inverse limits of finite state automata - reh.math.uni...
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![Page 1: Inverse limits of finite state automata - reh.math.uni ...reh.math.uni-duesseldorf.de/~internet/trees2018/slides/Ferov.pdf · Inverse limits of finite state automata Michal Ferov](https://reader030.vdocument.in/reader030/viewer/2022041201/5d4a2a8b88c9931f708b8e99/html5/thumbnails/1.jpg)
Inverse limits of finite state automata
Michal Ferov
University of Technology, Sydney
Trees, dynamics and locally compact groupsDusseldorf, Germany
June 29, 2018
Michal Ferov Inverse limits of finite state automata
![Page 2: Inverse limits of finite state automata - reh.math.uni ...reh.math.uni-duesseldorf.de/~internet/trees2018/slides/Ferov.pdf · Inverse limits of finite state automata Michal Ferov](https://reader030.vdocument.in/reader030/viewer/2022041201/5d4a2a8b88c9931f708b8e99/html5/thumbnails/2.jpg)
Formal languages in discrete groups
When a finitely generated group is given by a presentationhXkRi we work with sequences of symbols (words) over thefinite alphabet X [ X
�1 (assuming X \ X
�1 = ;).Sets of words are called formal languages.
Example
Some languages are of general interest in group theory:
word problem:WP(G ) = {wkw =G 1},coword problem: coWP(G ) = {wkw 6=G 1},multiplication table:mult(G ) = {(u, v ,w)kuv =G w},geodesics: geo(G ) = {wk8w 0 : w =G w
0 ) |w | |w 0|}.
Michal Ferov Inverse limits of finite state automata
![Page 3: Inverse limits of finite state automata - reh.math.uni ...reh.math.uni-duesseldorf.de/~internet/trees2018/slides/Ferov.pdf · Inverse limits of finite state automata Michal Ferov](https://reader030.vdocument.in/reader030/viewer/2022041201/5d4a2a8b88c9931f708b8e99/html5/thumbnails/3.jpg)
Formal languages in discrete groups
When a finitely generated group is given by a presentationhXkRi we work with sequences of symbols (words) over thefinite alphabet X [ X
�1 (assuming X \ X
�1 = ;).Sets of words are called formal languages.
Example
Some languages are of general interest in group theory:
word problem:WP(G ) = {wkw =G 1},coword problem: coWP(G ) = {wkw 6=G 1},multiplication table:mult(G ) = {(u, v ,w)kuv =G w},geodesics: geo(G ) = {wk8w 0 : w =G w
0 ) |w | |w 0|}.
Michal Ferov Inverse limits of finite state automata
![Page 4: Inverse limits of finite state automata - reh.math.uni ...reh.math.uni-duesseldorf.de/~internet/trees2018/slides/Ferov.pdf · Inverse limits of finite state automata Michal Ferov](https://reader030.vdocument.in/reader030/viewer/2022041201/5d4a2a8b88c9931f708b8e99/html5/thumbnails/4.jpg)
Chomsky hierarchy of languages
A natural way to understand the complexity of a formal language isby quantifying the computational strength of a machine thatrecognises it.
We say that a machine M accepts language L if somecomputation ends up in a accepting state after reading a wordw 2 L.
Machine Memory Language
Finite state automaton N/A RegPush-down automaton Push-down stack CFLinear bounded automaton Linearly bounded tape CSTuring machine Infinite tape RE
Michal Ferov Inverse limits of finite state automata
![Page 5: Inverse limits of finite state automata - reh.math.uni ...reh.math.uni-duesseldorf.de/~internet/trees2018/slides/Ferov.pdf · Inverse limits of finite state automata Michal Ferov](https://reader030.vdocument.in/reader030/viewer/2022041201/5d4a2a8b88c9931f708b8e99/html5/thumbnails/5.jpg)
Chomsky hierarchy of languages
A natural way to understand the complexity of a formal language isby quantifying the computational strength of a machine thatrecognises it.
We say that a machine M accepts language L if somecomputation ends up in a accepting state after reading a wordw 2 L.
