The group of reversible Turing machines Sebastin Barbieri, Jarkko - PowerPoint PPT Presentation

The group of reversible Turing machines Sebastián Barbieri, Jarkko Kari and Ville Salo LIP, ENS de Lyon – CNRS – INRIA – UCBL – Université de Lyon University of Turku Center for Mathematical Modeling, University of Chile AUTOMATA June, 2016

Motivation Recall that a Turing machine is defined by a rule : δ T : Σ × Q → Σ × Q × {− 1 , 0 , 1 }

Motivation Recall that a Turing machine is defined by a rule : δ T : Σ × Q → Σ × Q × {− 1 , 0 , 1 } q δ T ( , q ) = ( , r , − 1)

Motivation Recall that a Turing machine is defined by a rule : δ T : Σ × Q → Σ × Q × {− 1 , 0 , 1 } r δ T ( , q ) = ( , r , − 1)

Motivation This defines a natural action T : Σ Z × Q → Σ Z × Q q T r

Motivation This defines a natural action T : Σ Z × Q → Σ Z × Q Such that if ( x , q ) ∈ Σ Z × Q and δ T ( x 0 , q ) = ( a , r , d ) then : x ) , q ′ ) T ( x , q ) = ( σ − d (˜ where σ : Σ Z → Σ Z is the shift action given by σ d ( x ) z = x z − d , ˜ x 0 = a and ˜ x | Z \{ 0 } = x | Z \{ 0 } .

Motivation The composition of two actions T ◦ T ′ is not necessarily an action generated by a Turing machine. if the action T is a bijection then the inverse it not necessarily an action generated by a Turing machine.

Motivation The composition of two actions T ◦ T ′ is not necessarily an action generated by a Turing machine. if the action T is a bijection then the inverse it not necessarily an action generated by a Turing machine. As in cellular automata, the class of CA with radius bounded by some k ∈ N is not closed under composition or inverses.

Definition Let’s get rid of these constrains. Given F , F ′ finite subsets of Z d , consider instead of δ T a function : f T : Σ F × Q → Σ F ′ × Q × Z d ,

Definition Let’s get rid of these constrains. Given F , F ′ finite subsets of Z d , consider instead of δ T a function : f T : Σ F × Q → Σ F ′ × Q × Z d , Let F = F ′ = { 0 , 1 , 2 } 2 , then f T ( p , q ) = ( p ′ , q ′ ,� d ) means : p p ′ Turn state q into state q ′ Move head by � d .

Moving head model f T defines naturally an action T � Σ Z d × Q × Z d T q 2 q 1 f ( , q 1 ) = ( , q 2 , (1 , 1)) F = { (0 , 0) , (1 , 0) , (1 , 1) }

Moving head model f T defines naturally an action T � Σ Z d × Q × Z d T q 2 q 1 f ( , q 1 ) = ( , q 2 , (1 , 1)) F = { (0 , 0) , (1 , 0) , (1 , 1) } Let | Σ | = n and | Q | = k . ( TM ( Z d , n , k ) , ◦ ) is the monoid of all such T with the composition operation ; ( RTM ( Z d , n , k ) , ◦ ) is the group of all such T which are bijective .

Moving head model : As cellular automata Let Q = { 1 , . . . , k } and Σ = { 0 , . . . , n − 1 } . Σ Z d = { x : Z d → Σ } X k = { x ∈ { 0 , 1 , . . . , k } Z d | 0 / ⇒ � u = � ∈ { x � u , x � v } = v } Let X n , k = Σ Z d × X k and Y = Σ Z d × { 0 Z d } . Then :

Moving head model : As cellular automata Let Q = { 1 , . . . , k } and Σ = { 0 , . . . , n − 1 } . Σ Z d = { x : Z d → Σ } X k = { x ∈ { 0 , 1 , . . . , k } Z d | 0 / ⇒ � u = � ∈ { x � u , x � v } = v } Let X n , k = Σ Z d × X k and Y = Σ Z d × { 0 Z d } . Then : TM ( Z d , n , k ) = { φ ∈ End ( X n , k ) | φ | Y = id , φ − 1 ( Y ) = Y } RTM ( Z d , n , k ) = { φ ∈ Aut ( X n , k ) | φ | Y = id }

