Linear two-sorted higher type terms Linear two-sorted arithmetic Linear two-sorted arithmetic Helmut Schwichtenberg Mathematisches Institut, LMU, M¨ unchen Kyoto University, Japan, 8. May 2009 Helmut Schwichtenberg Linear two-sorted arithmetic
Linear two-sorted higher type terms Linear two-sorted arithmetic Feasible computation with higher types odel’s T (1958) “¨ G¨ Uber eine bisher noch nicht ben¨ utzte Erweiterung des finiten Standpunkts”: finitely typed λ -terms with structural recursion. LT (;) (Bellantoni, Niggl, S. 2000, 2002): restriction such that the definable functions are exactly the polynomial time computable ones. Here: Heyting Arithmetic = LA (;) G¨ odel’s T LT (;) Helmut Schwichtenberg Linear two-sorted arithmetic
Linear two-sorted higher type terms Linear two-sorted arithmetic Related work ◮ Hofmann (1998): similar results with a very different proof technique. Ramification concepts have been considered earlier e.g. by Simmons (1988), Bellantoni & Cook (1992), Leivant & Marion (1998, 2001), and Pfenning (2001). ◮ The “tiered” typed λ -calculi of Leivant & Marion (1993) depend heavily on different representations of data (as words and as Church-like abstraction terms), which is not necessary in the LT (;)-approach. ◮ Algebraic context semantics (Dal Lago 2006). ◮ Arai & Moser (2005), Beckmann & Weiermann (1996): Analysis (importance) of reduction strategies for R . ◮ Baillot & Terui (2004): DLAL. Similar results; they notice that one can drop ! and have ֒ → and → instead. Helmut Schwichtenberg Linear two-sorted arithmetic
Higher type input and output variables Linear two-sorted higher type terms Examples Linear two-sorted arithmetic Polynomial time Extending “Bellantoni/Cook” to higher types ◮ input → σ ) ( ρ ֒ n , ¯ x , ¯ y , . . . (BC: normal) Recurse on. Use many times ◮ output ( ρ → σ ) (BC: safe) a , p , x , y , . . . Cannot recurse on. Base type: use many times. Higher type: use at most once. Helmut Schwichtenberg Linear two-sorted arithmetic
Higher type input and output variables Linear two-sorted higher type terms Examples Linear two-sorted arithmetic Polynomial time Typing of recursion ◮ Recursion with value type τ has type R τ N : N ֒ → τ → ( N → τ → τ ) ֒ → τ with τ safe . ◮ A type is safe if it does not contain the input arrow ֒ → . Helmut Schwichtenberg Linear two-sorted arithmetic
Higher type input and output variables Linear two-sorted higher type terms Examples Linear two-sorted arithmetic Polynomial time Terms ◮ built from (typed) input/output variables and constants by introduction and elimination for ֒ → and → : x ρ | x ρ | C ρ ¯ (constant) | → σ | ( r ρ֒ → σ s ρ ) σ ( λ ¯ x ρ r σ ) ρ֒ ( s input term: FV ( s ) input) | ( λ x ρ r σ ) ρ → σ | ( r ρ → σ s ρ ) σ (higher type output vars in r , s distinct) , ◮ The restriction on output variables in r ρ → σ s ensures that every higher type output variable can occur at most once. ◮ A function f is definable in LT (;) if there is a closed term t f : N ։ . . . N ։ N ( ։ ∈ { ֒ → , →} ) denoting this function. Helmut Schwichtenberg Linear two-sorted arithmetic
Higher type input and output variables Linear two-sorted higher type terms Examples Linear two-sorted arithmetic Polynomial time Numerals Let W := L ( B ), and S 0 := λ v (ff :: v W ) , S 1 := λ v (tt :: v W ) . 1 := nil B , Particular lists are S i 1 ( . . . ( S i n 1 ) . . . ), called binary numerals (or words), denoted by v , w . . . . Helmut Schwichtenberg Linear two-sorted arithmetic
Higher type input and output variables Linear two-sorted higher type terms Examples Linear two-sorted arithmetic Polynomial time Polynomials ◮ ⊕ : W ֒ → W → W concatenates | | v | | bits onto w : 1 ⊕ w = S 0 w , ( S i v ) ⊕ w = S 0 ( v ⊕ w ) . The representing term is vS 0 λ , , p , w ( S 0 ( p W → W w )) w . v ⊕ w := R W → W ¯ ¯ ◮ ⊙ : W ֒ → W ֒ → W has output length | | v | | · | | w | | : v ⊙ 1 = v , v ⊙ ( S i w ) = v ⊕ ( v ⊙ w ) . The representing term is ¯ v ⊙ ¯ w := R W ¯ w ¯ v λ , , p (¯ v ⊕ p ) . Helmut Schwichtenberg Linear two-sorted arithmetic
Higher type input and output variables Linear two-sorted higher type terms Examples Linear two-sorted arithmetic Polynomial time A non-example: exponentiation ◮ Notice that ⊕ : W ֒ → W → W , and the value type for the recursion was W → W , which is safe. ◮ If we try to go on and define exponentiation from multiplication just as ⊙ was defined from ⊕ , we find that we cannot go ahead, because of the different typing ⊙ : W ֒ → W ֒ → W . Helmut Schwichtenberg Linear two-sorted arithmetic
