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• Generalised transfinite Turing machines andstrategies for games. P.D.Welch, University of Bristol. Chicheley Hall 2012

• Theme: Connections between inductive operators, discrete transfinitemachine models of computation, and determinacy.

Part I: ITTM description. Part II: Fixed points of operators yielding some strategies. Part III: Generalising Operators and Machines

• Theme: Connections between inductive operators, discrete transfinitemachine models of computation, and determinacy.

Part I: ITTM description. Part II: Fixed points of operators yielding some strategies. Part III: Generalising Operators and Machines

• Part I: ITTM description1

Allow a standard Turing machine to run transfinitely using one of the usualprograms Pe | e N. Alphabet: {0, 1}; Enumerate the cells of the tape Ck | k N.

Let the current instruction about to be performed at time be Ii();Let the current cell being inspected be Cp().

Behaviour at successor stages + 1: as normal.At limit times : (a) we specify cell values by:

Ck() = LiminfCk()

(where the value in Ck at time is Ck()).(b) we also (i) put the R/W to cell Cp() where

p() = Liminf 7{p() | < < };

(ii) seti() = Liminf7{i() | < < }.

1[HL] Hamkins & Lewis Infinite Time Turing Machines, JSL, vol. 65, 2000.

• Part I: ITTM description1

Allow a standard Turing machine to run transfinitely using one of the usualprograms Pe | e N. Alphabet: {0, 1}; Enumerate the cells of the tape Ck | k N.

Let the current instruction about to be performed at time be Ii();Let the current cell being inspected be Cp().

Behaviour at successor stages + 1: as normal.At limit times : (a) we specify cell values by:

Ck() = LiminfCk()

(where the value in Ck at time is Ck()).

(b) we also (i) put the R/W to cell Cp() where

p() = Liminf 7{p() | < < };

(ii) seti() = Liminf7{i() | < < }.

1[HL] Hamkins & Lewis Infinite Time Turing Machines, JSL, vol. 65, 2000.

• Part I: ITTM description1

Allow a standard Turing machine to run transfinitely using one of the usualprograms Pe | e N. Alphabet: {0, 1}; Enumerate the cells of the tape Ck | k N.

Let the current instruction about to be performed at time be Ii();Let the current cell being inspected be Cp().

Behaviour at successor stages + 1: as normal.At limit times : (a) we specify cell values by:

Ck() = LiminfCk()

(where the value in Ck at time is Ck()).(b) we also (i) put the R/W to cell Cp() where

p() = Liminf 7{p() | < < };

(ii) seti() = Liminf7{i() | < < }.

1[HL] Hamkins & Lewis Infinite Time Turing Machines, JSL, vol. 65, 2000.

• Hamkins & Lewis proved there is a universal machine, an Smn -Theorem,and a Recursion Theorem for ITTMs, and a wealth of results on theresulting ITTM-degree theory.We may define halting sets:

H = {(e, x) | e N, x 2N Pe(x) }

H0 = {(e, 0) | e N Pe(0) }

Q. What is H or H0?Q. How long do we have to wait to discover if e H0 or not?Q. What are the ITTM (semi)-decidable sets of integers? Or reals?

• Hamkins & Lewis proved there is a universal machine, an Smn -Theorem,and a Recursion Theorem for ITTMs, and a wealth of results on theresulting ITTM-degree theory.We may define halting sets:

H = {(e, x) | e N, x 2N Pe(x) }

H0 = {(e, 0) | e N Pe(0) }

Q. What is H or H0?Q. How long do we have to wait to discover if e H0 or not?Q. What are the ITTM (semi)-decidable sets of integers? Or reals?

• A Kleene Normal Form Theorem?

Wed like some type of a ITTM Normal Form Theorem:

TheoremThere is a universal predicate T which satisfies ex:

Pe(x) z y 2N[T(e, x, y) Last(y) = z].

However for this to occur we need to know whether the ordinal length ofany computation is capable of being output or written by a(nother)computation.

• A Kleene Normal Form Theorem?

Wed like some type of a ITTM Normal Form Theorem:

TheoremThere is a universal predicate T which satisfies ex:

Pe(x) z y 2N[T(e, x, y) Last(y) = z].

However for this to occur we need to know whether the ordinal length ofany computation is capable of being output or written by a(nother)computation.

• The , ,-Theorem2

TheoremLet be the least ordinal so that there exists > with the property that

L 2 L ; ( is 2-extendible.)

(i) Then the universal ITTM on integer input first enters a loop at time .

Let be the least ordinal satisfying:

L 1 L .

(ii) Then = sup{ | e Pe(0) in steps}= sup{ | e Pe(0)y WO ||y|| = }.

As a corollary one derives the Normal Form Theorem and:

Corollary

H0 1-Th(L).

2Welch The length of ITTM computations, Bull. London Math. Soc. 2000

• The , ,-Theorem2

TheoremLet be the least ordinal so that there exists > with the property that

L 2 L ; ( is 2-extendible.)

(i) Then the universal ITTM on integer input first enters a loop at time .Let be the least ordinal satisfying:

L 1 L .

