Recursive Analysis And Real Recursive Functions Emmanuel Hainry - PowerPoint PPT Presentation

Recursive Analysis And Real Recursive Functions Emmanuel Hainry Joint work with Olivier Bournez LORIA/INPL, Nancy, France Feb. 13, 2006 1/29

Introduction Definitions Recursive functions Limit operator Conclusion 1. Introduction Context Purpose 2. Definitions and known results Discrete computation Computable analysis Analytic classes 3. From elementary to recursive functions A real µ A class characterizing R ec ( N ) 4. Limit operator Definition of LIM Results 5. Conclusion 2/29

Introduction Definitions Recursive functions Limit operator Conclusion Computing on the reals ◮ There are several machines, models for computing over the reals ◮ Differential Analyzer ◮ General Purpose Analog Computer (GPAC) ◮ Computable Analysis (or Recursive Analysis ) ◮ R -recursive functions (or real recursive functions ) ◮ Polynomial Differential Equations (PolyDE) 3/29

Introduction Definitions Recursive functions Limit operator Conclusion Discrete Case ◮ There are several models for computation over integers ◮ Computable functions ◮ Turing machines ◮ λ -calculus ◮ ... ◮ But those models are “equivalent”. Church-Turing thesis All reasonable discrete models of computation compute exactly the same functions. 4/29

Introduction Definitions Recursive functions Limit operator Conclusion Linking models of “real” computation The models of computable analysis and R -recursive functions deal with similar functions but lack real relations between their classes. Investigating such links can help giving an analog characterization of what may be considered reasonable in computation over the reals. A step towards a Church Thesis for computation over the reals? 5/29

Introduction Definitions Recursive functions Limit operator Conclusion Purpose of this talk We are going to present ◮ some background results: characterizing elementary functions and all levels of the Grzegorczyk hierarchy as restrictions of real recursive functions; ◮ a zero-finding operator to extend this result and obtain a characterization of recursive functions; ◮ a limit operator that bridges the step from discrete functions to real recursive functions. 6/29

Introduction Definitions Recursive functions Limit operator Conclusion Recursive and Sub-recursive functions R ec = [0 , S , U ; COMP , REC , MU ] PR = [0 , U , S ; COMP , REC ] E = [0 , S , U , + , ⊖ ; COMP , BSUM , BPROD ] E n = [0 , S , U , + , ⊖ , E n − 1 ; COMP , BSUM , BPROD ] With E 2 ( x ) = 2 x E n +1 ( x ) = E [ x ] f [0] ( x ) = x n (1) for n ≥ 2 with f [ d +1] ( x ) = f ( f [ d ] ( x )) 7/29

Introduction Definitions Recursive functions Limit operator Conclusion Computable analysis: representing a real number Definition [Representation of a real] A real number is represented by a sequence of integers: Let x ∈ R . There exists ( x n ) ∈ Q N such that ∀ i , | x − x i | < 1 2 i . Let ν Q be a representation of the rational numbers. ( m n ) ∈ N N represents x iff ∀ i , | x − ν Q ( m i ) | < 1 2 i . Definition [Notation] We will write ( m n ) � x if the sequence ( m n ) represents x . 8/29

Introduction Definitions Recursive functions Limit operator Conclusion Computable functions [Weihrauch00,Ko91] Definition [Computable functions] A function f : [ a , b ] → R with a , b ∈ Q is computable (resp: elementarily computable) iff there exists φ : N N → N N recursive (resp: elementary) such that ∀ X � x , ( φ ( X )) � f ( x ) . 9/29

Introduction Definitions Recursive functions Limit operator Conclusion Complexity for functions from recursive analysis Definition [Complexity] Let f : G → R a computable function. t : G × N → N is a complexity bound for f if there exists φ computing f such that ∀ x ∈ G , ∀ n > 0 , t ( x , n ) ≥ sup { T (( φ ( X )) n ) } X � x where T ((( φ ( X )) n )) is the time taken to compute ( φ ( X )) n . Definition [Uniform Complexity] t ′ : N → N is a uniform complexity bound if ∀ x ∈ G ∩ [ − 2 n , 2 n ] , t ( x , n ) ≤ t ′ ( n ) In other words, the complexity of a function is the number t ( n ) of bits of the inputs that need to be read to get n bits of the output. 10/29

Introduction Definitions Recursive functions Limit operator Conclusion Real recursive functions [Moore96] Definition [ G ] G = [0 , 1 , U ; COMP , INT , MU ] ◮ INT : given g , h , INT ( g , h ) is the solution of the differential f ( − → g ( − → � x , 0) = x ) equation ∂ y ( − → h ( − → x , y , f ( − → ∂ f x , y ) = x , y )) ◮ MU : given f : D ⊂ R n +1 → R , � y − = sup y ≤ 0 { f ( − → x , y ) = 0 } if | y − | ≤ | y + | MU ( f ) : − → x �→ y + = inf y > 0 { f ( − → x , y ) = 0 } if | y + | < | y − | 11/29

