Cobham Recursive Set Functions Moritz M¨ uller Kurt G¨ odel Research Center, Vienna, Austria. joint with A. Beckmann, S. Buss, S.-D. Friedman and N. Thapen BASICS 2015 Summer School Logic Summer School in China 2015 Zhejiang Normal University
Computations on arbitrary sets There are many equivalent definitions of the class of recursive func- tions on the natural numbers. Different definitions have different uses while the equivalence of all the notions provides evidence for Church’s thesis, the thesis that the concept of recursive function is the most reasonable explication of our intuitive notion of effectively calculable function. As the various definitions are lifted to domains other than the integers (e. g., admissible sets) some of the equiva- lences break down. This break-down provides us with a laboratory for the study of recursion theory. Barwise, 1975
Computations on arbitrary sets There are many equivalent definitions of the class of recursive func- tions on the natural numbers. Different definitions have different uses while the equivalence of all the notions provides evidence for Church’s thesis, the thesis that the concept of recursive function is the most reasonable explication of our intuitive notion of effectively calculable function. As the various definitions are lifted to domains other than the integers (e. g., admissible sets) some of the equiva- lences break down. This break-down provides us with a laboratory for the study of recursion theory. Barwise, 1975 A computable function over N is one Recursion theoretic view obtainable from certain initial functions by means of composition, primitive recursion and the µ -operator. Definability theoretic view Σ 1 -definable in the language of arithmetic.
Primitive recursive set functions (Jensen, Karp 1971) are obtained from initial functions constant 0 = ∅ projections pair x, y �→ { x, y } union x �→ � x conditional cond ∈ ( x, y, u, v ) := if u ∈ v then x else y by composition and ǫ -recursion if g ( z, y, � x ) is PRSF, then so is f ( y, � x ) = g ( { f ( u, � x ) : u ∈ y } , y, � x )
Primitive recursive set functions (Jensen, Karp 1971) are obtained from initial functions constant 0 = ∅ projections pair x, y �→ { x, y } union x �→ � x conditional cond ∈ ( x, y, u, v ) := if u ∈ v then x else y by composition and ǫ -recursion if g ( z, y, � x ) is PRSF, then so is f ( y, � x ) = g ( { f ( u, � x ) : u ∈ y } , y, � x ) Generalizes primitive recursive computations to arbitrary sets Goal Similarly generalize polynomial time computations to arbitrary sets Need Recursion theoretic definition of PTIME
Cobham’s definition of polynomial time 1965 The polynomial time function over N are those obtained from initial functions constant 0, projections, successors s 0 ( x ) = 2 x and s 1 ( x ) = 2 x + 1 x # y = 2 | x | · | y | smash where | x | = ⌈ log( x + 1) ⌉ by composition and limited recursion on notation if h, g 0 , g 1 , t are polynomial time, then so is f (0 , � x ) = h ( � x ) where b ∈ { 0 , 1 } f ( sb ( y ) , � x ) = gb ( f ( y, � x ) , y, � x )
Cobham’s definition of polynomial time 1965 The polynomial time function over N are those obtained from initial functions constant 0, projections, successors s 0 ( x ) = 2 x and s 1 ( x ) = 2 x + 1 x # y = 2 | x | · | y | smash where | x | = ⌈ log( x + 1) ⌉ by composition and limited recursion on notation if h, g 0 , g 1 , t are polynomial time, then so is f (0 , � x ) = h ( � x ) where b ∈ { 0 , 1 } f ( sb ( y ) , � x ) = gb ( f ( y, � x ) , y, � x ) provided f ( y, � x ) ≤ t ( y, � x ) for all y, � x .
Cobham’s definition of polynomial time 1965 The polynomial time function over N are those obtained from initial functions constant 0, projections, successors s 0 ( x ) = 2 x and s 1 ( x ) = 2 x + 1 x # y = 2 | x | · | y | smash where | x | = ⌈ log( x + 1) ⌉ by composition and limited recursion on notation if h, g 0 , g 1 , t are polynomial time, then so is f (0 , � x ) = h ( � x ) f ( sb ( y ) , � x ) = gb ( f ( y, � x ) , y, � x ) where b ∈ { 0 , 1 } provided f ( y, � x ) ≤ t ( y, � x ) for all y, � x . Equivalent proviso: • t a #-term: built from variables, 1 = s 1 (0) and #. • | f ( y, x 1 , x 2 . . . ) | ≤ p ( | y | , | x 1 | , | x 2 | . . . ) for some polynomial p
Set composition and set smash � y if x = 0 Set composition x ⊙ y := { u ⊙ y : u ∈ x } if x � = 0
Set composition and set smash � y if x = 0 Set composition x ⊙ y := { u ⊙ y : u ∈ x } if x � = 0 Set smash x # y := y ⊙ { u # y : u ∈ x }
Set composition and set smash � y if x = 0 Set composition x ⊙ y := { u ⊙ y : u ∈ x } if x � = 0 Set smash x # y := y ⊙ { u # y : u ∈ x } #-term t ( � x ) built from variables � x , 1= { 0 } , ⊙ and # • There are polynomials p, q such that for all � x rk( t ( x 1 , x 2 . . . )) ≤ p (rk( x 1 ) , rk( x 2 ) . . . ) | tc( t ( x 1 , x 2 . . . )) | ≤ q ( | tc( x 1 ) | , | tc( x 2 ) | . . . )
Set composition and set smash � y if x = 0 Set composition x ⊙ y := { u ⊙ y : u ∈ x } if x � = 0 Set smash x # y := y ⊙ { u # y : u ∈ x } #-term t ( � x ) built from variables � x , 1= { 0 } , ⊙ and # • There are polynomials p, q such that for all � x rk( t ( x 1 , x 2 . . . )) ≤ p (rk( x 1 ) , rk( x 2 ) . . . ) | tc( t ( x 1 , x 2 . . . )) | ≤ q ( | tc( x 1 ) | , | tc( x 2 ) | . . . ) Intuition #-terms play the role of polynomial length bounds. Intuition consider only hereditarily finite x : finite Mostowski graph, one can compute f ( x ) = g ( { f ( u ) : u ∈ x } , x ) with oracle g in parallel time ≈ rk( x ) and total work ≈ | tc( x ) | .
