MealyAutomaton(
table[,
names[,
alphabet]] ) O
MealyAutomaton(
string ) O
MealyAutomaton(
autom ) O
Creates the Mealy automaton (see Short math background) defined by the argument table, string
or autom. Format of the argument table is
the following: it is a list of states, where each state is a list of
positive integers which represent transition function at the given state and a
permutation or transformation which represent the output function at this
state. Format of the string string is the same as in AutomatonGroup
(see AutomatonGroup).
The third form of this operation takes a tree homomorphism autom as its argument.
It returns noninitial automaton constructed from the sections of autom, whose first state
corresponds to autom itself.
gap> A := MealyAutomaton([[1,2,(1,2)],[3,1,()],[3,3,(1,2)]], ["a","b","c"]); <automaton> gap> Print(A, "\n"); a = (a, b)(1,2), b = (c, a), c = (c, c)(1,2) gap> B:=MealyAutomaton([[1,2,Transformation([1,1])],[3,1,()],[3,3,(1,2)]],["a","b","c"]); <automaton> gap> Print(B, "\n"); a = (a, b)[ 1, 1 ], b = (c, a), c = (c, c)[ 2, 1 ] gap> D := MealyAutomaton("a=(a,b)(1,2), b=(b,a)"); <automaton> gap> Basilica := AutomatonGroup( "u=(v,1)(1,2), v=(u,1)" ); < u, v > gap> M := MealyAutomaton(u*v*u^-3); <automaton> gap> Print(M); a1 = (a2, a5), a2 = (a3, a4), a3 = (a4, a2)(1,2), a4 = (a4, a4), a5 = (a6, a3) (1,2), a6 = (a7, a4), a7 = (a6, a4)(1,2)
IsMealyAutomaton(
A ) C
A category of non-initial finite Mealy automata with the same input and output alphabet.
NumberOfStates(
A ) A
Returns the number of states of the automaton A.
SizeOfAlphabet(
A ) A
Returns the number of letters in the alphabet the automaton A acts on.
AutomatonList(
A ) A
Returns the list of A acceptible by MealyAutomaton
(see MealyAutomaton)
IsTrivial(
A ) O
Computes whether the automaton A is equivalent to the trivial automaton.
gap> A := MealyAutomaton("a=(c,c), b=(a,b), c=(b,a)"); <automaton> gap> IsTrivial(A); true
IsInvertible(
A ) P
Is true
if A is invertible and false
otherwise.
MinimizationOfAutomaton(
A ) F
Returns the automaton obtained from automaton A by minimization.
gap> B := MealyAutomaton("a=(1,a)(1,2), b=(1,a)(1,2), c=(a,b), d=(a,b)"); <automaton> gap> C := MinimizationOfAutomaton(B); <automaton> gap> Print(C); a = (1, a)(1,2), c = (a, a), 1 = (1, 1)
MinimizationOfAutomatonTrack(
A ) F
Returns the list [A_new, new_via_old, old_via_new]
, where A_new
is an
automaton obtained from automaton A by minimization,
new_via_old
describes how new states are expressed in terms of the old ones, and
old_via_new
describes how old states are expressed in terms of the new ones.
gap> B := MealyAutomaton("a=(1,a)(1,2), b=(1,a)(1,2), c=(a,b), d=(a,b)"); <automaton> gap> B_min := MinimizationOfAutomatonTrack(B); [ <automaton>, [ 1, 3, 5 ], [ 1, 1, 2, 2, 3 ] ] gap> Print(B_min[1]); a = (1, a)(1,2), c = (a, a), 1 = (1, 1)
IsOfPolynomialGrowth(
A ) P
Determines whether the automaton A has polynomial growth in terms of Sidki Sid00.
See also IsBounded
(IsBounded) and
PolynomialDegreeOfGrowth
(PolynomialDegreeOfGrowth).
gap> B := MealyAutomaton("a=(b,1)(1,2), b=(a,1)"); <automaton> gap> IsOfPolynomialGrowth(B); true gap> D := MealyAutomaton("a=(a,b)(1,2), b=(b,a)"); <automaton> gap> IsOfPolynomialGrowth(D); false
IsBounded(
A ) P
Determines whether the automaton A is bounded in terms of Sidki Sid00.
