Quotient complexity of ideal languages

📝 Abstract

We study the state complexity of regular operations in the class of ideal languages. A language L over an alphabet Sigma is a right (left) ideal if it satisfies L = L Sigma* (L = Sigma* L). It is a two-sided ideal if L = Sigma* L Sigma , and an all-sided ideal if it is the shuffle of Sigma with L. We prefer the term “quotient complexity” instead of “state complexity”, and we use derivatives to calculate upper bounds on quotient complexity, whenever it is convenient. We find tight upper bounds on the quotient complexity of each type of ideal language in terms of the complexity of an arbitrary generator and of its minimal generator, the complexity of the minimal generator, and also on the operations union, intersection, set difference, symmetric difference, concatenation, star and reversal of ideal languages.

💡 Analysis

We study the state complexity of regular operations in the class of ideal languages. A language L over an alphabet Sigma is a right (left) ideal if it satisfies L = L Sigma* (L = Sigma* L). It is a two-sided ideal if L = Sigma* L Sigma , and an all-sided ideal if it is the shuffle of Sigma with L. We prefer the term “quotient complexity” instead of “state complexity”, and we use derivatives to calculate upper bounds on quotient complexity, whenever it is convenient. We find tight upper bounds on the quotient complexity of each type of ideal language in terms of the complexity of an arbitrary generator and of its minimal generator, the complexity of the minimal generator, and also on the operations union, intersection, set difference, symmetric difference, concatenation, star and reversal of ideal languages.

📄 Content

arXiv:0908.2083v1 [cs.FL] 14 Aug 2009 Quotient Complexity of Ideal Languages Janusz Brzozowski1, Galina Jir´askov´a2, and Baiyu Li1 1 David R. Cheriton School of Computer Science, University of Waterloo, Waterloo, ON, Canada N2L 3G1 {brzozo@, b5li@student.cs.}uwaterloo.ca 2 Mathematical Institute, Slovak Academy of Science, Greˇs´akova 6, 040 01 Koˇsice, Slovakia {jiraskov@saske.sk} September 18, 2021 Abstract. We study the state complexity of regular operations in the class of ideal languages. A language L ⊆Σ∗is a right (left) ideal if it satisfies L = LΣ∗(L = Σ∗L). It is a two-sided ideal if L = Σ∗LΣ∗, and an all-sided ideal if L = Σ∗ L, the shuffle of Σ∗with L. We prefer the term “quotient complexity” instead of “state complexity”, and we use derivatives to calculate upper bounds on quotient complexity, whenever it is convenient. We find tight upper bounds on the quotient complexity of each type of ideal language in terms of the complexity of an arbitrary generator and of its minimal generator, the complexity of the minimal generator, and also on the operations union, intersection, set difference, symmetric difference, concatenation, star and reversal of ideal languages. Keywords: automaton, complexity, derivative, ideal, language, quo- tient, state complexity, regular expression, regular operation, upper bound 1 Ideal Languages We assume that the reader is familiar with basic concepts of regular languages and finite automata, as described in [14, 18], for example, or in many textbooks. For general properties of ideal languages see [11, 16], for example. If Σ is a non-empty finite alphabet, then Σ+ is the free semigroup generated by Σ, and Σ∗is the free monoid generated by Σ, with empty word ε. A word is any element of Σ∗. The length of a word w ∈Σ∗is |w|, and |w|a denotes the number of a’s in w, where a ∈Σ. A language over Σ is any subset of Σ∗. The left quotient, or simply quotient, of a language L by a word w is the language w−1L = {x ∈Σ∗| wx ∈L}. Right quotient is defined similarly: Lw−1 = {x ∈Σ∗| xw ∈L}. If u, v, w ∈Σ∗and w = uv, then u is a prefix of w and v is a suffix of w. If w = uxv for some u, v, x ∈Σ∗, then x is a factor of w. Note that a prefix or suffix of w is also a factor of w. If w = w0a1w1 · · · anwn, where a1, . . . , an ∈Σ, and w0, . . . , wn ∈Σ∗, then v = a1 · · · an is a subword3 of w; note that every factor of w is a subword of w. A language L is prefix-convex if u, w ∈L with u a prefix of w implies that every word v must also be in L if u is a prefix of v and v is a prefix of w. L is prefix-free if w ∈L implies that no proper prefix of w is in L. L is prefix-closed if w ∈L implies that every prefix of w is also in L. In the same way, we define suffix-convex, factor-convex, and subword-convex, and the corresponding free and closed versions. A language L ⊆Σ∗is a right ideal (respectively, left ideal, two-sided ideal) if it is non-empty and satisfies L = LΣ∗(respectively, L = Σ∗L, L = Σ∗LΣ∗). We also study special two-sided ideals which satisfy L = Σ∗ L = [ a1···an∈L Σ∗a1Σ∗· · · Σ∗anΣ∗, where is the shuffle operator. We have not found a name for such an ideal in the literature, so we introduce the term all-sided ideal. We refer to all four types as ideal languages or simply ideals. They have the following properties: – If L is a right ideal, any K ⊆Σ∗such that L = KΣ∗is a generator of L. The minimal generator of L is G = L \ (LΣ+), and G is prefix-free. – If L is a left ideal, any K ⊆Σ∗such that L = Σ∗K is a generator of L. The minimal generator of L is G = L \ (Σ+L), and G is suffix-free. – If L is a two-sided ideal, any K ⊆Σ∗such that L = Σ∗KΣ∗is a generator of L. The minimal generator of L is G = L \ (Σ+LΣ∗∪Σ∗LΣ+), and G is factor-free. – If L is an all-sided ideal, any K ⊆Σ∗such that L = Σ∗ K is a generator of L. The minimal generator of L is G = L \ {w ∈L | a proper subword of w is in L}, and G is subword-free. An ideal L is principal if it is generated by a language {w} consisting of a single word w ∈Σ∗. In that case we write L = wΣ∗(rather than L = {w}Σ∗), L = Σ∗w, etc. Our main interest is in ideal languages that are regular. Left and right ideals were studied by Paz and Peleg [13] in 1965 under the names “ultimate definite” and “reverse ultimate definite events”. The results in [13] include closure prop- erties, decision procedures, and canonical representations for these languages. All-sided ideals were used by Haines [8] (not under that name) in 1969 in con- nection with subword-free and subword-closed languages, and by Thierrin [17] in 1973 in connection with subword-convex languages. De Luca and Varricchio [10] showed in 1990 that a language is factor-closed (also called “factorial”) if and only if it is the complement of a two-sided ideal. In 2001 Shyr [16] studied right, left, and two-sided ideals and their generators in connection with codes. In 2008 3 ‘Subword’ is often used to mean ‘factor’; here ‘subword’ means subsequence. 2 all four types of ideals were considered by Ang and Brzozowski [1, 2]

This content is AI-processed based on ArXiv data.