arXiv subjects

Janusz Brzozowski

Publications and source records attributed to Janusz Brzozowski.

At least 19 recordsLinked to original sources

arXiv2018

State Complexity of Overlap Assembly

The \emph{state complexity} of a regular language $L_m$ is the number $m$ of states in a minimal deterministic finite automaton (DFA) accepting $L_m$. The state complexity of a regularity-preserving binary operation on regular languages is defined as the maximal state complexity of the result of the operation where the two operands range over all languages of state complexities $\le m$ and $\le n$, respectively. We find a tight upper bound on the state complexity of the binary operation \emph{overlap assembly} on regular languages. This operation was introduced by Csuhaj-Varjú, Petre, and Vaszil to model the process of self-assembly of two linear DNA strands into a longer DNA strand, provided that their ends "overlap". We prove that the state complexity of the overlap assembly of languages $L_m$ and $L_n$, where $m\ge 2$ and $n\ge1$, is at most $2 (m-1) 3^{n-1} + 2^n$. Moreover, for $m \ge 2$ and $n \ge 3$ there exist languages $L_m$ and $L_n$ over an alphabet of size $n$ whose overlap assembly meets the upper bound and this bound cannot be met with smaller alphabets. Finally, we prove that $m+n$ is a tight upper bound on the overlap assembly of unary languages, and that there are binary languages whose overlap assembly has exponential state complexity at least $m(2^{n-1}-2)+2$.

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arXiv2017

Unrestricted State Complexity of Binary Operations on Regular and Ideal Languages

We study the state complexity of binary operations on regular languages over different alphabets. It is known that if $L'_m$ and $L_n$ are languages of state complexities $m$ and $n$, respectively, and restricted to the same alphabet, the state complexity of any binary boolean operation on $L'_m$ and $L_n$ is $mn$, and that of product (concatenation) is $m 2^n - 2^{n-1}$. In contrast to this, we show that if $L'_m$ and $L_n$ are over different alphabets, the state complexity of union and symmetric difference is $(m+1)(n+1)$, that of difference is $mn+m$, that of intersection is $mn$, and that of product is $m2^n+2^{n-1}$. We also study unrestricted complexity of binary operations in the classes of regular right, left, and two-sided ideals, and derive tight upper bounds. The bounds for product of the unrestricted cases (with the bounds for the restricted cases in parentheses) are as follows: right ideals $m+2^{n-2}+2^{n-1}$ ($m+2^{n-2}$); left ideals $mn+m+n$ ($m+n-1$); two-sided ideals $m+2n$ ($m+n-1$). The state complexities of boolean operations on all three types of ideals are the same as those of arbitrary regular languages, whereas that is not the case if the alphabets of the arguments are the same. Finally, we update the known results about most complex regular, right-ideal, left-ideal, and two-sided-ideal languages to include the unrestricted cases.

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arXiv2016

Complexity of Suffix-Free Regular Languages

We study various complexity properties of suffix-free regular languages. The quotient complexity of a regular language $L$ is the number of left quotients of $L$; this is the same as the state complexity of $L$. A regular language $L'$ is a dialect of a regular language $L$ if it differs only slightly from $L$. The quotient complexity of an operation on regular languages is the maximal quotient complexity of the result of the operation expressed as a function of the quotient complexities of the operands. A sequence $(L_k,L_{k+1},\dots)$ of regular languages in some class ${\mathcal C}$, where $n$ is the quotient complexity of $L_n$, is called a stream. A stream is most complex in class ${\mathcal C}$ if its languages $L_n$ meet the complexity upper bounds for all basic measures. It is known that there exist such most complex streams in the class of regular languages, in the class of prefix-free languages, and also in the classes of right, left, and two-sided ideals. In contrast to this, we prove that there does not exist a most complex stream in the class of suffix-free regular languages. However, we do exhibit one ternary suffix-free stream that meets the bound for product and whose restrictions to binary alphabets meet the bounds for star and boolean operations. We also exhibit a quinary stream that meets the bounds for boolean operations, reversal, size of syntactic semigroup, and atom complexities. Moreover, we solve an open problem about the bound for the product of two languages of quotient complexities $m$ and $n$ in the binary case by showing that it can be met for infinitely many $m$ and $n$.

