This chapter describes the representations used in this package for finite automata and some functions to determine information about them.
We have to remark that the states of an automaton are always named \(1,2,3,\ldots;\) the alphabet may be given by the user. By default it is \(\{a,b,c,\ldots\}\) (or \(\{a_1,a_2,a_3,\ldots\}\) in the case of alphabets with more than \(26\) letters).
The transition function of an automaton with \(q\) states over an alphabet with \( n\) letters is represented by a (not necessarily dense) \(n\times q\) matrix. Each row of the matrix describes the action of the corresponding letter on the states. In the case of a deterministic automaton (DFA) the entries of the matrix are non-negative integers. When all entries of the transition table are positive integers, the automaton is said to be dense or complete. In the case of a non deterministic automaton (NFA) the entries of the matrix may be lists of non-negative integers. Automata with \(\epsilon\)-transitions are also allowed: the last letter of the alphabet is assumed to be \(\epsilon\) and is represented by @.
The way to create an automaton in GAP is the following
‣ Automaton( Type, Size, Alphabet, TransitionTable, Initial, Accepting ) | ( function ) |
Type may be "det", "nondet" or "epsilon" according to whether the automaton is deterministic, non deterministic or an automaton with \(\epsilon\)-transitions. Size is a positive integer representing the number of states of the automaton. Alphabet is the number of letters of the alphabet or a list with the letters of the ordered alphabet. TransitionTable is the transition matrix. The entries are non-negative integers not greater than the size of the automaton. In the case of non deterministic automata, lists of non-negative integers not greater than the size of the automaton are also allowed. Initial and Accepting are, respectively, the lists of initial and accepting states.
gap> aut:=Automaton("det",4,2,[[3,,3,4],[3,4,0,4]],[1],[4]); < deterministic automaton on 2 letters with 4 states > gap> Display(aut); | 1 2 3 4 ----------------- a | 3 3 4 b | 3 4 4 Initial state: [ 1 ] Accepting state: [ 4 ]
The alphabet of the automaton may be specified:
gap> aut:=Automaton("det",4,"01",[[3,,3,4],[3,4,0,4]],[1],[4]); < deterministic automaton on 2 letters with 4 states > gap> Display(aut); | 1 2 3 4 ----------------- 0 | 3 3 4 1 | 3 4 4 Initial state: [ 1 ] Accepting state: [ 4 ]
Instead of leaving a hole in the transition matrix, we may write a 0 to mean that no transition is present. Non deterministic automata may be given the same way.
gap> ndaut:=Automaton("nondet",4,2,[[3,[1,2],3,0],[3,4,0,[2,3]]],[1],[4]); < non deterministic automaton on 2 letters with 4 states > gap> Display(ndaut); | 1 2 3 4 ----------------------------------------- a | [ 3 ] [ 1, 2 ] [ 3 ] b | [ 3 ] [ 4 ] [ 2, 3 ] Initial state: [ 1 ] Accepting state: [ 4 ]
Also in the same way can be given \(\epsilon\)-automata. The letter \(\epsilon\) is written @ instead.
gap> x:=Automaton("epsilon",3,"01@",[[,[2],[3]],[[1,3],,[1]],[[1],[2], > [2]]],[2],[2,3]); < epsilon automaton on 3 letters with 3 states > gap> Display(x); | 1 2 3 ------------------------------ 0 | [ 2 ] [ 3 ] 1 | [ 1, 3 ] [ 1 ] @ | [ 1 ] [ 2 ] [ 2 ] Initial state: [ 2 ] Accepting states: [ 2, 3 ]
Bigger automata are displayed in another form:
gap> aut:=Automaton("det",16,2,[[4,0,0,6,3,1,4,8,7,4,3,0,6,1,6,0], > [3,4,0,0,6,1,0,6,1,6,1,6,6,4,8,7,4,5]],[1],[4]); < deterministic automaton on 2 letters with 16 states > gap> Display(aut); 1 a 4 1 b 3 2 b 4 ... some more lines 15 a 6 15 b 8 16 b 7 Initial state: [ 1 ] Accepting state: [ 4 ]
‣ IsAutomaton( O ) | ( function ) |
In the presence of an object O, one may want to test whether O is an automaton. This may be done using the function IsAutomaton.
