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Characterizing Computable Analysis with Differential Equations Kerry - - PowerPoint PPT Presentation

Dec 13, 2023 105 likes •861 views

Basic Background Our model and our result Discussion of the Proof Conclusion Characterizing Computable Analysis with Differential Equations Kerry Ojakian 1 (with Manuel L. Campagnolo 2 ) 1 SQIG/IT Lisbon and IST, Portugal

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Basic Background Our model and our result Discussion of the Proof Conclusion

Characterizing Computable Analysis with Differential Equations

Kerry Ojakian1 (with Manuel L. Campagnolo2)

1SQIG/IT Lisbon and IST, Portugal

  • jakian@math.ist.utl.pt

2DM/ISA, Lisbon University of Technology and SQIG/IT Lisbon

CUNY Set Theory Seminar 2009

Ojakian, Campagnolo Characterizing Computable Analysis

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Basic Background Our model and our result Discussion of the Proof Conclusion

Outline

1

Basic Background

2

Our model and our result

3

Discussion of the Proof

4

Conclusion

Ojakian, Campagnolo Characterizing Computable Analysis

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Basic Background Our model and our result Discussion of the Proof Conclusion

Outline

1

Basic Background

2

Our model and our result

3

Discussion of the Proof

4

Conclusion

Ojakian, Campagnolo Characterizing Computable Analysis

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Basic Background Our model and our result Discussion of the Proof Conclusion

Computability over the Reals?

Suppose f : R → R. When is f “computable”? Let f(x) = 0, if x ≤ 0; 1, if x > 0. Is it computable? f is computable according to the BSS Model, and not according to Computable Analysis. Is ex computable? ex is not computable according to the BSS Model, and is computable according to Computable Analysis.

Ojakian, Campagnolo Characterizing Computable Analysis

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Basic Background Our model and our result Discussion of the Proof Conclusion

Computability over the Reals?

Suppose f : R → R. When is f “computable”? Let f(x) = 0, if x ≤ 0; 1, if x > 0. Is it computable? f is computable according to the BSS Model, and not according to Computable Analysis. Is ex computable? ex is not computable according to the BSS Model, and is computable according to Computable Analysis.

Ojakian, Campagnolo Characterizing Computable Analysis

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Basic Background Our model and our result Discussion of the Proof Conclusion

Computability over the Reals?

Suppose f : R → R. When is f “computable”? Let f(x) = 0, if x ≤ 0; 1, if x > 0. Is it computable? f is computable according to the BSS Model, and not according to Computable Analysis. Is ex computable? ex is not computable according to the BSS Model, and is computable according to Computable Analysis.

Ojakian, Campagnolo Characterizing Computable Analysis

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Basic Background Our model and our result Discussion of the Proof Conclusion

Computability over the Reals?

Suppose f : R → R. When is f “computable”? Let f(x) = 0, if x ≤ 0; 1, if x > 0. Is it computable? f is computable according to the BSS Model, and not according to Computable Analysis. Is ex computable? ex is not computable according to the BSS Model, and is computable according to Computable Analysis.

Ojakian, Campagnolo Characterizing Computable Analysis

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Basic Background Our model and our result Discussion of the Proof Conclusion

Computability over the Reals?

Suppose f : R → R. When is f “computable”? Let f(x) = 0, if x ≤ 0; 1, if x > 0. Is it computable? f is computable according to the BSS Model, and not according to Computable Analysis. Is ex computable? ex is not computable according to the BSS Model, and is computable according to Computable Analysis.

Ojakian, Campagnolo Characterizing Computable Analysis

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Basic Background Our model and our result Discussion of the Proof Conclusion

Computable Analysis

f is computable according to “Naive” Computable Analysis iff: There is a computable function F x(n) with an oracle for the real number x such that F x(n) → f(x), as n → ∞. Definition f ∈ CR (Computable Analysis) iff: There is a computable function F x(n) with an oracle for the real number x such that |f(x) − F x(n)| ≤ 1/n. Examples of functions in CR: ex, π, sin x, log x, ...

Ojakian, Campagnolo Characterizing Computable Analysis

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Basic Background Our model and our result Discussion of the Proof Conclusion

Computable Analysis

f is computable according to “Naive” Computable Analysis iff: There is a computable function F x(n) with an oracle for the real number x such that F x(n) → f(x), as n → ∞. Definition f ∈ CR (Computable Analysis) iff: There is a computable function F x(n) with an oracle for the real number x such that |f(x) − F x(n)| ≤ 1/n. Examples of functions in CR: ex, π, sin x, log x, ...