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arXiv2016

Most Complex Regular Ideal Languages

A right ideal (left ideal, two-sided ideal) is a non-empty language $L$ over an alphabet $Σ$ such that $L=LΣ^*$ ($L=Σ^*L$, $L=Σ^*LΣ^*$). Let $k=3$ for right ideals, 4 for left ideals and 5 for two-sided ideals. We show that there exist sequences ($L_n \mid n \ge k $) of right, left, and two-sided regular ideals, where $L_n$ has quotient complexity (state complexity) $n$, such that $L_n$ is most complex in its class under the following measures of complexity: the size of the syntactic semigroup, the quotient complexities of the left quotients of $L_n$, the number of atoms (intersections of complemented and uncomplemented left quotients), the quotient complexities of the atoms, and the quotient complexities of reversal, star, product (concatenation), and all binary boolean operations. In that sense, these ideals are "most complex" languages in their classes, or "universal witnesses" to the complexity of the various operations.

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arXiv2016

Complexity of Left-Ideal, Suffix-Closed and Suffix-Free Regular Languages

A language $L$ over an alphabet $Σ$ is suffix-convex if, for any words $x,y,z\inΣ^*$, whenever $z$ and $xyz$ are in $L$, then so is $yz$. Suffix-convex languages include three special cases: left-ideal, suffix-closed, and suffix-free languages. We examine complexity properties of these three special classes of suffix-convex regular languages. In particular, we study the quotient/state complexity of boolean operations, product (concatenation), star, and reversal on these languages, as well as the size of their syntactic semigroups, and the quotient complexity of their atoms.

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arXiv2016

On the State Complexity of the Shuffle of Regular Languages

We investigate the shuffle operation on regular languages represented by complete deterministic finite automata. We prove that $f(m,n)=2^{mn-1} + 2^{(m-1)(n-1)}(2^{m-1}-1)(2^{n-1}-1)$ is an upper bound on the state complexity of the shuffle of two regular languages having state complexities $m$ and $n$, respectively. We also state partial results about the tightness of this bound. We show that there exist witness languages meeting the bound if $2\le m\le 5$ and $n\ge2$, and also if $m=n=6$. Moreover, we prove that in the subset automaton of the NFA accepting the shuffle, all $2^{mn}$ states can be distinguishable, and an alphabet of size three suffices for that. It follows that the bound can be met if all $f(m,n)$ states are reachable. We know that an alphabet of size at least $mn$ is required provided that $m,n \ge 2$. The question of reachability, and hence also of the tightness of the bound $f(m,n)$ in general, remains open.

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arXiv2016

Complexity of Prefix-Convex Regular Languages

A language $L$ over an alphabet $Σ$ is prefix-convex if, for any words $x,y,z\inΣ^*$, whenever $x$ and $xyz$ are in $L$, then so is $xy$. Prefix-convex languages include right-ideal, prefix-closed, and prefix-free languages. We study complexity properties of prefix-convex regular languages. In particular, we find the quotient/state complexity of boolean operations, product (concatenation), star, and reversal, the size of the syntactic semigroup, and the quotient complexity of atoms. For binary operations we use arguments with different alphabets when appropriate; this leads to higher tight upper bounds than those obtained with equal alphabets. We exhibit most complex prefix-convex languages that meet the complexity bounds for all the measures listed above.

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arXiv2016

Unrestricted State Complexity of Binary Operations on Regular Languages

I study the state complexity of binary operations on regular languages over different alphabets. It is well known that if $L'_m$ and $L_n$ are languages restricted to be over the same alphabet, with $m$ and $n$ quotients, respectively, the state complexity of any binary boolean operation on $L'_m$ and $L_n$ is $mn$, and that of the product (concatenation) is $(m-1)2^n +2^{n-1}$. In contrast to this, I show that if $L'_m$ and $L_n$ are over their own different alphabets, the state complexity of union and symmetric difference is $mn+m+n+1$, that of intersection is $mn$, that of difference is $mn+m$, and that of the product is $m2^n+2^{n-1}$.