gap> x:=Automaton("det",3,2,[ [ 0, 2, 0 ], [ 0, 1, 0 ] ],[ 3 ],[ 2 ]);; gap> IsAutomaton(x); true
‣ IsDeterministicAutomaton( aut ) | ( function ) |
Returns true when aut is a deterministic automaton and false otherwise.
gap> x:=Automaton("det",3,2,[ [ 0, 2, 0 ], [ 0, 1, 0 ] ],[ 3 ],[ 2 ]);; gap> IsDeterministicAutomaton(x); true
‣ IsNonDeterministicAutomaton( aut ) | ( function ) |
Returns true when aut is a non deterministic automaton and false otherwise.
gap> y:=Automaton("nondet",3,2,[[,[1,3],],[,[2,3],[1,3]]],[1,2],[1,3]);; gap> IsNonDeterministicAutomaton(y); true
‣ IsEpsilonAutomaton( aut ) | ( function ) |
Returns true when aut is an \(\epsilon\)-automaton and false otherwise.
gap> z:=Automaton("epsilon",2,2,[[[1,2],],[[2],[1]]],[1,2],[1,2]);; gap> IsEpsilonAutomaton(z); true
‣ String( aut ) | ( function ) |
The way GAP displays an automaton is quite natural, but when one wants to do small changes, for example using copy/paste, the use of the function String (possibly followed by Print) may be useful.
gap> x:=Automaton("det",3,2,[ [ 0, 2, 0 ], [ 0, 1, 0 ] ],[ 3 ],[ 2 ]);; gap> String(x); "Automaton(\"det\",3,\"ab\",[ [ 0, 2, 0 ], [ 0, 1, 0 ] ],[ 3 ],[ 2 ]);;" gap> Print(String(x)); Automaton("det",3,"ab",[ [ 0, 2, 0 ], [ 0, 1, 0 ] ],[ 3 ],[ 2 ]);;
gap> z:=Automaton("epsilon",2,2,[[[1,2],],[[2],[1]]],[1,2],[1,2]);; gap> Print(String(z)); Automaton("epsilon",2,"a@",[ [ [ 1, 2 ], [ ] ], [ [ 2 ], [ 1 ] ] ],[ 1, 2 ],[ \ 1, 2 ]);;
‣ RandomAutomaton( Type, Size, Alphabet ) | ( function ) |
Given the Type, the Size (i.e. the number of states) and the Alphabet (a positive integer or a list), returns a pseudo random automaton with these parameters.
gap> RandomAutomaton("det",5,"ac"); < deterministic automaton on 2 letters with 5 states > gap> Display(last); | 1 2 3 4 5 -------------------- a | 2 3 c | 2 3 Initial state: [ 4 ] Accepting states: [ 3, 4 ] gap> RandomAutomaton("nondet",3,["a","b","c"]); < non deterministic automaton on 3 letters with 3 states > gap> RandomAutomaton("epsilon",2,"abc"); < epsilon automaton on 4 letters with 2 states > gap> RandomAutomaton("epsilon",2,2); < epsilon automaton on 3 letters with 2 states > gap> Display(last); | 1 2 ---------------------- a | [ 1, 2 ] b | [ 2 ] [ 1 ] @ | [ 1, 2 ] Initial state: [ 2 ] Accepting states: [ 1, 2 ] gap> a:=RandomTransformation(3);; gap> b:=RandomTransformation(3);; gap> aut:=RandomAutomaton("det",4,[a,b]); < deterministic automaton on 2 letters with 4 states >
In this section we describe the functions used to access the internals of an automaton.