Machine Memory Language
Finite state automaton N/A RegPush-down automaton Push-down stack CFLinear bounded automaton Linearly bounded tape CSTuring machine Infinite tape RE
Michal Ferov Inverse limits of finite state automata
![Page 6: Inverse limits of finite state automata - reh.math.uni ...reh.math.uni-duesseldorf.de/~internet/trees2018/slides/Ferov.pdf · Inverse limits of finite state automata Michal Ferov](https://reader030.vdocument.in/reader030/viewer/2022041201/5d4a2a8b88c9931f708b8e99/html5/thumbnails/6.jpg)
Chomsky hierarchy of languages
A natural way to understand the complexity of a formal language isby quantifying the computational strength of a machine thatrecognises it.
We say that a machine M accepts language L if somecomputation ends up in a accepting state after reading a wordw 2 L.
Machine Memory Language
Finite state automaton N/A RegPush-down automaton Push-down stack CFLinear bounded automaton Linearly bounded tape CSTuring machine Infinite tape RE
Michal Ferov Inverse limits of finite state automata
![Page 7: Inverse limits of finite state automata - reh.math.uni ...reh.math.uni-duesseldorf.de/~internet/trees2018/slides/Ferov.pdf · Inverse limits of finite state automata Michal Ferov](https://reader030.vdocument.in/reader030/viewer/2022041201/5d4a2a8b88c9931f708b8e99/html5/thumbnails/7.jpg)
Groups and Chomsky hierarchy
Some languages in group theory have been classified withinChomsky hierarchy:
regular (co)word problem i↵ finite (Anisimov),
context-free word problem i↵ virtually free (Muller & Schupp),
context-free multiplication table i↵ hyperbolic (Gilman),
Question
What about totally disconnected locally compact groups?Is there a computational model?
Michal Ferov Inverse limits of finite state automata
![Page 8: Inverse limits of finite state automata - reh.math.uni ...reh.math.uni-duesseldorf.de/~internet/trees2018/slides/Ferov.pdf · Inverse limits of finite state automata Michal Ferov](https://reader030.vdocument.in/reader030/viewer/2022041201/5d4a2a8b88c9931f708b8e99/html5/thumbnails/8.jpg)
Groups and Chomsky hierarchy
Some languages in group theory have been classified withinChomsky hierarchy:
regular (co)word problem i↵ finite (Anisimov),
context-free word problem i↵ virtually free (Muller & Schupp),
context-free multiplication table i↵ hyperbolic (Gilman),
Question
What about totally disconnected locally compact groups?Is there a computational model?
Michal Ferov Inverse limits of finite state automata
![Page 9: Inverse limits of finite state automata - reh.math.uni ...reh.math.uni-duesseldorf.de/~internet/trees2018/slides/Ferov.pdf · Inverse limits of finite state automata Michal Ferov](https://reader030.vdocument.in/reader030/viewer/2022041201/5d4a2a8b88c9931f708b8e99/html5/thumbnails/9.jpg)
An inspiration...
A group is residually finite if for every g 2 G there is N E G offinite index such that g /2 G .
Theorem
Mal’cev If G = hX | Ri is a finitely presented residually finite
group then G has solvable word problem.
Proof.
Run two algorithms in parallel:
first to enumerate all w 0 2 (X [ X
�1)⇤ such that w =G 1;
second to enumerate all Cay(G/N,X ) where N Ef G ;
Given a word w 2 (X [ X
�1)⇤,
first algorithm will stop if it finds w ,
second algorithm will stop if it finds N E G such that w is nota label of a closed loop in Cay(G/N,X ).
Exactly one of the algorithms will stop.
Michal Ferov Inverse limits of finite state automata
![Page 10: Inverse limits of finite state automata - reh.math.uni ...reh.math.uni-duesseldorf.de/~internet/trees2018/slides/Ferov.pdf · Inverse limits of finite state automata Michal Ferov](https://reader030.vdocument.in/reader030/viewer/2022041201/5d4a2a8b88c9931f708b8e99/html5/thumbnails/10.jpg)
An inspiration...
A group is residually finite if for every g 2 G there is N E G offinite index such that g /2 G .
Theorem
Mal’cev If G = hX | Ri is a finitely presented residually finite
group then G has solvable word problem.
Proof.
