Misplaced Pages

#866133

75-413: In the theory of computation , a branch of theoretical computer science , a deterministic finite automaton ( DFA )—also known as deterministic finite acceptor ( DFA ), deterministic finite-state machine ( DFSM ), or deterministic finite-state automaton ( DFSA )—is a finite-state machine that accepts or rejects a given string of symbols, by running through a state sequence uniquely determined by

150-555: A δ {\displaystyle \delta } , and so the two descriptions are equivalent. DFAs are one of the most practical models of computation, since there is a trivial linear time, constant-space, online algorithm to simulate a DFA on a stream of input. Also, there are efficient algorithms to find a DFA recognizing: Because DFAs can be reduced to a canonical form ( minimal DFAs ), there are also efficient algorithms to determine: DFAs are equivalent in computing power to nondeterministic finite automata (NFAs). This

225-482: A {\displaystyle a} , the number of words of length d m + a {\displaystyle dm+a} is asymptotically C a m p a λ a m {\displaystyle C_{a}m^{p_{a}}\lambda _{a}^{m}} . The zeta function of a language L is The zeta function of a regular language is not in general rational, but that of an arbitrary cyclic language is. The notion of

300-533: A ∈ Σ {\displaystyle a\in \Sigma } , one may construct a transition function δ a : Q → Q {\displaystyle \delta _{a}:Q\rightarrow Q} by defining δ a ( q ) = δ ( q , a ) {\displaystyle \delta _{a}(q)=\delta (q,a)} for all q ∈ Q {\displaystyle q\in Q} . (This trick

375-538: A characteristic set of the regular language; otherwise, the constructed DFA will be inconsistent either with S + {\displaystyle S^{+}} or S − {\displaystyle S^{-}} . Other notable DFA identification algorithms include the RPNI algorithm, the Blue-Fringe evidence-driven state-merging algorithm, and Windowed-EDSM. Another research direction

450-536: A finite automaton . The equivalence of regular expressions and finite automata is known as Kleene's theorem (after American mathematician Stephen Cole Kleene ). In the Chomsky hierarchy , regular languages are the languages generated by Type-3 grammars . The collection of regular languages over an alphabet Σ is defined recursively as follows: See regular expression for syntax and semantics of regular expressions. All finite languages are regular; in particular

525-404: A regular language (also called a rational language ) is a formal language that can be defined by a regular expression , in the strict sense in theoretical computer science (as opposed to many modern regular expression engines, which are augmented with features that allow the recognition of non-regular languages). Alternatively, a regular language can be defined as a language recognised by

600-430: A start state (denoted graphically by an arrow coming in from nowhere) where computations begin, and a set of accept states (denoted graphically by a double circle) which help define when a computation is successful. A DFA is defined as an abstract mathematical concept, but is often implemented in hardware and software for solving various specific problems such as lexical analysis and pattern matching . For example,

675-421: A DFA can model software that decides whether or not online user input such as email addresses are syntactically valid. DFAs have been generalized to nondeterministic finite automata (NFA) which may have several arrows of the same label starting from a state. Using the powerset construction method, every NFA can be translated to a DFA that recognizes the same language. DFAs, and NFAs as well, recognize exactly

750-504: A DFA-like automaton needs to have a state to represent any possible number of "currently open" parentheses, meaning it would need an unbounded number of states. Another simpler example is the language consisting of strings of the form ab for some finite but arbitrary number of a ' s, followed by an equal number of b ' s. Given a set of positive words S + ⊂ Σ ∗ {\displaystyle S^{+}\subset \Sigma ^{*}} and

825-413: A Turing machine is one of the most important results in computability theory, as it is an example of a concrete problem that is both easy to formulate and impossible to solve using a Turing machine. Much of computability theory builds on the halting problem result. Another important step in computability theory was Rice's theorem , which states that for all non-trivial properties of partial functions, it

SECTION 10

#1732772316867

900-563: A Turing machine will always require only a finite amount of memory. So in principle, any problem that can be solved (decided) by a Turing machine can be solved by a computer that has a finite amount of memory. The theory of computation can be considered the creation of models of all kinds in the field of computer science. Therefore, mathematics and logic are used. In the last century, it separated from mathematics and became an independent academic discipline with its own conferences such as FOCS in 1960 and STOC in 1969, and its own awards such as

975-487: A defined subclass of the recursive functions. Different models of computation have the ability to do different tasks. One way to measure the power of a computational model is to study the class of formal languages that the model can generate; in such a way to the Chomsky hierarchy of languages is obtained. (There are many textbooks in this area; this list is by necessity incomplete.) Regular language In theoretical computer science and formal language theory ,