(ii) Then = sup{ | e Pe(0) in steps}= sup{ | e Pe(0)y WO ||y|| = }.

As a corollary one derives the Normal Form Theorem and:

Corollary

H0 1-Th(L).

2Welch The length of ITTM computations, Bull. London Math. Soc. 2000

• The , ,-Theorem2

TheoremLet be the least ordinal so that there exists > with the property that

L 2 L ; ( is 2-extendible.)

(i) Then the universal ITTM on integer input first enters a loop at time .Let be the least ordinal satisfying:

L 1 L .

(ii) Then = sup{ | e Pe(0) in steps}= sup{ | e Pe(0)y WO ||y|| = }.

As a corollary one derives the Normal Form Theorem and:

Corollary

H0 1-Th(L).

2Welch The length of ITTM computations, Bull. London Math. Soc. 2000

• Part II: Over N 11-IND = a01

The game quantifier a:

DefinitionA set A N is a if there is B N R so that:

n A Player I has a winning strategy in GBn

where Bn = {x R | B(n, x)}.

Theorem (Folklore)A set A N is (mon.-11)-IND iff it is a01 iff it is 1(Lck1 ).

Theorem (Svenonius)Any 01 game, if it is a win for Player I then she has a (mon.-

11)-IND

winning strategy. Hence this is also 1(Lck1 ).

• Part II: Over N 11-IND = a01

The game quantifier a:

DefinitionA set A N is a if there is B N R so that:

n A Player I has a winning strategy in GBn

where Bn = {x R | B(n, x)}.

Theorem (Folklore)A set A N is (mon.-11)-IND iff it is a01 iff it is 1(Lck1 ).

Theorem (Svenonius)Any 01 game, if it is a win for Player I then she has a (mon.-

11)-IND

winning strategy. Hence this is also 1(Lck1 ).

• Part II: Over N 11-IND = a01

The game quantifier a:

DefinitionA set A N is a if there is B N R so that:

n A Player I has a winning strategy in GBn

where Bn = {x R | B(n, x)}.

Theorem (Folklore)A set A N is (mon.-11)-IND iff it is a01 iff it is 1(Lck1 ).

Theorem (Svenonius)Any 01 game, if it is a win for Player I then she has a (mon.-

11)-IND

winning strategy. Hence this is also 1(Lck1 ).

• Part II: Over N 11-IND = a02

Theorem (Solovay)A set A N is 11-IND iff it is a02 iff it is 1(L11 ).

(where 11 is the closure ordinal of 11-inductive definitions).

Theorem (Solovay)Any 02 game, if it is a win for Player I then she has a

11-IND winning

strategy. Hence this is also 1(L11 ).

Q. What about 03?

Theorem (Friedman)Z2 0 04-Det.

Q Are strategies for 03 sets ITTM-semi-decidable? Thus: are they 1(L)?

• Part II: Over N 11-IND = a02

Theorem (Solovay)A set A N is 11-IND iff it is a02 iff it is 1(L11 ).

(where 11 is the closure ordinal of 11-inductive definitions).

Theorem (Solovay)Any 02 game, if it is a win for Player I then she has a

11-IND winning

strategy. Hence this is also 1(L11 ).

Q. What about 03?

Theorem (Friedman)Z2 0 04-Det.

Q Are strategies for 03 sets ITTM-semi-decidable? Thus: are they 1(L)?

• Part II: Over N 11-IND = a02

Theorem (Solovay)A set A N is 11-IND iff it is a02 iff it is 1(L11 ).

(where 11 is the closure ordinal of 11-inductive definitions).

Theorem (Solovay)Any 02 game, if it is a win for Player I then she has a

11-IND winning

strategy. Hence this is also 1(L11 ).

Q. What about 03?

Theorem (Friedman)Z2 0 04-Det.

Q Are strategies for 03 sets ITTM-semi-decidable? Thus: are they 1(L)?

• Part II: Over N 11-IND = a02

Theorem (Solovay)A set A N is 11-IND iff it is a02 iff it is 1(L11 ).

(where 11 is the closure ordinal of 11-inductive definitions).

Theorem (Solovay)Any 02 game, if it is a win for Player I then she has a

11-IND winning

strategy. Hence this is also 1(L11 ).

Q. What about 03?

Theorem (Friedman)Z2 0 04-Det.

Q Are strategies for 03 sets ITTM-semi-decidable? Thus: are they 1(L)?

• Part II: Over N 11-IND = a02

Theorem (Solovay)A set A N is 11-IND iff it is a02 iff it is 1(L11 ).

(where 11 is the closure ordinal of 11-inductive definitions).

Theorem (Solovay)Any 02 game, if it is a win for Player I then she has a

11-IND winning

strategy. Hence this is also 1(L11 ).

Q. What about 03?

Theorem (Friedman)Z2 0 04-Det.

Q Are strategies for 03 sets ITTM-semi-decidable? Thus: are they 1(L)?

• DefinitionLet ITTM abbreviate: X(HX0 exists )(the complete ITTM-semi-decidable-in-X set exists).

TheoremThe theories:

13-CA0,13-CA0 +

03-Det,

13

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