‣ AlphabetOfAutomaton( aut ) | ( function ) |
Returns the number of symbols in the alphabet of automaton aut.
gap> aut:=Automaton("det",4,2,[[3,,3,4],[3,4,0,]],[1],[4]);; gap> AlphabetOfAutomaton(aut); 2
‣ AlphabetOfAutomatonAsList( aut ) | ( function ) |
Returns the alphabet of automaton aut always as a list. Note that when the alphabet of the automaton is given as an integer (meaning the number of symbols) not greater than 26 it returns the list "abcd....". If the alphabet is given by means of an integer greater than 26, the function returns [ "a1", "a2", "a3", "a4", ... ].
gap> a:=RandomAutomaton("det",5,"cat"); < deterministic automaton on 3 letters with 5 states > gap> AlphabetOfAutomaton(a); 3 gap> AlphabetOfAutomatonAsList(a); "cat" gap> a:=RandomAutomaton("det",5,20); < deterministic automaton on 20 letters with 5 states > gap> AlphabetOfAutomaton(a); 20 gap> AlphabetOfAutomatonAsList(a); "abcdefghijklmnopqrst" gap> a:=RandomAutomaton("det",5,30); < deterministic automaton on 30 letters with 5 states > gap> AlphabetOfAutomaton(a); 30 gap> AlphabetOfAutomatonAsList(a); [ "a1", "a2", "a3", "a4", "a5", "a6", "a7", "a8", "a9", "a10", "a11", "a12", "a13", "a14", "a15", "a16", "a17", "a18", "a19", "a20", "a21", "a22", "a23", "a24", "a25", "a26", "a27", "a28", "a29", "a30" ]
‣ TransitionMatrixOfAutomaton( aut ) | ( function ) |
Returns the transition matrix of automaton aut.
gap> aut:=Automaton("det",4,2,[[3,,3,4],[3,4,0,]],[1],[4]);; gap> TransitionMatrixOfAutomaton(aut); [ [ 3, 0, 3, 4 ], [ 3, 4, 0, 0 ] ]
‣ InitialStatesOfAutomaton( aut ) | ( function ) |
Returns the initial states of automaton aut.
gap> aut:=Automaton("det",4,2,[[3,,3,4],[3,4,0,]],[1],[4]);; gap> InitialStatesOfAutomaton(aut); [ 1 ]
‣ SetInitialStatesOfAutomaton( aut, I ) | ( function ) |
Sets the initial states of automaton aut. I may be a positive integer or a list of positive integers.
gap> aut:=Automaton("det",4,2,[[3,,3,4],[3,4,0,]],[1],[4]);; gap> SetInitialStatesOfAutomaton(aut,4); gap> InitialStatesOfAutomaton(aut); [ 4 ] gap> SetInitialStatesOfAutomaton(aut,[2,3]); gap> InitialStatesOfAutomaton(aut); [ 2, 3 ]
‣ FinalStatesOfAutomaton( aut ) | ( function ) |
Returns the final states of automaton aut.
gap> aut:=Automaton("det",4,2,[[3,,3,4],[3,4,0,]],[1],[4]);; gap> FinalStatesOfAutomaton(aut); [ 4 ]
‣ SetFinalStatesOfAutomaton( aut, F ) | ( function ) |
Sets the final states of automaton aut. F may be a positive integer or a list of positive integers.
gap> aut:=Automaton("det",4,2,[[3,,3,4],[3,4,0,]],[1],[4]);; gap> FinalStatesOfAutomaton(aut); [ 4 ] gap> SetFinalStatesOfAutomaton(aut,2); gap> FinalStatesOfAutomaton(aut); [ 2 ]
‣ NumberStatesOfAutomaton( aut ) | ( function ) |
Returns the number of states of automaton aut.
gap> aut:=Automaton("det",4,2,[[3,,3,4],[3,4,0,]],[1],[4]);; gap> NumberStatesOfAutomaton(aut); 4
Although there is no standard way to compare automata it is useful to be able to do some kind of comparison. Doing so, one can consider sets of automata. We just compare the strings of the automata.
gap> x:=Automaton("det",3,2,[ [ 0, 2, 0 ], [ 0, 1, 0 ] ],[ 3 ],[ 2 ]);; gap> y:=Automaton("det",3,2,[ [ 2, 0, 0 ], [ 1, 3, 0 ] ],[ 3 ],[ 2, 3 ]);; gap> x=y; false gap> Size(Set([y,x,x])); 2
This section describes some useful tests involving automata.
