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18
Syntactic complexity of ideal and closed languages
, 2011
"... The state complexity of a regular language is the number of states in the minimal deterministic automaton accepting the language. The syntactic complexity of a regular language is the cardinality of its syntactic semigroup. The syntactic complexity of a subclass of regular languages is the worstc ..."
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The state complexity of a regular language is the number of states in the minimal deterministic automaton accepting the language. The syntactic complexity of a regular language is the cardinality of its syntactic semigroup. The syntactic complexity of a subclass of regular languages is the worstcase syntactic complexity taken as a function of the state complexity n of languages in that class. We prove that nn−1 is a tight upper bound on the complexity of right ideals and prefixclosed languages, and that there exist left ideals and suffixclosed languages of syntactic complexity nn−1+n−1, and twosided ideals and factorclosed languages of syntactic complexity nn−2 + (n − 2)2n−2 + 1.
State complexity and the monoid of transformations of a finite set
, 2003
"... In this paper we consider the state complexity of an operation on formal languages, root(L). This naturally entails the study of the monoid of transformations of a finite set. We obtain lower bounds on the state complexity of root(L) and the size of the largest submonoid generated by two elements. 1 ..."
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Cited by 11 (0 self)
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In this paper we consider the state complexity of an operation on formal languages, root(L). This naturally entails the study of the monoid of transformations of a finite set. We obtain lower bounds on the state complexity of root(L) and the size of the largest submonoid generated by two elements. 1
Syntactic Complexity of Prefix, Suffix, and BifixFree Regular Languages
"... Abstract. The syntactic complexity of a regular language is the cardinality of its syntactic semigroup. The syntactic complexity of a subclass of the class of regular languages is the maximal syntactic complexity of languages in that class, taken as a function of the state complexity n of these lan ..."
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Abstract. The syntactic complexity of a regular language is the cardinality of its syntactic semigroup. The syntactic complexity of a subclass of the class of regular languages is the maximal syntactic complexity of languages in that class, taken as a function of the state complexity n of these languages. We study the syntactic complexity of prefix, suffix, and bifixfree regular languages. We prove that nn−2 is a tight upper bound for prefixfree regular languages. We present properties of the syntactic semigroups of suffix and bifixfree regular languages, and conjecture tight upper bounds on their size.
Quotient complexity of starfree languages
, 2010
"... The quotient complexity, also known as state complexity, of a regular language is the number of distinct left quotients of the language. The quotient complexity of an operation is the maximal quotient complexity of the language resulting from the operation, as a function of the quotient complexiti ..."
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Cited by 7 (3 self)
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The quotient complexity, also known as state complexity, of a regular language is the number of distinct left quotients of the language. The quotient complexity of an operation is the maximal quotient complexity of the language resulting from the operation, as a function of the quotient complexities of the operands. The class of starfree languages is the smallest class containing the finite languages and closed under boolean operations and concatenation. We prove that the tight bounds on the quotient complexities of union, intersection, difference, symmetric difference, concatenation, and star for starfree languages are the same as those for regular languages, with some small exceptions, whereas the bound for reversal is 2n − 1.
Syntactic Complexities of Some Classes of StarFree Languages
"... Abstract. The syntactic complexity of a regular language is the cardinality of its syntactic semigroup. The syntactic complexity of a subclass of regular languages is the maximal syntactic complexity of languages in that subclass, taken as a function of the state complexity n of these languages. W ..."
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Abstract. The syntactic complexity of a regular language is the cardinality of its syntactic semigroup. The syntactic complexity of a subclass of regular languages is the maximal syntactic complexity of languages in that subclass, taken as a function of the state complexity n of these languages. We study the syntactic complexity of three subclasses of starfree languages. We find tight upper bounds for languages accepted by monotonic, partially monotonic and “nearly monotonic ” automata; all three of these classes are starfree. We conjecture that the bound for nearly monotonic languages is also a tight upper bound for starfree languages.
Regular Languages, Sizes of Syntactic Monoids, Graph Colouring, State Complexity Results, and How These Topics are Related to Each Other
"... We invite the reader to join our quest for the largest subsemigroup of a transformation monoid on n elements generated by two transformations. Some of the presented results were independently obtained by the authors [6, 7, 8] and Krawetz, Lawrence, and Shallit [12, 13]. In particular, we will see ho ..."
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Cited by 2 (0 self)
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We invite the reader to join our quest for the largest subsemigroup of a transformation monoid on n elements generated by two transformations. Some of the presented results were independently obtained by the authors [6, 7, 8] and Krawetz, Lawrence, and Shallit [12, 13]. In particular, we will see how a surprising connection to graph colouring and chromatic polynomials is very helpful to count the elements of the investigated subsemigroup of transformations. At the end of our search, we will present some applications of these results to state complexity problems for one and twoway finite automata.
LARGEST 2GENERATED SUBSEMIGROUPS OF THE SYMMETRIC INVERSE SEMIGROUP
, 2006
"... The symmetric inverse monoid In is the set of all partial permutations of an nelement set. The largest possible size of a 2generated subsemigroup of In is determined. Examples of semigroups with these sizes are given. Consequently, if M(n) denotes this maximum, it is shown that M(n)/In  → 1 as ..."
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The symmetric inverse monoid In is the set of all partial permutations of an nelement set. The largest possible size of a 2generated subsemigroup of In is determined. Examples of semigroups with these sizes are given. Consequently, if M(n) denotes this maximum, it is shown that M(n)/In  → 1 as n → ∞. Furthermore, we may deduce, the already known fact, that In embeds as a local submonoid of an inverse 2generated subsemigroup of In+1.