Moving tape model f T defines naturally an action T � Σ Z d × Q T f q 1 q 2 f ( , q 1 ) = ( , q 2 , (1 , 1)) F = { (0 , 0) , (1 , 0) , (1 , 1) }

Moving tape model f T defines naturally an action T � Σ Z d × Q T f q 1 q 2 f ( , q 1 ) = ( , q 2 , (1 , 1)) F = { (0 , 0) , (1 , 0) , (1 , 1) } Let | Σ | = n and | Q | = k . ( TM fix ( Z d , n , k ) , ◦ ) is the monoid of all such T with the composition operation ; ( RTM fix ( Z d , n , k ) , ◦ ) is the group of all such T which are bijective .

Moving tape model : dynamical definition Let x , y ∈ Σ Z d . x and y are asymptotic , and write x ∼ y , if they differ in finitely many coordinates. We write x ∼ m y if x � v = y � v for all | � v | ≥ m .

Moving tape model : dynamical definition Let x , y ∈ Σ Z d . x and y are asymptotic , and write x ∼ y , if they differ in finitely many coordinates. We write x ∼ m y if x � v = y � v for v | ≥ m . Let T : Σ Z d × Q → Σ Z d × Q be a function. all | � T is a moving tape Turing machine ⇐ ⇒ T is continuous, and for a continuous function s : Σ Z d × Q → Z d and a ∈ N we have T ( x , q ) 1 ∼ a σ s ( x , q ) ( x ) for all ( x , q ) ∈ Σ Z d × Q .

Moving tape model : dynamical definition Let x , y ∈ Σ Z d . x and y are asymptotic , and write x ∼ y , if they differ in finitely many coordinates. We write x ∼ m y if x � v = y � v for v | ≥ m . Let T : Σ Z d × Q → Σ Z d × Q be a function. all | � T is a moving tape Turing machine ⇐ ⇒ T is continuous, and for a continuous function s : Σ Z d × Q → Z d and a ∈ N we have T ( x , q ) 1 ∼ a σ s ( x , q ) ( x ) for all ( x , q ) ∈ Σ Z d × Q . s : Σ Z d × Q → Z d is the shift indicator function

Equivalence of the models = S k and Z d ֒ RTM fix ( Z d , 1 , k ) ∼ → RTM ( Z d , 1 , k ).

Equivalence of the models = S k and Z d ֒ RTM fix ( Z d , 1 , k ) ∼ → RTM ( Z d , 1 , k ). Proposition If n ≥ 2 then : TM fix ( Z d , n , k ) ∼ = TM ( Z d , n , k ) RTM fix ( Z d , n , k ) ∼ = RTM ( Z d , n , k ) .

Properties of RTM Proposition Let T ∈ TM fix ( Z d , n , k ). Then the following are equivalent : 1 T is injective. 2 T is surjective. 3 T ∈ RTM fix ( Z d , n , k ). 4 T preserves the uniform measure ( µ ( T − 1 ( A )) = µ ( A ) for all Borel sets A ). 5 µ ( T ( A )) = µ ( A ) for all Borel sets A .

Properties of RTM Proposition If n ≥ 2 RTM ( Z d , n , k ) is not finitely generated.

Properties of RTM Proposition If n ≥ 2 RTM ( Z d , n , k ) is not finitely generated. Proof : We find an epimorphism from RTM to a non-finitely generated group. Let T ∈ RTM fix ( Z d , n , k ), therefore, it has a shift indicator s : Σ Z d × Q → Z d . Define � α ( T ) := E µ ( s ) = s ( x , q ) d µ, Σ Z d × Q One can check that α ( T 1 ◦ T 2 ) = α ( T 1 ) + α ( T 2 ). Therefore α : RTM ( Z d , n , k ) → Q d is an homomorphism

Properties of RTM Now consider the machine T SURF , m where for all a ∈ Σ and q ∈ Q : q 0 0 a 0 0 0 0 0 T SURF , m q a 0 0 0 0 0 0 0 f (0 m a , q ) = ( a 0 m , q , 1) . Otherwise f ( u , q ) = ( u , q , 0) .

