, L(F) ⊆ L(F \ {F }).
Conversely, assume that w ∈ L(F \ {F }). Suppose that w ncon F . Then, w ∈
C(F ) and so, there exists G = F such that w ∈ C(G). This contradicts the
assumption that w con G. Therefore, w con F and L(F \ {F }) ⊆ L(F).
Definition 25. A forbidding set F is called connecting reduced if for every F ∈ F
there exists w ∈ C(F ), such that w ∈ C(G) for every G ∈ F, G = F .
Proposition 26. Every connecting reduced forbidding set is subword incomparable.
Proof. Let F be connecting reduced. Let F1 , F2 ∈ F with F1 = F2 . Since F
is connecting reduced, there exists w ∈ C(F2 ), such that w ∈ C(F1 ), i.e., F1 ⊆
sub (w). Moreover, since w ∈ C(F2 ), by Remark 12, we have F2 ⊆ sub (w) and so,
sub (F2 ) ⊆ sub (w). Thus, F1 ⊆ sub (F2 ), otherwise F1 ⊆ sub (F2 ) ⊆ sub (w), a
contradiction. Hence, sub (F1 ) ⊆ sub (F2 ). Similarly, sub (F2 ) ⊆ sub (F1 ). Thus,
F is subword incomparable.
Example 23 shows that the converse of the above proposition does not hold,
since this forbidding set is subword incomparable, but not connecting reduced.
Lemma 27. Let F be a connecting reduced forbidding set. Then, L(F) ⊂ L(F \
{F }) for any F ∈ F.
Proof. Assume that F is connecting reduced and let F ∈ F. Clearly, L(F) ⊆
L(F \ {F }). Since F is connecting reduced, there exists w ∈ C(F ) such that
w ∈ C(G) for every G ∈ F, G = F . Then, w ∈ L(F \ {F }), but w ∈ L(F).
Therefore, L(F) ⊂ L(F \ {F }).
Example 23 also shows that a connecting free forbidding set is not necessarily
connecting reduced.
Definition 28. A forbidding set F is reduced if it is both connecting free and
connecting reduced.
The forbidding set from Example 2 is reduced.
The next theorem states that every reduced forbidding set is minimal, i.e.
removal of only one forbidder or only one word in a forbidder, changes the set
of words that the forbidding set defines. It follows from Lemmas 22 and 27.
Theorem 29. Every reduced forbidding set is minimal.
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Normal Forms for Strict Forbidding Sets
Proposition 30. Let F be a forbidding set. For every F ∈ F choose one connecting word sF ∈ C(F ) and consider F = {{sF } | F ∈ F}. Then, L(F) ⊆ L(F ).
Proof. Let w ∈ L(F) and let {sF } ∈ F . Then, sF is a connecting word for some
F ∈ F. Then F ⊆ sub (w) implies that {sF } ⊆ sub (w). Thus, L(F) ⊆ L(F ). Note that the converse is not necessarily true even if F is reduced. For example,
consider A = {a, b}, F = {{aa, bb}}, and F = {{aabb}}. Then w = bbaa is
such that w ∈ L(F ) and w ∈ L(F). However, if all minimal connecting words
of all forbidders are considered as singleton forbidders, then the converse holds.
Moreover, we have the following normal form.
Theorem 31. For every forbidding set there exists an equivalent strict forbidding set.
Proof. Let F be a forbidding set. For every F ∈ F construct the forbidding set
FF = {{s} | s ∈ Cmin (F )} and consider F = ∪F ∈F FF . We show that F ∼ F .
Assume that w con F. Note that by definition of F , for every {s} ∈ F there
exists F ∈ F such that F ⊆ sub (s). Since w con F, we have that F ⊆ sub (w). It
follows that {s} ⊆ sub (w). Hence, L(F) ⊆ L(F ). Conversely, let w con F and
let F ∈ F. Suppose F ⊆ sub (w). Then, w ∈ C(F ) and there is s ∈ Cmin (F )
such that s ∈ sub (w), which contradicts the assumption that w con F . Hence,
F ⊆ sub (w) and L(F ) ⊆ L(F). Consequently, F ∼ F .
