MealyAutomaton(
table[,
names[,
alphabet]] ) O
MealyAutomaton(
string ) O
MealyAutomaton(
autom ) O
MealyAutomaton(
tree_hom_list ) 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. The fourth form creates a noninitial automaton constructed
of the states of all tree homomorphisms from the tree_hom_list.
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 ) P
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
AdjacencyMatrix(
A ) A
Returns the adjacency matrix of a Mealy automaton A, in which the ij-th entry contains the number of arrows in the Moore diagram of A from state i to state j.
gap> A:=MealyAutomaton("a=(a,a,b)(1,2,3),b=(a,c,b)(1,2),c=(a,a,a)"); <automaton> gap> AdjacencyMatrix(A); [ [ 2, 1, 0 ], [ 1, 1, 1 ], [ 3, 0, 0 ] ]
IsAcyclic(
A ) P
Computes whether or not an automaton A is acyclic in the sense of Sidki Sid00.
I.e. returns true
if the Moore diagram of A does not contain cycles with two or more
states and false
otherwise.
gap> A:=MealyAutomaton("a=(a,a,b)(1,2,3),b=(c,c,b)(1,2),c=(d,c,1),d=(d,1,d)"); <automaton> gap> IsAcyclic(A); true gap> A:=MealyAutomaton("a=(a,a,b)(1,2,3),b=(c,c,d)(1,2),c=(d,c,1),d=(b,1,d)"); <automaton> gap> IsAcyclic(A); false
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 ) P
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
IsReversible(
A ) P
Computes whether or not the automaton A is reversible, i.e. the dual of A is invertible.
IsIRAutomaton(
A ) P
Computes whether or not the automaton A is an IR-automaton, i.e. A and its dual are invertible. The example below shows that the automaton generating lamplighter group is an IR-automaton.
gap> L := MealyAutomaton("a=(b,a)(1,2), b=(a,b)"); <automaton> gap> IsIRAutomaton(L); true
The next three commands deal with a procedure called MD-reduction developed in AKL. Particularly, according to KLI, a 2-letter or 2-state invertible reversible automaton generates a finite group if and only if the MD-reduction procedure terminates with a trivial automaton. In the last case the automaton is called MD-trivial.
MDReduction(
A ) O
Performs the process of MD-reduction of automaton A (alternating applications of minimization and dualization procedures) until a pair of minimal automata dual to each other is reached. Returns this pair. The main point of this procedure is in the fact that the (semi)group generated by the original automaton is finite if and only each of the (semi)groups generated by the output automata is finite.
gap> A:=MealyAutomaton("a=(d,d,d,d)(1,2)(3,4),b=(b,b,b,b)(1,4)(2,3),\\ > c=(a,c,a,c), d=(c,a,c,a)"); <automaton> gap> NumberOfStates(MinimizationOfAutomaton(A)); 4 gap> MDR:=MDReduction(A); [ <automaton>, <automaton> ] gap> Print(MDR[1]); d1 = (d2, d2, d1, d1)(1,4,3), d2 = (d1, d1, d2, d2)(1,4) gap> Print(MDR[2]); d1 = (d4, d4)(1,2), d2 = (d2, d2)(1,2), d3 = (d1, d3), d4 = (d3, d1)
IsMDTrivial(
A ) P
Returns true
if A is MD-trivial (i.e. if MD-reduction proedure returns the
trivial automaton) and false
otherwise.
IsMDReduced(
A ) P
Returns true
if A is MD-reduced and false
otherwise.
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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