1050-401: A finite sequence of 0s and 1s as input. For each state, there is a transition arrow leading out to a next state for both 0 and 1. Upon reading a symbol, a DFA jumps deterministically from one state to another by following the transition arrow. For example, if the automaton is currently in state S 0 and the current input symbol is 1, then it deterministically jumps to state S 1 . A DFA has

1125-437: A finite automaton, since a finite automaton has finite memory and it cannot remember the exact number of a's. Techniques to prove this fact rigorously are given below . A regular language satisfies the following equivalent properties: Properties 10. and 11. are purely algebraic approaches to define regular languages; a similar set of statements can be formulated for a monoid M ⊆ Σ . In this case, equivalence over M leads to

1200-516: A finite representation of a formal language that may be an infinite set. Automata are used as theoretical models for computing machines, and are used for proofs about computability. Language theory is a branch of mathematics concerned with describing languages as a set of operations over an alphabet . It is closely linked with automata theory, as automata are used to generate and recognize formal languages. There are several classes of formal languages, each allowing more complex language specification than

1275-415: A fixed finite alphabet, the theory of the set of all languages — together with strings, membership of a string in a language, and for each character, a function to append the character to a string (and no other operations) — is decidable, and its minimal elementary substructure consists precisely of regular languages. For a binary alphabet, the theory is called S2S . In computational complexity theory ,

1350-520: A pair of letters a , b ∈ Σ {\displaystyle a,b\in \Sigma } , one may define a new function δ ^ a b = δ a ∘ δ b {\displaystyle {\widehat {\delta }}_{ab}=\delta _{a}\circ \delta _{b}} , where ∘ {\displaystyle \circ } denotes function composition. Clearly, this process may be recursively continued, giving

1425-408: A problem can be solved at all on a computer, but also how efficiently the problem can be solved. Two major aspects are considered: time complexity and space complexity, which are respectively how many steps it takes to perform a computation, and how much memory is required to perform that computation. In order to analyze how much time and space a given algorithm requires, computer scientists express

1500-463: A regular language has been generalized to infinite words (see ω-automata ) and to trees (see tree automaton ). Rational set generalizes the notion (of regular/rational language) to monoids that are not necessarily free . Likewise, the notion of a recognizable language (by a finite automaton) has namesake as recognizable set over a monoid that is not necessarily free. Howard Straubing notes in relation to these facts that “The term "regular language"

1575-495: A set of negative words S − ⊂ Σ ∗ {\displaystyle S^{-}\subset \Sigma ^{*}} one can construct a DFA that accepts all words from S + {\displaystyle S^{+}} and rejects all words from S − {\displaystyle S^{-}} : this problem is called DFA identification (synthesis, learning). While some DFA can be constructed in linear time,

SECTION 20

#1732772316867

1650-424: A singly infinite tape is a 7- tuple where The machine always accepts a regular language. There must exist at least one element of the set F (a HALT state) for the language to be nonempty. Theory of computation In order to perform a rigorous study of computation, computer scientists work with a mathematical abstraction of computers called a model of computation . There are several models in use, but

1725-433: A slightly different notion: an automaton that defines at most one transition for each state and each input symbol; the transition function is allowed to be partial . When no transition is defined, such an automaton halts. A local automaton is a DFA, not necessarily complete, for which all edges with the same label lead to a single vertex. Local automata accept the class of local languages , those for which membership of

1800-420: A word in the language is determined by a "sliding window" of length two on the word. A Myhill graph over an alphabet A is a directed graph with vertex set A and subsets of vertices labelled "start" and "finish". The language accepted by a Myhill graph is the set of directed paths from a start vertex to a finish vertex: the graph thus acts as an automaton. The class of languages accepted by Myhill graphs

1875-445: Is PSPACE complete since there are small NFAs with shortest rejecting word in exponential size. A DFA is universal if and only if all states are final states, but this does not hold for NFAs. The Equality, Inclusion and Minimization Problems are also PSPACE complete since they require forming the complement of an NFA which results in an exponential blow up of size. On the other hand, finite-state automata are of strictly limited power in

1950-594: Is constant-recursive ; that is, there exist an integer constant n 0 {\displaystyle n_{0}} , complex constants λ 1 , … , λ k {\displaystyle \lambda _{1},\,\ldots ,\,\lambda _{k}} and complex polynomials p 1 ( x ) , … , p k ( x ) {\displaystyle p_{1}(x),\,\ldots ,\,p_{k}(x)} such that for every n ≥ n 0 {\displaystyle n\geq n_{0}}