Remark 32. Note that the above theorem does not make general forbidding sets
obsolete. Example 14 shows that even a simple forbidder F = {aa, bb} may have
an infinite number of minimal connecting words and replacing a finite number
of forbidders with an infinite number of forbidders maybe undesirable.
Remark 33. Any strict forbidding set is connecting (subword) free. Also, connecting reduced is equivalent to subword incomparable for such a set.
Corollary 34. For every forbidding set there exists an equivalent minimal strict
forbidding set.
Proof. Let F be given and construct an equivalent F as in the proof of Theorem
31. Then, from Corollary 9 for F there exists an equivalent connecting reduced
(subword incomparable) forbidding set F . By Lemma 27, F is minimal, i.e.,
L(F ) ⊂ L(F \ {F }) for any F ∈ F .
Lemma 35. Let F be a strict forbidding set and F1 and F2 be two minimal
strict forbidding sets equivalent to F. Then, F1 = F2 .
Proof. Let {u} ∈ F1 and since F1 ∼ F2 , u ∈ L(F1 ) = L(F2 ). It follows that
there exists {v} ∈ F2 such that {v} ⊆ sub (u). Hence, sub ({v}) ⊆ sub (u).
Similarly, since v ∈ L(F1 ), there exists {w} ∈ F1 such that {w} ⊆ sub (v),
which implies that sub ({w}) ⊆ sub (v). Since both {w} and {u} are in F1 ,
Normal Forms for Language fe-Systems
299
sub ({w}) ⊆ sub ({u}), and F1 is subword incomparable, it follows that {w} =
{u}. Hence, sub ({w}) = sub ({u}) and thus, sub ({v}) = sub ({u}). Since both
v and u are subword free, we have that {v} = {u} and {u} ∈ F2 . Thus, F1 ⊆ F2 .
Similarly, F2 ⊆ F1 . Consequently, F1 = F2 .
The next result shows that if a strict forbidding set is reduced, then it is both
minimal and unique.
Theorem 36. For every forbidding set there exists an equivalent unique minimal strict forbidding set.
Proof. Let F be given and let F be the strict forbidding set constructed as in the
proof of Theorem 31 consisting of singleton forbidders of all minimal connecting
words of the forbidders in F. From Corollary 9 there exists a connecting reduced
(subword incomparable) set F̂ equivalent to F . Then, F̂ is reduced. From Corollary 34, it is minimal and Lemma 35 establishes that it is unique.
In [7] it was shown that a language is local if and only if it is an f-language.
Since the characterization was obtained using strict forbidders, i.e. for every
local language L with a set of forbidden words H = {h1 , . . . , hn } it holds that
L = A∗ \A∗ HA∗ if and only if L = L(F), where F = {{h1 }, . . . , {hn }}, we have
the obvious corollary for local languages. The above theorem shows that the set
of forbidden words for L can be reduced.
Corollary 37. For every local language L with a set of forbidden words H, there
exists a unique minimal set of forbidden words H such that L = A∗ \A∗ H A∗ .
6
Concluding Remarks
This paper presented an investigation of normal forms for forbidding sets defining
fe-languages. It was shown that every normal form for forbidding sets defining
fe-families is also a normal form for forbidding sets defining fe-languages. However, such normal forms are not necessarily minimal nor unique for fe-languages
as they are known to be for fe-families. Thus, connecting free and connecting
reduced forbidding sets were introduced and it was shown that they are minimal
for fe-language forbidding sets and coincide with the subword free and subword
incomparable normal form for fe-families for strict forbidding sets. Further, investigation of normal forms for enforcing sets is needed to enhance the study
of language fe-systems and their applications to molecular computing. The relationship between language fe-systems and graph fe-systems [6] should be investigated further. Some similarity exists between the notion of connecting words
and that of connecting graphs that may lead to common properties of subwords
and subgraphs.
Acknowledgement. This work has been supported in part by a UNF Faculty
Development Scholarship Grant.
300
D. Genova
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