Properties of RTM Now consider the machine T SURF , m where for all a ∈ Σ and q ∈ Q : q 0 0 a 0 0 0 0 0 T SURF , m q a 0 0 0 0 0 0 0 f (0 m a , q ) = ( a 0 m , q , 1) . Otherwise f ( u , q ) = ( u , q , 0) . T SURF , m ∈ RTM ( Z , n , k ) and α ( T SURF , m ) = 1 / n m

Properties of RTM Now consider the machine T SURF , m where for all a ∈ Σ and q ∈ Q : q 0 0 a 0 0 0 0 0 T SURF , m q a 0 0 0 0 0 0 0 f (0 m a , q ) = ( a 0 m , q , 1) . Otherwise f ( u , q ) = ( u , q , 0) . T SURF , m ∈ RTM ( Z , n , k ) and α ( T SURF , m ) = 1 / n m � (1 / n m ) m ∈ N � ⊂ α ( RTM ( Z , n , k )) which is thus a non-finitely generated subgroup of Q .

Interesting subgroups of RTM ⊲ LP ( Z d , n , k ) − → Local permutations. q 0 0 1 0 0 1 0 0 1 1 1 0 1 0 0 1 0 0 T π r 0 0 1 0 0 1 0 1 0 0 1 1 0 0 0 1 0 0

Interesting subgroups of RTM ⊲ LP ( Z d , n , k ) − → Local permutations. ⊲ RFA ( Z d , n , k ) − → Reversible finite-state automata. q 0 0 1 0 0 1 0 0 1 1 1 0 1 0 0 1 0 0 T r 0 0 1 0 0 1 0 0 1 1 1 0 1 0 0 1 0 0

Interesting subgroups of RTM ⊲ LP ( Z d , n , k ) − → Local permutations. ⊲ RFA ( Z d , n , k ) − → Reversible finite-state automata. ⊲ OB ( Z d , n , k ) − → Oblivous machines � LP , Shift � .

Interesting subgroups of RTM ⊲ LP ( Z d , n , k ) − → Local permutations. ⊲ RFA ( Z d , n , k ) − → Reversible finite-state automata. ⊲ OB ( Z d , n , k ) − → Oblivous machines � LP , Shift � . ⊲ EL( Z d , n , k ) − → Elementary machines � LP , RFA � .

Small group theory roadmap Residually Finite LEF Surjunctive Abelian Sofic Amenable LEA Res. finite groups are those where every non-identity element can be mapped to a non-identity element by a homomorphism to a finite group Amenable groups admit left invariant finitely additive measures. LEF and LEA stand for locally embeddable into (finite/amenable) groups. Sofic groups are generalizations of LEF and LEA. Surjunctive groups satisfy that all injective CA are surjective.

Small group theory roadmap Residually Finite LEF Surjunctive Abelian Sofic Amenable LEA Theorem ∀ n ≥ 2 , RTM ( Z d , n , k ) is LEF but neither amenable nor residually finite.

Some properties : LP ( Z d , n , k ) → LP ( Z d , n , k ). For n ≥ 2, we have S ∞ ֒

Some properties : LP ( Z d , n , k ) → LP ( Z d , n , k ). For n ≥ 2, we have S ∞ ֒ This means that RTM is not residually finite, and that it contains all finite groups.

Some properties : LP ( Z d , n , k ) → LP ( Z d , n , k ). For n ≥ 2, we have S ∞ ֒ This means that RTM is not residually finite, and that it contains all finite groups. LP ( Z d , n , k ) is locally finite and amenable.