2025-528: Is undecidable whether a Turing machine computes a partial function with that property. Computability theory is closely related to the branch of mathematical logic called recursion theory , which removes the restriction of studying only models of computation which are reducible to the Turing model. Many mathematicians and computational theorists who study recursion theory will refer to it as computability theory. Complexity theory considers not only whether

2100-497: Is a bit unfortunate. Papers influenced by Eilenberg 's monograph often use either the term "recognizable language", which refers to the behavior of automata, or "rational language", which refers to important analogies between regular expressions and rational power series . (In fact, Eilenberg defines rational and recognizable subsets of arbitrary monoids; the two notions do not, in general, coincide.) This terminology, while better motivated, never really caught on, and "regular language"

2175-460: Is also true for the largest induced sub-digraph of minimum in-degree one, which can be seen as a directed version of 1 -core . If DFAs recognize the languages that are obtained by applying an operation on the DFA recognizable languages then DFAs are said to be closed under the operation. The DFAs are closed under the following operations. For each operation, an optimal construction with respect to

2250-515: Is because, firstly any DFA is also an NFA, so an NFA can do what a DFA can do. Also, given an NFA, using the powerset construction one can build a DFA that recognizes the same language as the NFA, although the DFA could have exponentially larger number of states than the NFA. However, even though NFAs are computationally equivalent to DFAs, the above-mentioned problems are not necessarily solved efficiently also for NFAs. The non-universality problem for NFAs

2325-447: Is called currying .) From this perspective, δ a {\displaystyle \delta _{a}} "acts" on a state in Q to yield another state. One may then consider the result of function composition repeatedly applied to the various functions δ a {\displaystyle \delta _{a}} , δ b {\displaystyle \delta _{b}} , and so on. Given

Deterministic finite automaton - Misplaced Pages Continue

2400-439: Is due to application of SAT solvers by Marjin J. H. Heule and S. Verwer: the minimal DFA identification problem is reduced to deciding the satisfiability of a Boolean formula. The main idea is to build an augmented prefix-tree acceptor (a trie containing all input words with corresponding labels) based on the input sets and reduce the problem of finding a DFA with C {\displaystyle C} states to coloring

2475-474: Is not of the form p ( n ) λ n {\displaystyle p(n)\lambda ^{n}} , witnessing the non-regularity of the Dyck language. Care must be taken since some of the eigenvalues λ i {\displaystyle \lambda _{i}} could have the same magnitude. For example, the number of words of length n {\displaystyle n} in

2550-417: Is of linear size and it can be reached by all vertices. It has also been proven that if k is allowed to increase as n increases, then the whole digraph has a phase transition for strong connectivity similar to Erdős–Rényi model for connectivity. In a random DFA, the maximum number of vertices reachable from one vertex is very close to the number of vertices in the largest SCC with high probability. This

2625-556: Is the language recognized by M and this language is denoted by L ( M ) . A deterministic finite automaton without accept states and without a starting state is known as a transition system or semiautomaton . For more comprehensive introduction of the formal definition see automata theory . The following example is of a DFA M , with a binary alphabet, which requires that the input contains an even number of 0s. M = ( Q , Σ, δ , q 0 , F ) where The state S 1 represents that there has been an even number of 0s in

2700-423: Is the application of evolutionary algorithms : the smart state labeling evolutionary algorithm allowed to solve a modified DFA identification problem in which the training data (sets S + {\displaystyle S^{+}} and S − {\displaystyle S^{-}} ) is noisy in the sense that some words are attributed to wrong classes. Yet another step forward

2775-437: Is the class of local languages. When the start state and accept states are ignored, a DFA of n states and an alphabet of size k can be seen as a digraph of n vertices in which all vertices have k out-arcs labeled 1, ..., k (a k -out digraph). It is known that when k ≥ 2 is a fixed integer, with high probability, the largest strongly connected component (SCC) in such a k -out digraph chosen uniformly at random

2850-416: Is the input size). In other words, DSPACE( o (log log  n )) equals the class of regular languages. In practice, most nonregular problems are solved by machines taking at least logarithmic space . To locate the regular languages in the Chomsky hierarchy , one notices that every regular language is context-free . The converse is not true: for example, the language consisting of all strings having