Ojakian, Campagnolo Characterizing Computable Analysis

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Basic Background Our model and our result Discussion of the Proof Conclusion

Computable Analysis

f is computable according to “Naive” Computable Analysis iff: There is a computable function F x(n) with an oracle for the real number x such that F x(n) → f(x), as n → ∞. Definition f ∈ CR (Computable Analysis) iff: There is a computable function F x(n) with an oracle for the real number x such that |f(x) − F x(n)| ≤ 1/n. Examples of functions in CR: ex, π, sin x, log x, ...

Ojakian, Campagnolo Characterizing Computable Analysis

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Basic Background Our model and our result Discussion of the Proof Conclusion

Other Models

1

Shannon’s circuit model.

2

Neural Networks.

3

Hybrid systems. There is not an agreed upon definition of computability over R. There is no “basic theorem” for computability over R! Our work: Showing certain models equivalent.

Ojakian, Campagnolo Characterizing Computable Analysis

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Basic Background Our model and our result Discussion of the Proof Conclusion

Other Models

1

Shannon’s circuit model.

2

Neural Networks.

3

Hybrid systems. There is not an agreed upon definition of computability over R. There is no “basic theorem” for computability over R! Our work: Showing certain models equivalent.

Ojakian, Campagnolo Characterizing Computable Analysis

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Basic Background Our model and our result Discussion of the Proof Conclusion

Other Models

1

Shannon’s circuit model.

2

Neural Networks.

3

Hybrid systems. There is not an agreed upon definition of computability over R. There is no “basic theorem” for computability over R! Our work: Showing certain models equivalent.

Ojakian, Campagnolo Characterizing Computable Analysis

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Basic Background Our model and our result Discussion of the Proof Conclusion

Outline

1

Basic Background

2

Our model and our result

3

Discussion of the Proof

4

Conclusion

Ojakian, Campagnolo Characterizing Computable Analysis

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Basic Background Our model and our result Discussion of the Proof Conclusion

Function Algebras and Real Recursive Functions

Definition A Function Algebra FA[f1, . . . , fk; op1, . . . , opn] is the smallest set of functions containing f1, . . . , fk, and closed under the

  • perations op1, . . . , opn.

Example: FA[0, 1, +, . , P; comp, , , µ], which is equal to the computable functions over the naturals. Real Recursive Functions: Function algebras over the reals, introduced by C. Moore 1996.

Ojakian, Campagnolo Characterizing Computable Analysis

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Basic Background Our model and our result Discussion of the Proof Conclusion

Function Algebras and Real Recursive Functions

Definition A Function Algebra FA[f1, . . . , fk; op1, . . . , opn] is the smallest set of functions containing f1, . . . , fk, and closed under the

  • perations op1, . . . , opn.

Example: FA[0, 1, +, . , P; comp, , , µ], which is equal to the computable functions over the naturals. Real Recursive Functions: Function algebras over the reals, introduced by C. Moore 1996.

Ojakian, Campagnolo Characterizing Computable Analysis

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Basic Background Our model and our result Discussion of the Proof Conclusion

Function Algebras and Real Recursive Functions

Definition A Function Algebra FA[f1, . . . , fk; op1, . . . , opn] is the smallest set of functions containing f1, . . . , fk, and closed under the

  • perations op1, . . . , opn.

Example: FA[0, 1, +, . , P; comp, , , µ], which is equal to the computable functions over the naturals. Real Recursive Functions: Function algebras over the reals, introduced by C. Moore 1996.

Ojakian, Campagnolo Characterizing Computable Analysis

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Basic Background Our model and our result Discussion of the Proof Conclusion

Some basic functions over the reals

Constant functions: 0, 1, −1 Projection functions “P” (example: U(x, y) = x) θk(x) = 0, x < 0; xk, x ≥ 0.

Ojakian, Campagnolo Characterizing Computable Analysis

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Basic Background Our model and our result Discussion of the Proof Conclusion

Some basic functions over the reals

Constant functions: 0, 1, −1 Projection functions “P” (example: U(x, y) = x) θk(x) = 0, x < 0; xk, x ≥ 0.