‣ IsDenseAutomaton( aut ) | ( function ) |
Tests whether a deterministic automaton aut is complete. (See also NullCompletionAutomaton (2.5-2).)
gap> aut:=Automaton("det",4,2,[[3,,3,4],[3,4,0,]],[1],[4]);; gap> IsDenseAutomaton(aut); false
‣ IsRecognizedByAutomaton( A, w ) | ( function ) |
The arguments are: an automaton A and a string (i.e. a word) w in the alphabet of the automaton. Returns true if the word is recognized by the automaton and false otherwise.
gap> aut:=Automaton("det",3,2,[[1,2,1],[2,1,3]],[1],[2]);; gap> IsRecognizedByAutomaton(aut,"bbb"); true gap> aut:=Automaton("det",3,"01",[[1,2,1],[2,1,3]],[1],[2]);; gap> IsRecognizedByAutomaton(aut,"111"); true
‣ IsPermutationAutomaton( aut ) | ( function ) |
The argument is a deterministic automaton. Returns true when each letter of the alphabet induces a permutation on the vertices and false otherwise.
gap> x:=Automaton("det",3,2,[ [ 1, 2, 3 ], [ 1, 2, 3 ] ],[ 1 ],[ 2, 3 ]);; gap> IsPermutationAutomaton(x); true
‣ IsInverseAutomaton( aut ) | ( function ) |
The argument is a deterministic automaton. Returns true when each letter of the alphabet induces an injective partial function on the vertices and false otherwise.
gap> x:=Automaton("det",3,2,[ [ 0, 1, 3 ], [ 0, 1, 2 ] ],[ 2 ],[ 1 ]);; gap> IsInverseAutomaton(x); true
Frequently an inverse automaton is thought as if the inverse edges (labeled by formal inverses of the letters of the alphabet) were present, although they are usually not explicited. They can be made explicit using the function AddInverseEdgesToInverseAutomaton
‣ AddInverseEdgesToInverseAutomaton( aut ) | ( function ) |
The argument is an inverse automaton over the alphabet \(\{a,b,c,\ldots\}\). Returns an automaton with the inverse edges added. (The formal inverse of a letter is represented by the corresponding capital letter.)
gap> x:=Automaton("det",3,2,[[ 0, 1, 3 ],[ 0, 1, 2 ]],[ 2 ],[ 1 ]);;Display(x); | 1 2 3 -------------- a | 1 3 b | 1 2 Initial state: [ 2 ] Accepting state: [ 1 ] gap> AddInverseEdgesToInverseAutomaton(x);Display(x); | 1 2 3 -------------- a | 1 3 b | 1 2 A | 2 3 B | 2 3 Initial state: [ 2 ] Accepting state: [ 1 ]
‣ IsReversibleAutomaton( aut ) | ( function ) |
The argument is a deterministic automaton. Returns true when aut is a reversible automaton, i.e. the automaton obtained by reversing all edges and switching the initial and final states (see also ReversedAutomaton (2.5-5)) is deterministic. Returns false otherwise.
gap> x:=Automaton("det",3,2,[ [ 0, 1, 2 ], [ 0, 1, 3 ] ],[ 2 ],[ 2 ]);; gap> IsReversibleAutomaton(x); true
‣ CopyAutomaton( aut ) | ( function ) |
Returns a new automaton, which is a copy of automaton aut.
‣ NullCompletionAutomaton( aut ) | ( function ) |
aut is a deterministic automaton. If it is complete returns aut, otherwise returns the completion (with a null state) of aut. Notice that the words recognized by aut and its completion are the same.
gap> aut:=Automaton("det",4,2,[[3,,3,4],[2,4,4,]],[1],[4]);; gap> IsDenseAutomaton(aut); false gap> y:=NullCompletionAutomaton(aut);;Display(y); | 1 2 3 4 5 -------------------- a | 3 5 3 4 5 b | 2 4 4 5 5 Initial state: [ 1 ] Accepting state: [ 4 ]
The state added is a sink state i.e. it is a state \(q\) which is not initial nor accepting and for all letter \(a\) in the alphabet of the automaton, \(q\) is the result of the action of \(a\) in \(q\). (Notice that reading a word, one does not go out of a sink state.)