See also IsOfPolynomialGrowth
(IsOfPolynomialGrowth)
and PolynomialDegreeOfGrowth
(PolynomialDegreeOfGrowth).
gap> B := MealyAutomaton("a=(b,1)(1,2), b=(a,1)"); <automaton> gap> IsBounded(B); true gap> C := MealyAutomaton("a=(a,b)(1,2), b=(b,c), c=(c,1)(1,2)"); <automaton> gap> IsBounded(C); false
PolynomialDegreeOfGrowth(
A ) A
For an automaton A of polynomial growth in terms of Sidki Sid00
determines its degree of
polynomial growth. This degree is 0 if and only if automaton is bounded.
If the growth of automaton is exponential returns fail
.
See also IsOfPolynomialGrowth
(IsOfPolynomialGrowth)
and IsBounded
(IsBounded).
gap> B := MealyAutomaton("a=(b,1)(1,2), b=(a,1)"); <automaton> gap> PolynomialDegreeOfGrowth(B); 0 gap> C := MealyAutomaton("a=(a,b)(1,2), b=(b,c), c=(c,1)(1,2)"); <automaton> gap> PolynomialDegreeOfGrowth(C); 2
DualAutomaton(
A ) O
Returns the automaton dual of A.
gap> A := MealyAutomaton("a=(b,a)(1,2), b=(b,a)"); <automaton> gap> D := DualAutomaton(A); <automaton> gap> Print(D); d1 = (d2, d1)[ 2, 2 ], d2 = (d1, d2)[ 1, 1 ]
InverseAutomaton(
A ) O
Returns the automaton inverse to A if A is invertible.
gap> A := MealyAutomaton("a=(b,a)(1,2), b=(b,a)"); <automaton> gap> B := InverseAutomaton(A); <automaton> gap> Print(B); a1 = (a1, a2)(1,2), a2 = (a2, a1)
IsBireversible(
A ) O
Computes whether or not the automaton A is bireversible, i.e. A, the dual of A and the dual of the inverse of A are invertible. The example below shows that the Bellaterra automaton is bireversible.
gap> Bellaterra := MealyAutomaton("a=(c,c)(1,2), b=(a,b), c=(b,a)"); <automaton> gap> IsBireversible(Bellaterra); true
DisjointUnion(
A,
B ) O
Constructs the disjoint union of automata A and B
gap> A := MealyAutomaton("a=(a,b)(1,2), b=(a,b)"); <automaton> gap> B := MealyAutomaton("c=(d,c), d=(c,e)(1,2), e=(e,d)"); <automaton> gap> Print(DisjointUnion(A, B)); a1 = (a1, a2)(1,2), a2 = (a1, a2), a3 = (a4, a3), a4 = (a3, a5) (1,2), a5 = (a5, a4)
A *
B
Constructs the product of 2 noninitial automata A and B.
gap> A := MealyAutomaton("a=(a,b)(1,2), b=(a,b)"); <automaton> gap> B := MealyAutomaton("c=(d,c), d=(c,e)(1,2), e=(e,d)"); <automaton> gap> Print(A*B); a1 = (a1, a5)(1,2), a2 = (a3, a4), a3 = (a2, a6) (1,2), a4 = (a2, a4), a5 = (a1, a6)(1,2), a6 = (a3, a5)
SubautomatonWithStates(
A,
states ) O
Returns the minimal subautomaton of the automaton A containing states states.
gap> A := MealyAutomaton("a=(e,d)(1,2),b=(c,c),c=(b,c)(1,2),d=(a,e)(1,2),e=(e,d)"); <automaton> gap> Print(SubautomatonWithStates(A, [1, 4])); a = (e, d)(1,2), d = (a, e)(1,2), e = (e, d)
AutomatonNucleus(
A ) O
Returns the nucleus of the automaton A, i.e. the minimal subautomaton containing all cycles in A.
gap> A := MealyAutomaton("a=(b,c)(1,2),b=(d,d),c=(d,b)(1,2),d=(d,b)(1,2),e=(a,d)"); <automaton> gap> Print(AutomatonNucleus(A)); b = (d, d), d = (d, b)(1,2)
AreEquivalentAutomata(
A,
B ) O
Returns true
if for every state s
of the automaton A there is a state of the automaton B
equivalent to s
and vice versa.
gap> A := MealyAutomaton("a=(b,a)(1,2), b=(a,c), c=(b,c)(1,2)"); <automaton> gap> B := MealyAutomaton("b=(a,c), c=(b,c)(1,2), a=(b,a)(1,2), d=(b,c)(1,2)"); <automaton> gap> AreEquivalentAutomata(A, B); true
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