Ojakian, Campagnolo Characterizing Computable Analysis

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Basic Background Our model and our result Discussion of the Proof Conclusion

Some basic functions over the reals

Constant functions: 0, 1, −1 Projection functions “P” (example: U(x, y) = x) θk(x) = 0, x < 0; xk, x ≥ 0.

Ojakian, Campagnolo Characterizing Computable Analysis

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Basic Background Our model and our result Discussion of the Proof Conclusion

The Differential Equation Operation

Definition ODE is the operation: Input: g(x), f(y, u, x). Output: The solution of the IVP h(0, x) = g(x), ∂ ∂y h = f(y, h, x) Definition LI is the operation defined like ODE, except that f must be linear in h.

Ojakian, Campagnolo Characterizing Computable Analysis

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Basic Background Our model and our result Discussion of the Proof Conclusion

The Differential Equation Operation

Definition ODE is the operation: Input: g(x), f(y, u, x). Output: The solution of the IVP h(0, x) = g(x), ∂ ∂y h = f(y, h, x) Definition LI is the operation defined like ODE, except that f must be linear in h.

Ojakian, Campagnolo Characterizing Computable Analysis

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Basic Background Our model and our result Discussion of the Proof Conclusion

A real function algebra and some examples

Definition Let ODE⋆

k be the total functions of

FA[0, 1, −1, θk, P; comp, ODE] Some functions in ODE⋆

k: ex, (x + y), (xy), sin x, . . .

Ojakian, Campagnolo Characterizing Computable Analysis

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Basic Background Our model and our result Discussion of the Proof Conclusion

A real function algebra and some examples

Definition Let ODE⋆

k be the total functions of

FA[0, 1, −1, θk, P; comp, ODE] Some functions in ODE⋆

k: ex, (x + y), (xy), sin x, . . .

Ojakian, Campagnolo Characterizing Computable Analysis

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Basic Background Our model and our result Discussion of the Proof Conclusion

The Limit Operation

Definition LIM is the operation: Input: f(t, ¯ x) Output: F(¯ x) = lim

t→∞ f(t, ¯

x), if the limit exists, and |F(¯ x) − f(t, ¯ x)| ≤ 1/t. Definition If F a set of functions, then F(LIM) is F closed under the

  • peration LIM.

Ojakian, Campagnolo Characterizing Computable Analysis

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Basic Background Our model and our result Discussion of the Proof Conclusion

The Limit Operation

Definition LIM is the operation: Input: f(t, ¯ x) Output: F(¯ x) = lim

t→∞ f(t, ¯

x), if the limit exists, and |F(¯ x) − f(t, ¯ x)| ≤ 1/t. Definition If F a set of functions, then F(LIM) is F closed under the

  • peration LIM.

Ojakian, Campagnolo Characterizing Computable Analysis

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Basic Background Our model and our result Discussion of the Proof Conclusion

Our main theorem

Theorem (Main Theorem) CR = ODE⋆

k(LIM) for k ≥ 2.

Ojakian, Campagnolo Characterizing Computable Analysis

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Basic Background Our model and our result Discussion of the Proof Conclusion

A function algebra based on searching

Definition The operation UMU takes as Input: f(t, ¯ x) such that

1

For any ¯ x, f(t, ¯ x) increases in t, and

2

For any ¯ x, there is a unique T such that f(T, ¯ x) = 0 (and at that T, ∂

∂t f > 0).

Output: Function F(¯ x) = the unique T such that f(T, ¯ x) = 0. Definition Let UMUk be FA[0, 1, θk, P; comp, LI, UMU]

Ojakian, Campagnolo Characterizing Computable Analysis

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Basic Background Our model and our result Discussion of the Proof Conclusion

A function algebra based on searching

Definition The operation UMU takes as Input: f(t, ¯ x) such that

1

For any ¯ x, f(t, ¯ x) increases in t, and

2

For any ¯ x, there is a unique T such that f(T, ¯ x) = 0 (and at that T, ∂

∂t f > 0).

Output: Function F(¯ x) = the unique T such that f(T, ¯ x) = 0. Definition Let UMUk be FA[0, 1, θk, P; comp, LI, UMU]

Ojakian, Campagnolo Characterizing Computable Analysis

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Basic Background Our model and our result Discussion of the Proof Conclusion

Our mainer theorem

Theorem For k ≥ 2, CR = ODE⋆

k(LIM) = UMUk(LIM).

