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An Analog Characterization of Elementarily Computable Functions over the Real Numbers Olivier Bournez and Emmanuel Hainry LORIA/INRIA, Nancy, France April 14, 2003 An Analog Characterization of Elementarily Computable Functions over the Real Numbers

Introduction Continuous models Extension of L Conclusion 1. Introduction The discrete world The continuous world 2. Continuous models Recursive analysis Class G Class L 3. Extension of L New schemata Properties of L ∗ Characterization of E ( R ) 4. Conclusion An Analog Characterization of Elementarily Computable Functions over the Real Numbers

Introduction Continuous models Extension of L Conclusion The discrete world The continuous world Discrete, Continuous ◮ Discrete world: computing over N in discrete time. (Turing machines, automata...) ◮ Continuous world: computing over R ◮ in discrete time. (Recursive analysis, BSS machines) ◮ in continuous time. (General Purpose Analog Computer, Neural networks...) An Analog Characterization of Elementarily Computable Functions over the Real Numbers

Introduction Continuous models Extension of L Conclusion The discrete world The continuous world Discrete World Church’s thesis: All reasonable discrete computational models compute the same functions. Turing machines, as well as 2-stack automata compute recursive functions ( R ec = [0 , S , U ; COMP , REC , MU ]). An Analog Characterization of Elementarily Computable Functions over the Real Numbers

Introduction Continuous models Extension of L Conclusion The discrete world The continuous world Sub-recursive functions E = [0 , S , U , + , ⊖ ; COMP , BSUM , BPROD ] E n = [0 , S , U , + , ⊖ , E n − 1 ; COMP , BSUM , BPROD ] PR = [0 , U , S ; COMP , REC ] With E 0 ( x , y ) = x + y E 1 ( x , y ) = ( x + 1) × ( y + 1) 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 )) An Analog Characterization of Elementarily Computable Functions over the Real Numbers

Introduction Continuous models Extension of L Conclusion The discrete world The continuous world Continuous World Several models: ◮ Recursive analysis ◮ GPAC ◮ R -recursive functions ◮ Optical models ◮ ... But no equivalent of Church thesis. An Analog Characterization of Elementarily Computable Functions over the Real Numbers

Introduction Continuous models Extension of L Conclusion Recursive analysis Class G Class L Recursive analysis: type 2 machines A tape represents a real number: Let ν Q be a representation of the rational numbers. ( x n ) � x iff ∀ i , | x − ν Q ( x i ) | < 1 2 i M behaves like a Turing Machine Read-only one-way input tapes Write-only one-way output tape. An Analog Characterization of Elementarily Computable Functions over the Real Numbers

Introduction Continuous models Extension of L Conclusion Recursive analysis Class G Class L Elementarily computable functions Definition (Elementarily computable functions on compact domains) A function f : [ a , b ] → R with a , b ∈ Q is elementarily computable iff there exists φ : N N → N N elementary such that ∀ X � x , ( φ ( X )) � f ( x ) . Definition (Elementarily computable functions on other domains) A function f : [ a , b ) → R with a , b ∈ Q is elementarily computable iff there exists φ : N N × N → N N elementary such that ∀ M < b , ∀ x ∈ [ a , M ] , ∀ X � x , ( φ ( X , M )) � f ( x ) . An Analog Characterization of Elementarily Computable Functions over the Real Numbers

Introduction Continuous models Extension of L Conclusion Recursive analysis Class G Class L Class G ([Moore96]) R ec G 0 0 1 S U U Comp Comp REC : f , g �→ h INT : f , g �→ h h ( x , 0) = f ( x ) h ( x , 0) = f ( x ) h ( x , n + 1) = g ( x , n , h ( x , n )) ∂ y h ( x , y ) = g ( x , y , h ( x , y )) MU : x , f �→ min { y ; f ( x , y ) = 0 } MU : x , f �→ inf { y | f ( x , y ) = 0 } An Analog Characterization of Elementarily Computable Functions over the Real Numbers

Introduction Continuous models Extension of L Conclusion Recursive analysis Class G Class L Troubles with G ◮ Not always well defined (0 × + ∞ = 0, integration on non integrable functions...) ◮ Contains bad functions ( χ Q which is nowhere continuous) ◮ Not physically reasonable (Zeno paradox ⇒ infinite energy required) An Analog Characterization of Elementarily Computable Functions over the Real Numbers

Introduction Continuous models Extension of L Conclusion Recursive analysis Class G Class L Class L ([Campagnolo00]) G L ′ 0 0 1 1, − 1, π U U Comp Comp INT : f , g �→ h LI : f , g �→ h h ( x , 0) = f ( x ) h ( x , 0) = f ( x ) ∂ y h ( x , y ) = g ( x , y , h ( x , y )) ∂ y h ( x , y ) = g ( x , y ) h ( x , y ) MU An Analog Characterization of Elementarily Computable Functions over the Real Numbers