‣ ListSinkStatesAut( aut ) | ( function ) |
Computes the list of all sink states of the automaton aut.
gap> x:=Automaton("det",3,2,[ [ 2, 3, 3 ], [ 1, 2, 3 ] ],[ 1 ],[ 2, 3 ]);; gap> ListSinkStatesAut(x); [ ] gap> y:=Automaton("det",3,2,[ [ 2, 3, 3 ], [ 1, 2, 3 ] ],[ 1 ],[ 2 ]);; gap> ListSinkStatesAut(y); [ 3 ]
‣ RemovedSinkStates( aut ) | ( function ) |
Removes all sink states of the automaton aut.
gap> y:=Automaton("det",3,2,[[ 2, 3, 3 ],[ 1, 2, 3 ]],[ 1 ],[ 2 ]);;Display(y); | 1 2 3 -------------- a | 2 3 3 b | 1 2 3 Initial state: [ 1 ] Accepting state: [ 2 ] gap> x := RemovedSinkStates(y);Display(x); < deterministic automaton on 2 letters with 2 states > | 1 2 ----------- a | 2 b | 1 2 Initial state: [ 1 ] Accepting state: [ 2 ]
‣ ReversedAutomaton( aut ) | ( function ) |
Inverts the arrows of the automaton aut.
gap> y:=Automaton("det",3,2,[ [ 2, 3, 3 ], [ 1, 2, 3 ] ],[ 1 ],[ 2 ]);; gap> z:=ReversedAutomaton(y);;Display(z); | 1 2 3 ------------------------------ a | [ 1 ] [ 2, 3 ] b | [ 1 ] [ 2 ] [ 3 ] Initial state: [ 2 ] Accepting state: [ 1 ]
‣ PermutedAutomaton( aut, p ) | ( function ) |
Given an automaton aut and a list p representing a permutation of the states, outputs the equivalent permuted automaton.
gap> y:=Automaton("det",4,2,[[2,3,4,2],[0,0,0,1]],[1],[3]);;Display(y); | 1 2 3 4 ----------------- a | 2 3 4 2 b | 1 Initial state: [ 1 ] Accepting state: [ 3 ] gap> Display(PermutedAutomaton(y, [3,2,4,1])); | 1 2 3 4 ----------------- a | 2 4 2 1 b | 3 Initial state: [ 3 ] Accepting state: [ 4 ]
‣ ListPermutedAutomata( aut ) | ( function ) |
Given an automaton aut, returns a list of automata with permuted states
gap> x:=Automaton("det",3,2,[ [ 0, 2, 3 ], [ 1, 2, 3 ] ],[ 1 ],[ 2, 3 ]);; gap> ListPermutedAutomata(x); [ < deterministic automaton on 2 letters with 3 states >, < deterministic automaton on 2 letters with 3 states >, < deterministic automaton on 2 letters with 3 states >, < deterministic automaton on 2 letters with 3 states >, < deterministic automaton on 2 letters with 3 states >, < deterministic automaton on 2 letters with 3 states > ]
‣ NormalizedAutomaton( A ) | ( function ) |
Produces an equivalent automaton but in which the initial state is numbered 1 and the accepting states have the greatest numbers.
gap> x:=Automaton("det",3,2,[[ 1, 2, 0 ],[ 0, 1, 2 ]],[2],[1, 2]);;Display(x); | 1 2 3 -------------- a | 1 2 b | 1 2 Initial state: [ 2 ] Accepting states: [ 1, 2 ] gap> Display(NormalizedAutomaton(x)); | 1 2 3 -------------- a | 1 3 b | 3 1 Initial state: [ 1 ] Accepting states: [ 3, 1 ]
‣ UnionAutomata( A, B ) | ( function ) |
Produces the disjoint union of the deterministic or non deterministic automata A and B. The output is a non-deterministic automaton.