Run two algorithms in parallel:
first to enumerate all w 0 2 (X [ X
�1)⇤ such that w =G 1;
second to enumerate all Cay(G/N,X ) where N Ef G ;
Given a word w 2 (X [ X
�1)⇤,
first algorithm will stop if it finds w ,
second algorithm will stop if it finds N E G such that w is nota label of a closed loop in Cay(G/N,X ).
Exactly one of the algorithms will stop.
Michal Ferov Inverse limits of finite state automata
![Page 11: Inverse limits of finite state automata - reh.math.uni ...reh.math.uni-duesseldorf.de/~internet/trees2018/slides/Ferov.pdf · Inverse limits of finite state automata Michal Ferov](https://reader030.vdocument.in/reader030/viewer/2022041201/5d4a2a8b88c9931f708b8e99/html5/thumbnails/11.jpg)
An inspiration...
A group is residually finite if for every g 2 G there is N E G offinite index such that g /2 G .
Theorem
Mal’cev If G = hX | Ri is a finitely presented residually finite
group then G has solvable word problem.
Proof.
Run two algorithms in parallel:
first to enumerate all w 0 2 (X [ X
�1)⇤ such that w =G 1;
second to enumerate all Cay(G/N,X ) where N Ef G ;
Given a word w 2 (X [ X
�1)⇤,
first algorithm will stop if it finds w ,
second algorithm will stop if it finds N E G such that w is nota label of a closed loop in Cay(G/N,X ).
Exactly one of the algorithms will stop.
Michal Ferov Inverse limits of finite state automata
![Page 12: Inverse limits of finite state automata - reh.math.uni ...reh.math.uni-duesseldorf.de/~internet/trees2018/slides/Ferov.pdf · Inverse limits of finite state automata Michal Ferov](https://reader030.vdocument.in/reader030/viewer/2022041201/5d4a2a8b88c9931f708b8e99/html5/thumbnails/12.jpg)
Finite state automaton over X
Definition (X -FSA)
A finite state automaton over a finite alphabet X is a tupleM = (Q, q
0
,A, �), where
Q is a finite set of states,
q
0
2 Q is the initial state,
; 6= A ✓ Q is the set of accepting states,
� ✓ Q ⇥ X ⇥ Q is the transition relation.
A word w = x
1
. . . xn 2 X
⇤ takes state q to q
0 if there is asequence of states q
1
, . . . , qn�1
2 Q such that(q, x
1
, q1
), (q1
, x2
, q2
), . . . , (qn�1
, xn, q0) 2 �. Denotew(q) = {q0 2 Q | w takes q to q
0}.The machine M accepts word w if w(q
0
) \ A 6= ;. The set ofwords accepted by M is denoted as L(M).
Michal Ferov Inverse limits of finite state automata
![Page 13: Inverse limits of finite state automata - reh.math.uni ...reh.math.uni-duesseldorf.de/~internet/trees2018/slides/Ferov.pdf · Inverse limits of finite state automata Michal Ferov](https://reader030.vdocument.in/reader030/viewer/2022041201/5d4a2a8b88c9931f708b8e99/html5/thumbnails/13.jpg)
Finite state automaton over X
Definition (X -FSA)
A finite state automaton over a finite alphabet X is a tupleM = (Q, q
0
,A, �), where
Q is a finite set of states,
q
0
2 Q is the initial state,
; 6= A ✓ Q is the set of accepting states,
� ✓ Q ⇥ X ⇥ Q is the transition relation.
A word w = x
1
. . . xn 2 X
⇤ takes state q to q
0 if there is asequence of states q
1
, . . . , qn�1
2 Q such that(q, x
1
, q1
), (q1
, x2
, q2
), . . . , (qn�1
, xn, q0) 2 �. Denotew(q) = {q0 2 Q | w takes q to q
0}.
The machine M accepts word w if w(q0
) \ A 6= ;. The set ofwords accepted by M is denoted as L(M).
Michal Ferov Inverse limits of finite state automata
![Page 14: Inverse limits of finite state automata - reh.math.uni ...reh.math.uni-duesseldorf.de/~internet/trees2018/slides/Ferov.pdf · Inverse limits of finite state automata Michal Ferov](https://reader030.vdocument.in/reader030/viewer/2022041201/5d4a2a8b88c9931f708b8e99/html5/thumbnails/14.jpg)
Finite state automaton over X
Definition (X -FSA)
A finite state automaton over a finite alphabet X is a tupleM = (Q, q
0
,A, �), where
Q is a finite set of states,
q
0
2 Q is the initial state,
; 6= A ✓ Q is the set of accepting states,
� ✓ Q ⇥ X ⇥ Q is the transition relation.