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arXiv2015

Syntactic Complexity of Suffix-Free Languages

We solve an open problem concerning syntactic complexity: We prove that the cardinality of the syntactic semigroup of a suffix-free language with $n$ left quotients (that is, with state complexity $n$) is at most $(n-1)^{n-2}+n-2$ for $n\ge 6$. Since this bound is known to be reachable, this settles the problem. We also reduce the alphabet of the witness languages reaching this bound to five letters instead of $n+2$, and show that it cannot be any smaller. Finally, we prove that the transition semigroup of a minimal deterministic automaton accepting a witness language is unique for each $n$.

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arXiv2015

Quotient Complexities of Atoms in Regular Ideal Languages

A (left) quotient of a language $L$ by a word $w$ is the language $w^{-1}L=\{x\mid wx\in L\}$. The quotient complexity of a regular language $L$ is the number of quotients of $L$; it is equal to the state complexity of $L$, which is the number of states in a minimal deterministic finite automaton accepting $L$. An atom of $L$ is an equivalence class of the relation in which two words are equivalent if for each quotient, they either are both in the quotient or both not in it; hence it is a non-empty intersection of complemented and uncomplemented quotients of $L$. A right (respectively, left and two-sided) ideal is a language $L$ over an alphabet $Σ$ that satisfies $L=LΣ^*$ (respectively, $L=Σ^*L$ and $L=Σ^*LΣ^*$). We compute the maximal number of atoms and the maximal quotient complexities of atoms of right, left and two-sided regular ideals.

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arXiv2014

Upper Bounds on Syntactic Complexity of Left and Two-Sided Ideals

We solve two open problems concerning syntactic complexity: We prove that the cardinality of the syntactic semigroup of a left ideal or a suffix-closed language with $n$ left quotients (that is, with state complexity $n$) is at most $n^{n-1}+n-1$, and that of a two-sided ideal or a factor-closed language is at most $n^{n-2}+(n-2)2^{n-2}+1$. Since these bounds are known to be reachable, this settles the problems.

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arXiv2014

Large Aperiodic Semigroups

The syntactic complexity of a regular language is the size of its syntactic semigroup. This semigroup is isomorphic to the transition semigroup of the minimal deterministic finite automaton accepting the language, that is, to the semigroup generated by transformations induced by non-empty words on the set of states of the automaton. In this paper we search for the largest syntactic semigroup of a star-free language having $n$ left quotients; equivalently, we look for the largest transition semigroup of an aperiodic finite automaton with $n$ states. We introduce two new aperiodic transition semigroups. The first is generated by transformations that change only one state; we call such transformations and resulting semigroups unitary. In particular, we study complete unitary semigroups which have a special structure, and we show that each maximal unitary semigroup is complete. For $n \ge 4$ there exists a complete unitary semigroup that is larger than any aperiodic semigroup known to date. We then present even larger aperiodic semigroups, generated by transformations that map a non-empty subset of states to a single state; we call such transformations and semigroups semiconstant. In particular, we examine semiconstant tree semigroups which have a structure based on full binary trees. The semiconstant tree semigroups are at present the best candidates for largest aperiodic semigroups. We also prove that $2^n-1$ is an upper bound on the state complexity of reversal of star-free languages, and resolve an open problem about a special case of state complexity of concatenation of star-free languages.

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arXiv2014

Maximally Atomic Languages

The atoms of a regular language are non-empty intersections of complemented and uncomplemented quotients of the language. Tight upper bounds on the number of atoms of a language and on the quotient complexities of atoms are known. We introduce a new class of regular languages, called the maximally atomic languages, consisting of all languages meeting these bounds. We prove the following result: If L is a regular language of quotient complexity n and G is the subgroup of permutations in the transition semigroup T of the minimal DFA of L, then L is maximally atomic if and only if G is transitive on k-subsets of 1,...,n for 0 <= k <= n and T contains a transformation of rank n-1.