Theorem ( Bournez and Hainry 2006 ) Roughly: C2 ∩ [CR] = [UMUk(LIM)], for k ≥ 3

Ojakian, Campagnolo Characterizing Computable Analysis

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Basic Background Our model and our result Discussion of the Proof Conclusion

Our mainer theorem

Theorem For k ≥ 2, CR = ODE⋆

k(LIM) = UMUk(LIM).

Theorem ( Bournez and Hainry 2006 ) Roughly: C2 ∩ [CR] = [UMUk(LIM)], for k ≥ 3

Ojakian, Campagnolo Characterizing Computable Analysis

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Basic Background Our model and our result Discussion of the Proof Conclusion

Our mainer theorem

Theorem For k ≥ 2, CR = ODE⋆

k(LIM) = UMUk(LIM).

Theorem ( Bournez and Hainry 2006 ) Roughly: C2 ∩ [CR] = [UMUk(LIM)], for k ≥ 3

Ojakian, Campagnolo Characterizing Computable Analysis

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Basic Background Our model and our result Discussion of the Proof Conclusion

More Motivation

Results of the form “CA = FA”. Interesting to provide alternative models for Computable Analysis. Connects discrete-time and continuous-time. “Basic Theorem” over the reals? Church-Turing thesis over the reals? Could alternative models facilitate technical work? (e.g. like showing a function or operation is or is not computable)

Ojakian, Campagnolo Characterizing Computable Analysis

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Basic Background Our model and our result Discussion of the Proof Conclusion

More Motivation

Results of the form “CA = FA”. Interesting to provide alternative models for Computable Analysis. Connects discrete-time and continuous-time. “Basic Theorem” over the reals? Church-Turing thesis over the reals? Could alternative models facilitate technical work? (e.g. like showing a function or operation is or is not computable)

Ojakian, Campagnolo Characterizing Computable Analysis

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Basic Background Our model and our result Discussion of the Proof Conclusion

More Motivation

Results of the form “CA = FA”. Interesting to provide alternative models for Computable Analysis. Connects discrete-time and continuous-time. “Basic Theorem” over the reals? Church-Turing thesis over the reals? Could alternative models facilitate technical work? (e.g. like showing a function or operation is or is not computable)

Ojakian, Campagnolo Characterizing Computable Analysis

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Basic Background Our model and our result Discussion of the Proof Conclusion

More Motivation

Results of the form “CA = FA”. Interesting to provide alternative models for Computable Analysis. Connects discrete-time and continuous-time. “Basic Theorem” over the reals? Church-Turing thesis over the reals? Could alternative models facilitate technical work? (e.g. like showing a function or operation is or is not computable)

Ojakian, Campagnolo Characterizing Computable Analysis

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Basic Background Our model and our result Discussion of the Proof Conclusion

More Motivation

Results of the form “CA = FA”. Interesting to provide alternative models for Computable Analysis. Connects discrete-time and continuous-time. “Basic Theorem” over the reals? Church-Turing thesis over the reals? Could alternative models facilitate technical work? (e.g. like showing a function or operation is or is not computable)

Ojakian, Campagnolo Characterizing Computable Analysis

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Basic Background Our model and our result Discussion of the Proof Conclusion

Outline

1

Basic Background

2

Our model and our result

3

Discussion of the Proof

4

Conclusion

Ojakian, Campagnolo Characterizing Computable Analysis

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Basic Background Our model and our result Discussion of the Proof Conclusion

The “Main Step”

CR ⊆ UMUk(LIM) By Turing Machine simulation (Bournez and Hainry 2006). Difficulties with this approach ... Forced into unnecessary restrictions. Appears less general. Our approach: Approximation ...

Ojakian, Campagnolo Characterizing Computable Analysis

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Basic Background Our model and our result Discussion of the Proof Conclusion

The “Main Step”

CR ⊆ UMUk(LIM) By Turing Machine simulation (Bournez and Hainry 2006). Difficulties with this approach ... Forced into unnecessary restrictions. Appears less general. Our approach: Approximation ...

