35 research outputs found
Deterministic blow-ups of minimal NFA\u27s
The paper treats the question whether there
always exists a minimal nondeterministic finite automaton of n states whose equivalent minimal deterministic finite automaton has α states for any integers n and α with n ≤ α ≤ 2n.
Partial answers to this question were given by Iwama, Kambayashi, and Takaki (2000) and by Iwama, Matsuura, and Paterson (2003).
In the present paper, the question is completely solved by presenting appropriate automata for all values of n and α. However,
in order to give an explicit construction of the automata, we increase the input alphabet to exponential sizes. Then we prove that 2n letters would be sufficient but we describe the related automata only implicitly. In the last section, we investigate the above question for automata over binary and unary alphabets
Operations on Boolean and Alternating Finite Automata
We examine the complexity of basic regular operations on languages
represented by Boolean and alternating finite automata. We get tight upper
bounds m+n and m+n+1 for union, intersection, and difference, 2^m+n and 2^m+n+1
for concatenation, 2^n+n and 2^n+n+1 for square, m and m+1 for left quotient,
2^m and 2^m+1 for right quotient. We also show that in both models, the
complexity of complementation and symmetric difference is n and m+n,
respectively, while the complexity of star and reversal is 2^n. All our
witnesses are described over a unary or binary alphabets, and whenever we use a
binary alphabet, it is always optimal.Comment: In Proceedings AFL 2023, arXiv:2309.0112
Operations on Automata with All States Final
We study the complexity of basic regular operations on languages represented
by incomplete deterministic or nondeterministic automata, in which all states
are final. Such languages are known to be prefix-closed. We get tight bounds on
both incomplete and nondeterministic state complexity of complement,
intersection, union, concatenation, star, and reversal on prefix-closed
languages.Comment: In Proceedings AFL 2014, arXiv:1405.527
Complexity in Prefix-Free Regular Languages
We examine deterministic and nondeterministic state complexities of regular
operations on prefix-free languages. We strengthen several results by providing
witness languages over smaller alphabets, usually as small as possible. We next
provide the tight bounds on state complexity of symmetric difference, and
deterministic and nondeterministic state complexity of difference and cyclic
shift of prefix-free languages.Comment: In Proceedings DCFS 2010, arXiv:1008.127
Quotient Complexity of Ideal Languages
The final publication is available at Elsevier via http://dx.doi.org/10.1016/j.tcs.2012.10.055 © 2013. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/A language L over an alphabet 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. Ideal languages are not only of interest from the theoretical point of view, but also have applications to pattern matching. We study the state complexity of common operations in the class of regular ideal languages, but prefer to use the equivalent term “quotient complexity”, which is the number of distinct left quotients of a language. We find tight upper bounds on the complexity of each type of ideal language in terms of the complexity of an arbitrary generator and of the minimal generator, and also on the complexity of the minimal generator in terms of the complexity of the language. Moreover, tight upper bounds on the complexity of union, intersection, set difference, symmetric difference, concatenation, star, and reversal of ideal languages are derived.Natural Sciences and Engineering Research Council of Canada grant [OGP0000871]VEGA grant 2/0111/0
Quotient Complexity Of Closed Languages
The final publication is available at Springer via http://dx.doi.org/10.1007/s00224-013-9515-7A language L is prefix-closed if, whenever a word w is in L, then every prefix of w is also in L. We define suffix-, factor-, and subword-closed languages in an analogous way, where by factor we mean contiguous subsequence, and by subword we mean scattered subsequence. We study the state complexity (which we prefer to call quotient complexity) of operations on prefix-, suffix-, factor-, and subword-closed languages. We find tight upper bounds on the complexity of the subword-closure of arbitrary languages, and on the complexity of boolean operations, concatenation, star, and reversal in each of the four classes of closed languages. We show that repeated applications of positive closure and complement to a closed language result in at most four distinct languages, while Kleene closure and complement give at most eight.Natural Sciences and Engineering Research Council of Canada [OGP0000871]VEGA grant [2/0183/11][APVV-0035-10
Quotient Complexity of Bifix-, Factor-, and Subword-Free Regular Language
A language is prefix-free if whenever words and are in and is a prefix of , then . Suffix-, factor-, and subword-free languages are defined similarly, where by ``subword" we mean ``subsequence", and a language is bifix-free if it is both prefix- and suffix-free. These languages have important applications in coding theory. The quotient complexity of an operation on regular languages is defined as the number of left quotients of the result of the operation as a function of the numbers of left quotients of the operands. The quotient complexity of a regular language is the same as its state complexity, which is the number of states in the complete minimal deterministic finite automaton accepting the language. The state/quotient complexity of operations in the classes of prefix- and suffix-free languages has been studied before. Here, we study the complexity of operations in the classes of bifix-, factor-, and subword-free languages. We find tight upper bounds on the quotient complexity of intersection, union, difference, symmetric difference, concatenation, star, and reversal in these three classes of languages.Natural Sciences and Engineering Research Council of Canada [OGP0000871]Slovak Research and Development Agency [APVV-0035-10]Algorithms, Automata, and Discrete Data Structures VEGA, [2/0183/11
A New Technique for Reachability of States in Concatenation Automata
We present a new technique for demonstrating the reachability of states in
deterministic finite automata representing the concatenation of two languages.
Such demonstrations are a necessary step in establishing the state complexity
of the concatenation of two languages, and thus in establishing the state
complexity of concatenation as an operation. Typically, ad-hoc induction
arguments are used to show particular states are reachable in concatenation
automata. We prove some results that seem to capture the essence of many of
these induction arguments. Using these results, reachability proofs in
concatenation automata can often be done more simply and without using
induction directly.Comment: 23 pages, 1 table. Added missing affiliation/funding informatio