Introduction Definitions Recursive functions Limit operator Conclusion Real recursive functions [Campagnolo01] Definition [ L ] L = [0 , 1 , − 1 , π, U , θ 3 ; COMP , CLI ] With ◮ U : projections � R → R ◮ θ 3 : max(0 , x 3 ) x �→ ◮ COMP : composition ◮ CLI : given g , h , c such that h ′ bounded by c . f = CLI ( g , h , c ) is the maximal solution of f ( − → g ( − → x , 0) = x ) ∂ y ( − → h ( − → x , y ) f ( − → ∂ f x , y ) = x , y ) 12/29

Introduction Definitions Recursive functions Limit operator Conclusion Properties of L Proposition [Campagnolo] All functions from L are continuous, defined everywhere and of class C 2 . For a class F of functions R → R , DP ( F ) is the set of functions N → N that have an extension in F . Proposition [Campagnolo] DP ( L ) = E Where E = [0 , S , U , ⊖ ; COMP , BSUM , BPROD ] is the class of discrete elementary functions. 13/29

Introduction Definitions Recursive functions Limit operator Conclusion What about recursive functions? This result gives a characterization of E (and has been extended to all levels of the Grzegorczyk hierarchy). We will now present an operator that will extend the discrete µ . 14/29

Introduction Definitions Recursive functions Limit operator Conclusion A real µ operator Remark: A naive “return the smallest root” operator yields unwanted functions (see [Moore96]). Definition [ UMU ] Given f : D × I ⊂ R k +1 → R differentiable such that: ◮ ∀− → x : y �→ f ( − → x ∈ D , the function g − x , y ) is non decreasing, → x ∈ I ◦ , ◮ g − x has a unique root y − → → ∂ y ( − → ◮ ∂ f x , y − x ) > 0. → � R k − → R UMU ( f ) = − → y such that f ( − → �→ x , y ) = 0 x Proposition UMU preserves C 2 . 15/29

Introduction Definitions Recursive functions Limit operator Conclusion H = L + UMU Definition [ H ] H = [0 , 1 , U , θ 3 ; COMP , CLI , UMU ] Proposition L ⊂ H Proof: ◮ − 1 = UMU ( x �→ x + 1) 1 x , y �→ (1 + x 2 ) y − 1 ◮ x �→ � � 1+ x 2 = UMU ; 1 arctan(0) = 0 and arctan ′ ( x ) = 1+ x 2 ; π = 4 arctan(1) 16/29

Introduction Definitions Recursive functions Limit operator Conclusion Result: DP ( H ) = R ec ( N ) Theorem DP ( H ) = R ec ( N ) Where R ec ( N ) denotes the set of discrete partial recursive functions. Proof: we have to demonstrate both directions. ◮ DP ( H ) ⊂ R ec ( N ) comes from the fact that UMU preserves computability (in the sense of recursive analysis). ◮ R ec ( N ) ⊂ DP ( H ) can be proven using a normal form theorem in R ec ( N ) and translating the discrete µ into our UMU . 17/29

Introduction Definitions Recursive functions Limit operator Conclusion Proof of R ec ( N ) ⊂ DP ( H ) Let φ ∈ R ec ( N ). There exists χ and ψ elementary such that φ = χ ◦ µ ( ψ ). � σ ( m , n ) = ψ ( m , z ) z < n κ ( m , n ) = 1 ⊖ (1 ⊖ (1 ⊖ σ ( m , n ) + σ ( m , n + 1))) ι ( m , n ) = 1 ⊖ κ + 2 × (1 ⊖ σ ) ι ∈ E has a single one and it coincides with ψ ’s first zero. We then extend ι into a function i from L and use some tricks so that UMU can be applied to i − 1. Finally, with h an extension to L of χ , COMP ( UMU ( i − 1) , h ) is an extension to H of φ . 18/29

Introduction Definitions Recursive functions Limit operator Conclusion Consequences Corollary L � H Corollary [“Normal Form”] A function from H can be written with at most 3 nested UMU . We may need 2 UMU to obtain π and − 1. The other UMU comes from the simulation of the discrete µ . 19/29

Introduction Definitions Recursive functions Limit operator Conclusion Characterizing computable analysis classes Those results give analog characterizations of E and R ec ( N ). With a limit operator, we can extend those characterizations to obtain characterizations of E ( R ) and R ec ( R ). 20/29

Introduction Definitions Recursive functions Limit operator Conclusion Operator LIM Definition [ LIM schema] Given f : R × D ⊂ R k +1 → R l , If there are K : D → R and β : D → R a polynomial such that x � , � ∂ f ∀− → x , ∀ t ≥ �− → ∂ t ( t , − → x ) � ≤ K ( − → x ) exp( − t β ( − → x )) . Then, F = LIM ( f , K , β ) is defined by F ( − → x ) = lim t →∞ f ( t , − → x ) provided it is C 2 . 21/29

Introduction Definitions Recursive functions Limit operator Conclusion Theorems We will write C ∗ where C = [ F ; O ] to denote the class [ F ; O , LIM ]. Theorem For functions of class C 2 defined on a compact domain, L ∗ = E ( R ) . Theorem For functions of class C 2 defined on a compact domain, H ∗ = R ec ( R ) . 22/29