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arXiv2013

Most Complex Regular Right-Ideal Languages

A right ideal is a language L over an alphabet A that satisfies L = LA*. We show that there exists a stream (sequence) (R_n : n \ge 3) of regular right ideal languages, where R_n has n left quotients and is most complex under the following measures of complexity: the state complexities of the left quotients, the number of atoms (intersections of complemented and uncomplemented left quotients), the state complexities of the atoms, the size of the syntactic semigroup, the state complexities of the operations of reversal, star, and product, and the state complexities of all binary boolean operations. In that sense, this stream of right ideals is a universal witness.

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arXiv2013

Symmetric Groups and Quotient Complexity of Boolean Operations

The quotient complexity of a regular language L is the number of left quotients of L, which is the same as the state complexity of L. Suppose that L and L' are binary regular languages with quotient complexities m and n, and that the transition semigroups of the minimal deterministic automata accepting L and L' are the symmetric groups S_m and S_n of degrees m and n, respectively. Denote by o any binary boolean operation that is not a constant and not a function of one argument only. For m,n >= 2 with (m,n) not in {(2,2),(3,4),(4,3),(4,4)} we prove that the quotient complexity of LoL' is mn if and only either (a) m is not equal to n or (b) m=n and the bases (ordered pairs of generators) of S_m and S_n are not conjugate. For (m,n)\in {(2,2),(3,4),(4,3),(4,4)} we give examples to show that this need not hold. In proving these results we generalize the notion of uniform minimality to direct products of automata. We also establish a non-trivial connection between complexity of boolean operations and group theory.

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arXiv2013

Theory of Atomata

We show that every regular language defines a unique nondeterministic finite automaton (NFA), which we call "átomaton", whose states are the "atoms" of the language, that is, non-empty intersections of complemented or uncomplemented left quotients of the language. We describe methods of constructing the átomaton, and prove that it is isomorphic to the reverse automaton of the minimal deterministic finite automaton (DFA) of the reverse language. We study "atomic" NFAs in which the right language of every state is a union of atoms. We generalize Brzozowski's double-reversal method for minimizing a deterministic finite automaton (DFA), showing that the result of applying the subset construction to an NFA is a minimal DFA if and only if the reverse of the NFA is atomic. We prove that Sengoku's claim that his method always finds a minimal NFA is false.

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arXiv2013

Maximal Syntactic Complexity of Regular Languages Implies Maximal Quotient Complexities of Atoms

We relate two measures of complexity of regular languages. The first is syntactic complexity, that is, the cardinality of the syntactic semigroup of the language. That semigroup is isomorphic to the semigroup of transformations of states induced by non-empty words in the minimal deterministic finite automaton accepting the language. If the language has n left quotients (its minimal automaton has n states), then its syntactic complexity is at most n^n and this bound is tight. The second measure consists of the quotient (state) complexities of the atoms of the language, where atoms are non-empty intersections of complemented and uncomplemented quotients. A regular language has at most 2^n atoms and this bound is tight. The maximal quotient complexity of any atom with r complemented quotients is 2^n-1, if r=0 or r=n, and 1+\sum_{k=1}^{r} \sum_{h=k+1}^{k+n-r} \binom{h}{n} \binom{k}{h}, otherwise. We prove that if a language has maximal syntactic complexity, then it has 2^n atoms and each atom has maximal quotient complexity, but the converse is false.

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arXiv2013

Syntactic Complexity of R- and J-Trivial Regular Languages

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 R- and J-trivial regular languages, and prove that n! and floor of [e(n-1)!] are tight upper bounds for these languages, respectively. We also prove that 2^{n-1} is the tight upper bound on the state complexity of reversal of J-trivial regular languages.

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