Ojakian, Campagnolo Characterizing Computable Analysis

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Basic Background Our model and our result Discussion of the Proof Conclusion

The “Main Step”

CR ⊆ UMUk(LIM) By Turing Machine simulation (Bournez and Hainry 2006). Difficulties with this approach ... Forced into unnecessary restrictions. Appears less general. Our approach: Approximation ...

Ojakian, Campagnolo Characterizing Computable Analysis

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Basic Background Our model and our result Discussion of the Proof Conclusion

The “Main Step”

CR ⊆ UMUk(LIM) By Turing Machine simulation (Bournez and Hainry 2006). Difficulties with this approach ... Forced into unnecessary restrictions. Appears less general. Our approach: Approximation ...

Ojakian, Campagnolo Characterizing Computable Analysis

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Basic Background Our model and our result Discussion of the Proof Conclusion

The “Main Step”

CR ⊆ UMUk(LIM) By Turing Machine simulation (Bournez and Hainry 2006). Difficulties with this approach ... Forced into unnecessary restrictions. Appears less general. Our approach: Approximation ...

Ojakian, Campagnolo Characterizing Computable Analysis

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Basic Background Our model and our result Discussion of the Proof Conclusion

Approximation

Definition f(¯ x) t f ∗(¯ x, t) means: |f(¯ x) − f ∗(¯ x, t)| < 1 t For classes of functions A and B, A B means: For any f ∈ A there is f ∗ ∈ B such that f f ∗. Goal: CR UMUk Implies: CR ⊆ UMUk(LIM)

Ojakian, Campagnolo Characterizing Computable Analysis

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Basic Background Our model and our result Discussion of the Proof Conclusion

Approximation

Definition f(¯ x) t f ∗(¯ x, t) means: |f(¯ x) − f ∗(¯ x, t)| < 1 t For classes of functions A and B, A B means: For any f ∈ A there is f ∗ ∈ B such that f f ∗. Goal: CR UMUk Implies: CR ⊆ UMUk(LIM)

Ojakian, Campagnolo Characterizing Computable Analysis

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Basic Background Our model and our result Discussion of the Proof Conclusion

Breaking up the proof

Lemma (Transitivity) Suppose A, B, and C are classes of functions and suppose C is

  • nice. Then A B and B C implies A C.

Definition Let CQ be {f|Q | f ∈ CR} CQ dCQ dMUQ MUQ UMUk Transitivity: CQ UMUk Implies: CR UMUk

Ojakian, Campagnolo Characterizing Computable Analysis

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Basic Background Our model and our result Discussion of the Proof Conclusion

Breaking up the proof

Lemma (Transitivity) Suppose A, B, and C are classes of functions and suppose C is

  • nice. Then A B and B C implies A C.

Definition Let CQ be {f|Q | f ∈ CR} CQ dCQ dMUQ MUQ UMUk Transitivity: CQ UMUk Implies: CR UMUk

Ojakian, Campagnolo Characterizing Computable Analysis

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Basic Background Our model and our result Discussion of the Proof Conclusion

Breaking up the proof

Lemma (Transitivity) Suppose A, B, and C are classes of functions and suppose C is

  • nice. Then A B and B C implies A C.

Definition Let CQ be {f|Q | f ∈ CR} CQ dCQ dMUQ MUQ UMUk Transitivity: CQ UMUk Implies: CR UMUk

Ojakian, Campagnolo Characterizing Computable Analysis

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Basic Background Our model and our result Discussion of the Proof Conclusion

Breaking up the proof

Lemma (Transitivity) Suppose A, B, and C are classes of functions and suppose C is

  • nice. Then A B and B C implies A C.

Definition Let CQ be {f|Q | f ∈ CR} CQ dCQ dMUQ MUQ UMUk Transitivity: CQ UMUk Implies: CR UMUk

Ojakian, Campagnolo Characterizing Computable Analysis

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Basic Background Our model and our result Discussion of the Proof Conclusion

Breaking up the proof

Lemma (Transitivity) Suppose A, B, and C are classes of functions and suppose C is

  • nice. Then A B and B C implies A C.