A word w = x
1
. . . xn 2 X
⇤ takes state q to q
0 if there is asequence of states q
1
, . . . , qn�1
2 Q such that(q, x
1
, q1
), (q1
, x2
, q2
), . . . , (qn�1
, xn, q0) 2 �. Denotew(q) = {q0 2 Q | w takes q to q
0}.The machine M accepts word w if w(q
0
) \ A 6= ;. The set ofwords accepted by M is denoted as L(M).
Michal Ferov Inverse limits of finite state automata
![Page 15: Inverse limits of finite state automata - reh.math.uni ...reh.math.uni-duesseldorf.de/~internet/trees2018/slides/Ferov.pdf · Inverse limits of finite state automata Michal Ferov](https://reader030.vdocument.in/reader030/viewer/2022041201/5d4a2a8b88c9931f708b8e99/html5/thumbnails/15.jpg)
Category of X -FSAs
Definition (morphism of X -FSAs)
Let M = (Q, q0
,A, �) and M
0 = (Q 0, q00
,A0, �0) be X -FSAs. A mapf : Q ! Q
0 is a morphism of X -FSAs if
f (q0
) = q
00
,
f (A) ✓ A
0,
(q1
, x , q2
) 2 � ) (f (q1
), x , f (q2
)) 2 �0
and we write f : M ! M
0. By definition, L(M) ✓ L(M 0).
We say that a pair of words w ,w 0 is f -compatible iff (w(q)) ✓ w
0(f (q)) for every q 2 Q.The set of pairs pair of f -compatible words is closed undercoordinate-wise concatenation.
Michal Ferov Inverse limits of finite state automata
![Page 16: Inverse limits of finite state automata - reh.math.uni ...reh.math.uni-duesseldorf.de/~internet/trees2018/slides/Ferov.pdf · Inverse limits of finite state automata Michal Ferov](https://reader030.vdocument.in/reader030/viewer/2022041201/5d4a2a8b88c9931f708b8e99/html5/thumbnails/16.jpg)
Inverse limit of X -FSAs
Definition (Profinite state automaton over X )
Let (I ,) be a directed poset and let
MI = ((Mi )i2I , (fi ,j : Mj ! Mi )ij)
be a directed system of X -FSAs indexed by I , i.e. i j k
implies that fi ,k = fi ,j � fj ,k .We say MI = lim �Mi is a profinite state automaton.
The automaton works with sequences of words
WI = {(wi )i2I | the pair (wj ,wi ) is fi ,j compatible whenever i j}.
We say that MI accepts w 2 WI if Mi accepts wi for every i 2 I .
Michal Ferov Inverse limits of finite state automata
![Page 17: Inverse limits of finite state automata - reh.math.uni ...reh.math.uni-duesseldorf.de/~internet/trees2018/slides/Ferov.pdf · Inverse limits of finite state automata Michal Ferov](https://reader030.vdocument.in/reader030/viewer/2022041201/5d4a2a8b88c9931f708b8e99/html5/thumbnails/17.jpg)
Profinite state automata from profinite groups
Lemma
If G = hX i is a finitely generated profinite group then there is a
profinite-state-automaton over X that accepts sequences of words
in X converging to the identity.
Proof.
Suppose that G = lim �Gi .
Then interpret Cay(Gi ,X ) as an X -FSA Mi and set MI = lim �Mi .
Obviously, MI accepts w 2 WI if and only if w represents theidentity in G
Michal Ferov Inverse limits of finite state automata
![Page 18: Inverse limits of finite state automata - reh.math.uni ...reh.math.uni-duesseldorf.de/~internet/trees2018/slides/Ferov.pdf · Inverse limits of finite state automata Michal Ferov](https://reader030.vdocument.in/reader030/viewer/2022041201/5d4a2a8b88c9931f708b8e99/html5/thumbnails/18.jpg)
Profinite state automata from profinite groups
Lemma
If G = hX i is a finitely generated profinite group then there is a
profinite-state-automaton over X that accepts sequences of words
in X converging to the identity.