Introduction Continuous models Extension of L Conclusion Recursive analysis Class G Class L Definition of L Definition ( θ 3 ) 1000 theta3(x) 800 � 0 if x < 0 600 θ 3 ( x ) = x 3 if x ≥ 0 400 200 0 -10 -5 0 5 10 Definition ( L ) L = [0 , 1 , − 1 , π, U , θ 3 ; COMP , LI ] An Analog Characterization of Elementarily Computable Functions over the Real Numbers

Introduction Continuous models Extension of L Conclusion Recursive analysis Class G Class L Properties of L Theorem (Campagnolo) L ⊂ E ( R ) Theorem (Campagnolo) E ⊂ DP ( L ) All discrete elementary functions have a real extension in L . An Analog Characterization of Elementarily Computable Functions over the Real Numbers

Introduction Continuous models Extension of L Conclusion Recursive analysis Class G Class L Definition of L n Definition ( ¯ E n ) exp 2 ( n ) = 2 n exp i +1 ( n ) = exp [ n ] i (1) ¯ E n is a monotonous real extension of exp n . Definition ( L n ) ¯ L n = [0 , 1 , − 1 , π, U , θ 3 , E n − 1 ; COMP , LI ] . An Analog Characterization of Elementarily Computable Functions over the Real Numbers

Introduction Continuous models Extension of L Conclusion Recursive analysis Class G Class L Properties of L n Theorem (Campagnolo) L n ⊂ E n ( R ) Theorem (Campagnolo) E n ⊂ DP ( L n ) All E n -computable functions over N have a real extension in L n . An Analog Characterization of Elementarily Computable Functions over the Real Numbers

Introduction Continuous models Extension of L Conclusion New schemata Properties of L ∗ Characterization of E ( R ) Observation L fails to characterize elementarily computable functions over the reals. Question: How can we characterize elementarily computable functions over the reals? Observation E ( R ) is not stable by composition. An Analog Characterization of Elementarily Computable Functions over the Real Numbers

Introduction Continuous models Extension of L Conclusion New schemata Properties of L ∗ Characterization of E ( R ) Definition of a weaker Composition schema Definition (COMP) COMP ( f , g ) is defined only if there exists a product of closed intervals C such that Range ( f ) ⊆ C ⊂ Domain ( g ). COMP ( f , g )( − → x ) = g ( f ( − → x )) . Remark: For total functions, this schema remains the classical one. An Analog Characterization of Elementarily Computable Functions over the Real Numbers

Introduction Continuous models Extension of L Conclusion New schemata Properties of L ∗ Characterization of E ( R ) Definition of a limit operator Definition (LIM) Let f : R × D → R and a polynomial β : D → R such that ∃ K such that ∀ t , x , � ∂ f ∂ t ( t , x ) � ≤ K exp( − t β ( x )) � ∂ 2 f ∂ t ∂ x ( t , x ) � ≤ K exp( − t β ( x )) Then, on an interval I ⊂ R where β ( x ) > 0, F = LIM ( f , β ) is defined by F ( x ) = t → + ∞ f ( t , x ) lim if this function is C 2 . An Analog Characterization of Elementarily Computable Functions over the Real Numbers

Introduction Continuous models Extension of L Conclusion New schemata Properties of L ∗ Characterization of E ( R ) New classes L ∗ = [0 , 1 , − 1 , U , θ 3 ; COMP , LI , LIM ] ¯ L ∗ n = [0 , 1 , − 1 , U , θ 3 , E n − 1 ; COMP , LI , LIM ] An Analog Characterization of Elementarily Computable Functions over the Real Numbers

Introduction Continuous models Extension of L Conclusion New schemata Properties of L ∗ Characterization of E ( R ) Basic properties of L ∗ � R > 0 → R ◮ 1 x : belongs to L ∗ : 1 �→ x x � 1 − exp( − tx )) if x � = 0 Let E = LI (0 , exp( − tx )). E ( t , x ) = x if x = 0 . t And 1 x = LIM ( E , x �→ x ). ◮ π ∈ L ∗ : 1 + x 2 ∈ L ∗ , 1 1+ x 2 ∈ L ∗ . 1 arctan = LI (0 , 1+ x 2 ) and π = 4 arctan(1). ◮ L � L ∗ An Analog Characterization of Elementarily Computable Functions over the Real Numbers

Introduction Continuous models Extension of L Conclusion New schemata Properties of L ∗ Characterization of E ( R ) Characterization of E ( R ) Proposition L ∗ ⊆ E ( R ) Proposition Let f a C 2 function defined on a compact domain, if f ∈ E ( R ), then f ∈ L ∗ . An Analog Characterization of Elementarily Computable Functions over the Real Numbers