2925-426: Is used almost universally.” Rational series is another generalization, this time in the context of a formal power series over a semiring . This approach gives rise to weighted rational expressions and weighted automata . In this algebraic context, the regular languages (corresponding to Boolean -weighted rational expressions) are usually called rational languages . Also in this context, Kleene's theorem finds

3000-943: The Clay Mathematics Institute in 2000. The Official Problem Description was given by Turing Award winner Stephen Cook . Aside from a Turing machine , other equivalent (See: Church–Turing thesis ) models of computation are in use. In addition to the general computational models, some simpler computational models are useful for special, restricted applications. Regular expressions , for example, specify string patterns in many contexts, from office productivity software to programming languages . Another formalism mathematically equivalent to regular expressions, Finite automata are used in circuit design and in some kinds of problem-solving. Context-free grammars specify programming language syntax. Non-deterministic pushdown automata are another formalism equivalent to context-free grammars. Primitive recursive functions are

3075-725: The IMU Abacus Medal (established in 1981 as the Rolf Nevanlinna Prize), the Gödel Prize , established in 1993, and the Knuth Prize , established in 1996. Some pioneers of the theory of computation were Ramon Llull , Alonzo Church , Kurt Gödel , Alan Turing , Stephen Kleene , Rózsa Péter , John von Neumann and Claude Shannon . Automata theory is the study of abstract machines (or more appropriately, abstract 'mathematical' machines or systems) and

Deterministic finite automaton - Misplaced Pages Continue

3150-512: The breadth-first search algorithm: the sought DFA's states are constrained to be numbered according to the BFS algorithm launched from the initial state. This approach reduces the search space by C ! {\displaystyle C!} by eliminating isomorphic automata. Read-only right-moving Turing machines are a particular type of Turing machine that only moves right; these are almost exactly equivalent to DFAs. The definition based on

3225-460: The complexity class of all regular languages is sometimes referred to as REGULAR or REG and equals DSPACE (O(1)), the decision problems that can be solved in constant space (the space used is independent of the input size). REGULAR ≠ AC , since it (trivially) contains the parity problem of determining whether the number of 1 bits in the input is even or odd and this problem is not in AC . On

3300-422: The empty string language {ε} = Ø* is regular. Other typical examples include the language consisting of all strings over the alphabet { a , b } which contain an even number of a 's, or the language consisting of all strings of the form: several a 's followed by several b 's. A simple example of a language that is not regular is the set of strings { a b | n ≥ 0}. Intuitively, it cannot be recognized with

3375-465: The regular expression (1*) (0 (1*) 0 (1*))* , where * is the Kleene star , e.g., 1* denotes any number (possibly zero) of consecutive ones. According to the above definition, deterministic finite automata are always complete : they define from each state a transition for each input symbol. While this is the most common definition, some authors use the term deterministic finite automaton for

3450-491: The DFA is a sequence of compositions of δ ^ {\displaystyle {\widehat {\delta }}} with itself. Repeated function composition forms a monoid . For the transition functions, this monoid is known as the transition monoid , or sometimes the transformation semigroup . The construction can also be reversed: given a δ ^ {\displaystyle {\widehat {\delta }}} , one can reconstruct

3525-676: The TB-algorithm that does not use any assumptions about S + {\displaystyle S^{+}} and S − {\displaystyle S^{-}} , the Traxbar algorithm. However, Traxbar does not guarantee the minimality of the constructed DFA. In his work E.M. Gold also proposed a heuristic algorithm for minimal DFA identification. Gold's algorithm assumes that S + {\displaystyle S^{+}} and S − {\displaystyle S^{-}} contain

3600-455: The computational problems that can be solved using these machines. These abstract machines are called automata. Automata comes from the Greek word (Αυτόματα) which means that something is doing something by itself. Automata theory is also closely related to formal language theory, as the automata are often classified by the class of formal languages they are able to recognize. An automaton can be

3675-430: The computer needs to perform a number of steps that grow linearly in the size of the problem. To simplify this problem, computer scientists have adopted Big O notation , which allows functions to be compared in a way that ensures that particular aspects of a machine's construction do not need to be considered, but rather only the asymptotic behavior as problems become large. So in our previous example, we might say that

3750-497: The concept of a recognizable language. Some authors use one of the above properties different from "1." as an alternative definition of regular languages. Some of the equivalences above, particularly those among the first four formalisms, are called Kleene's theorem in textbooks. Precisely which one (or which subset) is called such varies between authors. One textbook calls the equivalence of regular expressions and NFAs ("1." and "2." above) "Kleene's theorem". Another textbook calls