Definition Let CQ be {f|Q | f ∈ CR} CQ dCQ dMUQ MUQ UMUk Transitivity: CQ UMUk Implies: CR UMUk

Ojakian, Campagnolo Characterizing Computable Analysis

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Basic Background Our model and our result Discussion of the Proof Conclusion

A function algebra for the computable functions

Definition Let MUN be FA[0, 1, +, . , P; comp, , , MU]. Definition The operation MU. Input: f(t, ¯ x) (over N) satisfying: For each ¯ x, there is a unique T ≥ 1 such that f(T, ¯ x) = 0, and otherwise f(t, ¯ x) = −1, if t < T; 1, if t > T. Output: The function g(¯ x) = the unique T such that f(T, ¯ x) = 0.

Ojakian, Campagnolo Characterizing Computable Analysis

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Basic Background Our model and our result Discussion of the Proof Conclusion

A function algebra for the computable functions

Definition Let MUN be FA[0, 1, +, . , P; comp, , , MU]. Definition The operation MU. Input: f(t, ¯ x) (over N) satisfying: For each ¯ x, there is a unique T ≥ 1 such that f(T, ¯ x) = 0, and otherwise f(t, ¯ x) = −1, if t < T; 1, if t > T. Output: The function g(¯ x) = the unique T such that f(T, ¯ x) = 0.

Ojakian, Campagnolo Characterizing Computable Analysis

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A function algebra over Q

Definition Let MUQ be FA[0, 1, −1, +, P, ∗, div, θ1; comp,

Q, Q, MUQ, LinQ].

Definition Suppose OP takes a function f : Nk → N and returns a function g : Nm → N. Then OPQ is the following operation:

1

OPQ takes as input f : Qk → Q such that f|N : Nk → N.

2

OPQ then applies OP to f|N to get some g : Nm → N.

3

OPQ outputs LinQ(g).

Ojakian, Campagnolo Characterizing Computable Analysis

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Basic Background Our model and our result Discussion of the Proof Conclusion

A function algebra over Q

Definition Let MUQ be FA[0, 1, −1, +, P, ∗, div, θ1; comp,

Q, Q, MUQ, LinQ].

Definition Suppose OP takes a function f : Nk → N and returns a function g : Nm → N. Then OPQ is the following operation:

1

OPQ takes as input f : Qk → Q such that f|N : Nk → N.

2

OPQ then applies OP to f|N to get some g : Nm → N.

3

OPQ outputs LinQ(g).

Ojakian, Campagnolo Characterizing Computable Analysis

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Basic Background Our model and our result Discussion of the Proof Conclusion

A function algebra over Q

Definition Let MUQ be FA[0, 1, −1, +, P, ∗, div, θ1; comp,

Q, Q, MUQ, LinQ].

Definition Suppose OP takes a function f : Nk → N and returns a function g : Nm → N. Then OPQ is the following operation:

1

OPQ takes as input f : Qk → Q such that f|N : Nk → N.

2

OPQ then applies OP to f|N to get some g : Nm → N.

3

OPQ outputs LinQ(g).

Ojakian, Campagnolo Characterizing Computable Analysis

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Basic Background Our model and our result Discussion of the Proof Conclusion

A function algebra over Q

Definition Let MUQ be FA[0, 1, −1, +, P, ∗, div, θ1; comp,

Q, Q, MUQ, LinQ].

Definition Suppose OP takes a function f : Nk → N and returns a function g : Nm → N. Then OPQ is the following operation:

1

OPQ takes as input f : Qk → Q such that f|N : Nk → N.

2

OPQ then applies OP to f|N to get some g : Nm → N.

3

OPQ outputs LinQ(g).

Ojakian, Campagnolo Characterizing Computable Analysis

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Basic Background Our model and our result Discussion of the Proof Conclusion

A function algebra over Q

Definition Let MUQ be FA[0, 1, −1, +, P, ∗, div, θ1; comp,

Q, Q, MUQ, LinQ].

Definition Suppose OP takes a function f : Nk → N and returns a function g : Nm → N. Then OPQ is the following operation:

1

OPQ takes as input f : Qk → Q such that f|N : Nk → N.

2

OPQ then applies OP to f|N to get some g : Nm → N.

3

OPQ outputs LinQ(g).

Ojakian, Campagnolo Characterizing Computable Analysis

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The inductive proof

Proof proceeds inductively on the function algebra MUQ: Show the basic functions of MUQ are approximated by UMUk. Show that the operations of MUQ preserve the approximation: For f ∈ MUQ and f ∗ ∈ UMUk, suppose g = OP(f), and f f ∗. Then there is g∗ ∈ UMUk such that g g∗.