Proof.
Suppose that G = lim �Gi .
Then interpret Cay(Gi ,X ) as an X -FSA Mi and set MI = lim �Mi .
Obviously, MI accepts w 2 WI if and only if w represents theidentity in G
Michal Ferov Inverse limits of finite state automata
![Page 19: Inverse limits of finite state automata - reh.math.uni ...reh.math.uni-duesseldorf.de/~internet/trees2018/slides/Ferov.pdf · Inverse limits of finite state automata Michal Ferov](https://reader030.vdocument.in/reader030/viewer/2022041201/5d4a2a8b88c9931f708b8e99/html5/thumbnails/19.jpg)
Profinite groups from profinite state automata
Lemma
Let G = hX i be a finitely generated group and let MI = lim �Mi
what accepts w 2 WI if and only if w represents a Cauchy
sequence converging to the identity. Then G is a profinite group,
in particular G = lim �Gi .
Proof.
For every i 2 I can construct a X -FSA M
0i and a morphism
fi : Mi ! M
0i such that L(M) = L(M 0) and M
0i⇠= Cay(Gi ,X ) as a
decorated graph.Start at the bottom and consistently work your way upwards.
Michal Ferov Inverse limits of finite state automata
![Page 20: Inverse limits of finite state automata - reh.math.uni ...reh.math.uni-duesseldorf.de/~internet/trees2018/slides/Ferov.pdf · Inverse limits of finite state automata Michal Ferov](https://reader030.vdocument.in/reader030/viewer/2022041201/5d4a2a8b88c9931f708b8e99/html5/thumbnails/20.jpg)
Profinite groups from profinite state automata
Lemma
Let G = hX i be a finitely generated group and let MI = lim �Mi
what accepts w 2 WI if and only if w represents a Cauchy
sequence converging to the identity. Then G is a profinite group,
in particular G = lim �Gi .
Proof.
For every i 2 I can construct a X -FSA M
0i and a morphism
fi : Mi ! M
0i such that L(M) = L(M 0) and M
0i⇠= Cay(Gi ,X ) as a
decorated graph.
Start at the bottom and consistently work your way upwards.
Michal Ferov Inverse limits of finite state automata
![Page 21: Inverse limits of finite state automata - reh.math.uni ...reh.math.uni-duesseldorf.de/~internet/trees2018/slides/Ferov.pdf · Inverse limits of finite state automata Michal Ferov](https://reader030.vdocument.in/reader030/viewer/2022041201/5d4a2a8b88c9931f708b8e99/html5/thumbnails/21.jpg)
Profinite groups from profinite state automata
Lemma
Let G = hX i be a finitely generated group and let MI = lim �Mi
what accepts w 2 WI if and only if w represents a Cauchy
sequence converging to the identity. Then G is a profinite group,
in particular G = lim �Gi .
Proof.
For every i 2 I can construct a X -FSA M
0i and a morphism
fi : Mi ! M
0i such that L(M) = L(M 0) and M
0i⇠= Cay(Gi ,X ) as a
decorated graph.Start at the bottom and consistently work your way upwards.
Michal Ferov Inverse limits of finite state automata
![Page 22: Inverse limits of finite state automata - reh.math.uni ...reh.math.uni-duesseldorf.de/~internet/trees2018/slides/Ferov.pdf · Inverse limits of finite state automata Michal Ferov](https://reader030.vdocument.in/reader030/viewer/2022041201/5d4a2a8b88c9931f708b8e99/html5/thumbnails/22.jpg)
That’s all for now
Questions?
Thank you!
Michal Ferov Inverse limits of finite state automata
![Page 23: Inverse limits of finite state automata - reh.math.uni ...reh.math.uni-duesseldorf.de/~internet/trees2018/slides/Ferov.pdf · Inverse limits of finite state automata Michal Ferov](https://reader030.vdocument.in/reader030/viewer/2022041201/5d4a2a8b88c9931f708b8e99/html5/thumbnails/23.jpg)
That’s all for now
Questions?
Thank you!
Michal Ferov Inverse limits of finite state automata