3825-442: The equivalence of regular expressions and DFAs ("1." and "3." above) "Kleene's theorem". Two other textbooks first prove the expressive equivalence of NFAs and DFAs ("2." and "3.") and then state "Kleene's theorem" as the equivalence between regular expressions and finite automata (the latter said to describe "recognizable languages"). A linguistically oriented text first equates regular grammars ("4." above) with DFAs and NFAs, calls

SECTION 50

#1732772316867

3900-494: The following recursive definition of δ ^ : Q × Σ ⋆ → Q {\displaystyle {\widehat {\delta }}:Q\times \Sigma ^{\star }\rightarrow Q} : δ ^ {\displaystyle {\widehat {\delta }}} is defined for all words w ∈ Σ ∗ {\displaystyle w\in \Sigma ^{*}} . A run of

3975-428: The input so far, while S 2 signifies an odd number. A 1 in the input does not change the state of the automaton. When the input ends, the state will show whether the input contained an even number of 0s or not. If the input did contain an even number of 0s, M will finish in state S 1 , an accepting state, so the input string will be accepted. The language recognized by M is the regular language given by

4050-457: The language of all even binary words is not of the form p ( n ) λ n {\displaystyle p(n)\lambda ^{n}} , but the number of words of even or odd length are of this form; the corresponding eigenvalues are 2 , − 2 {\displaystyle 2,-2} . In general, for every regular language there exists a constant d {\displaystyle d} such that for all

4125-450: The languages K and L are regular, so is the result of the following operations: Given two deterministic finite automata A and B , it is decidable whether they accept the same language. As a consequence, using the above closure properties, the following problems are also decidable for arbitrarily given deterministic finite automata A and B , with accepted languages L A and L B , respectively: For regular expressions,

4200-415: The languages generated by (any of) these "regular", after which it introduces regular expressions which it terms to describe "rational languages", and finally states "Kleene's theorem" as the coincidence of regular and rational languages. Other authors simply define "rational expression" and "regular expressions" as synonymous and do the same with "rational languages" and "regular languages". Apparently,

4275-444: The languages they can recognize; many simple languages, including any problem that requires more than constant space to solve, cannot be recognized by a DFA. The classic example of a simply described language that no DFA can recognize is bracket or Dyck language , i.e., the language that consists of properly paired brackets such as word "(()())". Intuitively, no DFA can recognize the Dyck language because DFAs are not capable of counting:

4350-442: The machine starts in the start state q 0 . The second condition says that given each character of string w , the machine will transition from state to state according to the transition function δ . The last condition says that the machine accepts w if the last input of w causes the machine to halt in one of the accepting states. Otherwise, it is said that the automaton rejects the string. The set of strings that M accepts

4425-493: The most commonly examined is the Turing machine . Computer scientists study the Turing machine because it is simple to formulate, can be analyzed and used to prove results, and because it represents what many consider the most powerful possible "reasonable" model of computation (see Church–Turing thesis ). It might seem that the potentially infinite memory capacity is an unrealizable attribute, but any decidable problem solved by

4500-648: The number s L ( n ) {\displaystyle s_{L}(n)} of words of length n {\displaystyle n} in L {\displaystyle L} is s L ( n ) = p 1 ( n ) λ 1 n + ⋯ + p k ( n ) λ k n {\displaystyle s_{L}(n)=p_{1}(n)\lambda _{1}^{n}+\dotsb +p_{k}(n)\lambda _{k}^{n}} . Thus, non-regularity of certain languages L ′ {\displaystyle L'} can be proved by counting

4575-407: The number of states has been determined in state complexity research. Since DFAs are equivalent to nondeterministic finite automata (NFA), these closures may also be proved using closure properties of NFA. A run of a given DFA can be seen as a sequence of compositions of a very general formulation of the transition function with itself. Here we construct that function. For a given input symbol

SECTION 60

#1732772316867

4650-494: The number of words of length n {\displaystyle n} in L {\displaystyle L} . The ordinary generating function for L is the formal power series The generating function of a language L is a rational function if L is regular. Hence for every regular language L {\displaystyle L} the sequence s L ( n ) n ≥ 0 {\displaystyle s_{L}(n)_{n\geq 0}}

4725-439: The one before it, i.e. Chomsky hierarchy , and each corresponding to a class of automata which recognizes it. Because automata are used as models for computation, formal languages are the preferred mode of specification for any problem that must be computed. Computability theory deals primarily with the question of the extent to which a problem is solvable on a computer. The statement that the halting problem cannot be solved by