Ojakian, Campagnolo Characterizing Computable Analysis

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The inductive proof

Proof proceeds inductively on the function algebra MUQ: Show the basic functions of MUQ are approximated by UMUk. Show that the operations of MUQ preserve the approximation: For f ∈ MUQ and f ∗ ∈ UMUk, suppose g = OP(f), and f f ∗. Then there is g∗ ∈ UMUk such that g g∗.

Ojakian, Campagnolo Characterizing Computable Analysis

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Basic Functions

0, 1, −1, +, P, ∗ θ1, div ...

Ojakian, Campagnolo Characterizing Computable Analysis

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Composition

Generic and easy, but uses “modulus assumption” significantly .......

Ojakian, Campagnolo Characterizing Computable Analysis

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Linearization

Generic, but involved .......

Ojakian, Campagnolo Characterizing Computable Analysis

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Basic Background Our model and our result Discussion of the Proof Conclusion

Sums and Products

Specific, but uses earlier results .......

Ojakian, Campagnolo Characterizing Computable Analysis

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Search Operation MU

Specific and significant ....... Key Points: Use the particular shape. Need to process the approximation function. Can approximate linearization in well-behaved way.

Ojakian, Campagnolo Characterizing Computable Analysis

slide-66
SLIDE 66

Basic Background Our model and our result Discussion of the Proof Conclusion

Search Operation MU

Specific and significant ....... Key Points: Use the particular shape. Need to process the approximation function. Can approximate linearization in well-behaved way.

Ojakian, Campagnolo Characterizing Computable Analysis

slide-67
SLIDE 67

Basic Background Our model and our result Discussion of the Proof Conclusion

Search Operation MU

Specific and significant ....... Key Points: Use the particular shape. Need to process the approximation function. Can approximate linearization in well-behaved way.

Ojakian, Campagnolo Characterizing Computable Analysis

slide-68
SLIDE 68

Basic Background Our model and our result Discussion of the Proof Conclusion

Search Operation MU

Specific and significant ....... Key Points: Use the particular shape. Need to process the approximation function. Can approximate linearization in well-behaved way.

Ojakian, Campagnolo Characterizing Computable Analysis

slide-69
SLIDE 69

Basic Background Our model and our result Discussion of the Proof Conclusion

Outline

1

Basic Background

2

Our model and our result

3

Discussion of the Proof

4

Conclusion

Ojakian, Campagnolo Characterizing Computable Analysis

slide-70
SLIDE 70

Basic Background Our model and our result Discussion of the Proof Conclusion

Conclusion

We have a new characterization of Computable Analysis. Further improvements?: Show it is useful. Simplify to “analytic version” by removing θk. Does CR = ODE⋆(LIM)? Develop a general theory.

Ojakian, Campagnolo Characterizing Computable Analysis

slide-71
SLIDE 71

Basic Background Our model and our result Discussion of the Proof Conclusion

Conclusion

We have a new characterization of Computable Analysis. Further improvements?: Show it is useful. Simplify to “analytic version” by removing θk. Does CR = ODE⋆(LIM)? Develop a general theory.

Ojakian, Campagnolo Characterizing Computable Analysis

slide-72
SLIDE 72

Basic Background Our model and our result Discussion of the Proof Conclusion

Conclusion

We have a new characterization of Computable Analysis. Further improvements?: Show it is useful. Simplify to “analytic version” by removing θk. Does CR = ODE⋆(LIM)? Develop a general theory.

Ojakian, Campagnolo Characterizing Computable Analysis

slide-73
SLIDE 73

Basic Background Our model and our result Discussion of the Proof Conclusion

Conclusion

We have a new characterization of Computable Analysis. Further improvements?: Show it is useful. Simplify to “analytic version” by removing θk. Does CR = ODE⋆(LIM)? Develop a general theory.

Ojakian, Campagnolo Characterizing Computable Analysis

slide-74
SLIDE 74

Basic Background Our model and our result Discussion of the Proof Conclusion

Conclusion

We have a new characterization of Computable Analysis. Further improvements?: Show it is useful. Simplify to “analytic version” by removing θk. Does CR = ODE⋆(LIM)? Develop a general theory.

Ojakian, Campagnolo Characterizing Computable Analysis