gap> x:=Automaton("det",3,2,[ [ 1, 2, 0 ], [ 0, 1, 2 ] ],[ 2 ],[ 1, 2 ]);; gap> y:=Automaton("det",3,2,[ [ 0, 1, 3 ], [ 0, 0, 0 ] ],[ 1 ],[ 1, 2, 3 ]);; gap> UnionAutomata(x,y); < non deterministic automaton on 2 letters with 6 states > gap> Display(last); | 1 2 3 4 5 6 ------------------------------------------------ a | [ 1 ] [ 2 ] [ 4 ] [ 6 ] b | [ 1 ] [ 2 ] Initial states: [ 2, 4 ] Accepting states: [ 1, 2, 4, 5, 6 ]
‣ ProductAutomaton( A1, A2 ) | ( function ) |
The arguments must be deterministic automata. Returns the product of A1 and A2.
Note: \((p,q)->(p-1)m+q\) is a bijection from \(\{1,\ldots, n\}\times \{1,\ldots, m\}\) to \(\{1,\ldots,mn\}\).
gap> x := Automaton("det",3,"ab",[ [ 0, 1, 2 ], [ 1, 3, 0 ] ],[ 1 ],[ 1, 2 ]);; gap> Display(x); | 1 2 3 -------------- a | 1 2 b | 1 3 Initial state: [ 1 ] Accepting states: [ 1, 2 ] gap> y := Automaton("det",3,"ab",[ [ 0, 1, 2 ], [ 0, 2, 0 ] ],[ 2 ],[ 1, 2 ]);; gap> Display(y); | 1 2 3 -------------- a | 1 2 b | 2 Initial state: [ 2 ] Accepting states: [ 1, 2 ] gap> z:=ProductAutomaton(x, y);;Display(z); | 1 2 3 4 5 6 7 8 9 -------------------------------- a | 1 2 4 5 b | 2 8 Initial state: [ 2 ] Accepting states: [ 1, 2, 4, 5 ]
‣ ProductOfLanguages( A1, A2 ) | ( function ) |
Given two regular languages (as automata or rational expressions), returns an automaton that recognizes the concatenation of the given languages, that is, the set of words \(uv\) such that \(u\) belongs to the first language and \(v\) belongs to the second language.
gap> a1:=ListOfWordsToAutomaton("ab",["aa","bb"]); < deterministic automaton on 2 letters with 5 states > gap> a2:=ListOfWordsToAutomaton("ab",["a","b"]); < deterministic automaton on 2 letters with 3 states > gap> ProductOfLanguages(a1,a2); < deterministic automaton on 2 letters with 5 states > gap> FAtoRatExp(last); (bbUaa)(aUb)
Each letter of the alphabet of an automaton induces a partial transformation in its set of states. The semigroup generated by these transformations is called the transition semigroup of the automaton.
‣ TransitionSemigroup( aut ) | ( function ) |
Returns the transition semigroup of the deterministic automaton aut.
gap> aut := Automaton("det",10,2,[[7,5,7,5,4,9,10,9,10,9], > [8,6,8,9,9,1,3,1,9,9]],[2],[6,7,8,9,10]);; gap> s := TransitionSemigroup(aut);; gap> Size(s); 30
The transition semigroup of the minimal automaton recognizing a language is the syntactic semigroup of that language.
‣ SyntacticSemigroupAut( aut ) | ( function ) |
Returns the syntactic semigroup of the deterministic automaton aut (i.e. the transition semigroup of the equivalent minimal automaton) when it is non empty and returns fail otherwise.
gap> x:=Automaton("det",3,2,[ [ 1, 2, 0 ], [ 0, 1, 2 ] ],[ 2 ],[ 1, 2 ]);; gap> S:=SyntacticSemigroupAut(x);; gap> Size(S); 3
‣ SyntacticSemigroupLang( rat ) | ( function ) |
Returns the syntactic semigroup of the language given by the rational expression rat.
gap> rat := RationalExpression("a*ba*ba*(@Ub)");; gap> S:=SyntacticSemigroupLang(rat);; gap> Size(S); 7
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