4800-438: The other hand, REGULAR does not contain AC , because the nonregular language of palindromes , or the nonregular language { 0 n 1 n : n ∈ N } {\displaystyle \{0^{n}1^{n}:n\in \mathbb {N} \}} can both be recognized in AC . If a language is not regular, it requires a machine with at least Ω (log log  n ) space to recognize (where n

4875-587: The problem of identifying a DFA with the minimal number of states is NP-complete. The first algorithm for minimal DFA identification has been proposed by Trakhtenbrot and Barzdin and is called the TB-algorithm . However, the TB-algorithm assumes that all words from Σ {\displaystyle \Sigma } up to a given length are contained in either S + ∪ S − {\displaystyle S^{+}\cup S^{-}} . Later, K. Lang proposed an extension of

4950-416: The problem requires O ( n ) {\displaystyle O(n)} steps to solve. Perhaps the most important open problem in all of computer science is the question of whether a certain broad class of problems denoted NP can be solved efficiently. This is discussed further at Complexity classes P and NP , and P versus NP problem is one of the seven Millennium Prize Problems stated by

5025-495: The same number of a 's as b 's is context-free but not regular. To prove that a language is not regular, one often uses the Myhill–Nerode theorem and the pumping lemma . Other approaches include using the closure properties of regular languages or quantifying Kolmogorov complexity . Important subclasses of regular languages include Let s L ( n ) {\displaystyle s_{L}(n)} denote

5100-399: The set of regular languages . A deterministic finite automaton M is a 5- tuple , ( Q , Σ, δ , q 0 , F ) , consisting of Let w = a 1 a 2 ... a n be a string over the alphabet Σ . The automaton M accepts the string w if a sequence of states, r 0 , r 1 , ..., r n , exists in Q with the following conditions: In words, the first condition says that

5175-545: The size of input data increases. Therefore, Heule and Verwer's initial algorithm has later been augmented with making several steps of the EDSM algorithm prior to SAT solver execution: the DFASAT algorithm. This allows reducing the search space of the problem, but leads to loss of the minimality guarantee. Another way of reducing the search space has been proposed by Ulyantsev et al. by means of new symmetry breaking predicates based on

5250-488: The string. Deterministic refers to the uniqueness of the computation run. In search of the simplest models to capture finite-state machines, Warren McCulloch and Walter Pitts were among the first researchers to introduce a concept similar to finite automata in 1943. The figure illustrates a deterministic finite automaton using a state diagram . In this example automaton, there are three states: S 0 , S 1 , and S 2 (denoted graphically by circles). The automaton takes

5325-518: The term "regular" originates from a 1951 technical report where Kleene introduced "regular events" and explicitly welcomed "any suggestions as to a more descriptive term" . Noam Chomsky , in his 1959 seminal article, used the term "regular" in a different meaning at first (referring to what is called " Chomsky normal form " today), but noticed that his "finite state languages" were equivalent to Kleene's "regular events" . The regular languages are closed under various operations, that is, if

5400-443: The time or space required to solve the problem as a function of the size of the input problem. For example, finding a particular number in a long list of numbers becomes harder as the list of numbers grows larger. If we say there are n numbers in the list, then if the list is not sorted or indexed in any way we may have to look at every number in order to find the number we're seeking. We thus say that in order to solve this problem,

5475-458: The tree vertices with C {\displaystyle C} states in such a way that when vertices with one color are merged to one state, the generated automaton is deterministic and complies with S + {\displaystyle S^{+}} and S − {\displaystyle S^{-}} . Though this approach allows finding the minimal DFA, it suffers from exponential blow-up of execution time when

5550-461: The universality problem is NP-complete already for a singleton alphabet. For larger alphabets, that problem is PSPACE-complete . If regular expressions are extended to allow also a squaring operator , with " A " denoting the same as " AA ", still just regular languages can be described, but the universality problem has an exponential space lower bound, and is in fact complete for exponential space with respect to polynomial-time reduction. For

5625-621: The words of a given length in L ′ {\displaystyle L'} . Consider, for example, the Dyck language of strings of balanced parentheses. The number of words of length 2 n {\displaystyle 2n} in the Dyck language is equal to the Catalan number C n ∼ 4 n n 3 / 2 π {\displaystyle C_{n}\sim {\frac {4^{n}}{n^{3/2}{\sqrt {\pi }}}}} , which

#866133