Higher type input and output variables Linear two-sorted higher type terms Examples Linear two-sorted arithmetic Polynomial time Two recursions Consider D ( 1 ) := S 0 ( 1 ) , E ( 1 ) := 1 , D ( S i ( w )) := S 0 ( S 0 ( D ( w ))) , E ( S i ( w )) := D ( E ( w )) . The corresponding terms are D := λ ¯ w ( R W ¯ w ( S 0 1 ) λ , , p ( S 0 ( S 0 p ))) , E := λ ¯ w ( R W ¯ w 1 λ , , p ( Dp )) . Here D is legal, but E is not: the application Dp is not allowed. Helmut Schwichtenberg Linear two-sorted arithmetic
Higher type input and output variables Linear two-sorted higher type terms Examples Linear two-sorted arithmetic Polynomial time Recursion with parameter substitution Consider E ( 1 , v ) := S 0 ( v ) , E ( 1 ) := S 0 , or E ( S i ( w ) , v ) := E ( w , E ( w , v )) , E ( S i ( w )) := E ( w ) ◦ E ( w ) . The corresponding term wS 0 λ , , p , v ( p W → W ( pv ))) λ ¯ w ( R W → W ¯ does not satisfy the linearity condition: the higher type variable p occurs twice, and the typing of R requires p to be an output variable. Helmut Schwichtenberg Linear two-sorted arithmetic
Higher type input and output variables Linear two-sorted higher type terms Examples Linear two-sorted arithmetic Polynomial time Higher argument types ◮ Consider iteration I ( n , f ) = f n : I (0 , f , w ) := w , I (0 , f ) := id , or I ( n + 1 , f , w ) := I ( n , f , f ( w )) , I ( n + 1 , f ) := I ( n , f ) ◦ f . It is definable by a term with f a parameter of type W → W : I f := λ n ( R W → W n ( λ w w ) λ , p , w ( p W → W ( fw ))) . ◮ f must be an input variable, because the step argument of a recursion is by definition an input argument. Thus λ f I f may only be applied to input terms of type W → W . ◮ We cannot define the exponential function by λ n ( R W → W nS λ , p ( I p 2)) . The step type requires p to be an output variable, but I p is only correctly formed if p is an input variable. Helmut Schwichtenberg Linear two-sorted arithmetic
Higher type input and output variables Linear two-sorted higher type terms Examples Linear two-sorted arithmetic Polynomial time Normalization Let t be a closed LT (;)-term of type W ։ . . . W ։ W ( ։ ∈ { ֒ → , →} ). Then t denotes a polytime function. ◮ Let � z be new variables. Compute the normal form of t � z (needs constant time w.r.t. � n ). ◮ nf ( t � z ) is “simple” (i.e., no free or bound higher type input variables). ◮ Reduce to an R -free simple term rf ( nf ( t � z ); � z ; � n ) in time P t ( | | � n | | ), w.r.t. to a dag model of computation. ◮ Since the running time bounds the size of the produced term, | | rf ( nf ( t � z ); � z ; � n ) | | ≤ P t ( | | � n | | ). ◮ By Sharing Normalization one computes | ) 2 ). nf ( t � n ) = nf ( rf ( nf ( t � z ); � z ; � n )) in time O ( P t ( | | � n | Helmut Schwichtenberg Linear two-sorted arithmetic
Motivation Linear two-sorted higher type terms Proof terms Linear two-sorted arithmetic Example: Insertion sort Linear two-sorted arithmetic LA (;) Using the Curry-Howard correspondence, we transfer the term system LT (;) to a logical system LA (;) of arithmetic, with ◮ two arrow types, ρ ֒ → σ (input) and ρ → σ (output), ◮ two sorts of variables, input ones ¯ x and output ones x , and ◮ two implications, A ֒ → B (input) and A → B (output). Restrictions: ◮ Proofs of the premise of A ֒ → B are only allowed to use input assumptions or input variables. ◮ Proofs of the premise of A → B can only have at most one use of the hypothesis, in case its type is not a base type. Helmut Schwichtenberg Linear two-sorted arithmetic
Motivation Linear two-sorted higher type terms Proof terms Linear two-sorted arithmetic Example: Insertion sort Double use of assumptions ◮ Consider E ( 1 , y ) := S 0 ( y ) , E ( 1 ) := S 0 , or E ( S i ( x ) , y ) := E ( x , E ( x , y )) , E ( S i ( x )) := E ( x ) ◦ E ( x ) . Then E ( x ) = S (2 | | x | |− 1 ) , i.e., E grows exponentially. 0 ◮ Corresponding existence proof. Show by induction on x |− 1 + | | = 2 | | x | ∀ x , y ∃ v ( | | v | | y | | ) . ◮ Double use of the (“functional”) induction hypothesis is responsible for exponential growth. The linearity restriction on output implications will exclude such proofs. Helmut Schwichtenberg Linear two-sorted arithmetic
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