Bounding relation � A single-valued embedding of x into y is τ : tc( x ) → tc( y ) st if u � = v , then τ ( u ) � = τ ( v ) if u ∈ v , then τ ( u ) ∈ tc( τ ( v )) Example if x ⊆ y , then the identity is such an embedding.
Bounding relation � A single-valued embedding of x into y is τ : tc( x ) → tc( y ) st if u � = v , then τ ( u ) � = τ ( v ) if u ∈ v , then τ ( u ) ∈ tc( τ ( v )) Example if x ⊆ y , then the identity is such an embedding. A (multi-valued) embedding of x into y is τ : tc( x ) → P (tc( y )) \ { 0 } st if u � = v , then τ ( u ) ∩ τ ( v ) = 0 if u ∈ v and v ′ ∈ τ ( v ), then there exists u ′ ∈ τ ( u ) such that u ′ ∈ tc( v ′ )
Bounding relation � A single-valued embedding of x into y is τ : tc( x ) → tc( y ) st if u � = v , then τ ( u ) � = τ ( v ) if u ∈ v , then τ ( u ) ∈ tc( τ ( v )) Example if x ⊆ y , then the identity is such an embedding. A (multi-valued) embedding of x into y is τ : tc( x ) → P (tc( y )) \ { 0 } st if u � = v , then τ ( u ) ∩ τ ( v ) = 0 if u ∈ v and v ′ ∈ τ ( v ), then there exists u ′ ∈ τ ( u ) such that u ′ ∈ tc( v ′ ) • Then rk( x ) ≤ rk( y ) and | tc( x ) | ≤ | tc( y ) | . Intuition Then x is structurally no more complex than y .
Cobham recursive set functions are obtained from initial functions constant 0, projections, pair { x, y } , union � x , cond ∈ ( x, y, u, v ), smash x # y by composition and Cobham recursion if g, τ, t are CRSF, then so is f ( y, � x ) = g ( { f ( u, � x ) : u ∈ y } , y, � x )
Cobham recursive set functions are obtained from initial functions constant 0, projections, pair { x, y } , union � x , cond ∈ ( x, y, u, v ), smash x # y by composition and Cobham recursion if g, τ, t are CRSF, then so is f ( y, � x ) = g ( { f ( u, � x ) : u ∈ y } , y, � x ) provided τ ( · , y, � x ) : f ( y, � x ) � t ( y, � x ) for all y, � x . (i.e. u �→ τ ( u, y, � x ) is an embedding of f ( y, � x ) into t ( y, � x ))
Cobham recursive set functions are obtained from initial functions constant 0, projections, pair { x, y } , union � x , cond ∈ ( x, y, u, v ), smash x # y by composition and Cobham recursion if g, τ, t are CRSF, then so is f ( y, � x ) = g ( { f ( u, � x ) : u ∈ y } , y, � x ) provided τ ( · , y, � x ) : f ( y, � x ) � t ( y, � x ) for all y, � x . (i.e. u �→ τ ( u, y, � x ) is an embedding of f ( y, � x ) into t ( y, � x )) • equivalent: demand t to be a #-term. • equivalent: allow “impredicative” τ ( · , y, � x, f ( y, � x )).
Bootstrapping CRSF • Bounded replacement if g, τ, t are CRSF, then so is f ( x ) = { g ( u, x ) : u ∈ x } provided τ ( · , x ) : f ( x ) � t ( x )
Bootstrapping CRSF • Bounded replacement if g, τ, t are CRSF, then so is f ( x ) = { g ( u, x ) : u ∈ x } provided τ ( · , x ) : f ( x ) � t ( x ) • Separation if g is CRSF, then so is f ( x ) = { u ∈ x : g ( u ) � = 0 } ;
Bootstrapping CRSF • Bounded replacement if g, τ, t are CRSF, then so is f ( x ) = { g ( u, x ) : u ∈ x } provided τ ( · , x ) : f ( x ) � t ( x ) • Separation if g is CRSF, then so is f ( x ) = { u ∈ x : g ( u ) � = 0 } ; h ( u ) := if g ( u ) ∈ { 0 } then 0 else { u } Bounded replacement gives f ( x ) = { � h ( u ) : u ∈ x } Proviso satisfied since f ( x ) ⊆ x . �
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