Separability of the Lambda Calculus and Term Rewriting Systems - PowerPoint PPT Presentation

Separability of the Lambda Calculus and Term Rewriting Systems Department of Computer Science Kyungsung University Sugwoo Byun 19 August 2003

Contents 1. Head normal form in the λ -Calculus 2. B¨ ohm trees 3. Separability of the λ -calculus 4. Sequentiality of the λ -calculus 5. B¨ ohm-out transformation technique 6. Lambda Definability of Term Rewriting Systems 7. Separable Systems 8. Encoding of Separable Systems 9. Correctness Reference: H. Barendregt. The Lambda Calculus: Its Syntax and Semantics . North-Holland. 1984. Separability 03 PL Workshop - 1

Head Normal Form in the λ -calculus • M is a hnf ( head normal form ) if M has the form M ≡ λx 1 · · · x n .yN 1 · · · N k Ω ≡ ( λx.xx )( λx.xx ). λx.Ix Ω has no normal form, but has an hnf λx.x Ω. Each normal form is a hnf. • Consistent Theory Let T be a formal theory with equations as formulas. Then T is consistent if T does not prove every closed equation. • T = { M = N | M, N ∈ Λ K without normal form } . Then T is not consistent. Proof. Let M ≡ λx.xK Ω , N ≡ λx.xS Ω. Then M = N ∈ T . Hence T ⊢ K = MK = NK = S • In λ K-calculus meaningless ⇔ no hnf ⇒ no nf Separability 03 PL Workshop - 2

• In λ I-calculus meaningless ⇔ no nf • Head Reduction If M is of the form M ≡ λx 1 · · · x n . ( λx.M 0 ) M 1 · · · M k n ≥ 0 , k ≥ 1 ( λx.M 0 ) M 1 is called head redex . We write M − → N if a head redex is chosen to be reduced. If the selected h redex is not a head redex, it is an internal redex and its reduction is written as M − → N . i • Standard Reduction. Let ∆ 0 ∆ 1 ∆ 2 σ : M 0 − → M 1 − → M 2 − → · · · be a reduction. σ is called a standard reduction if ∀ i ∀ j < i ∆ i is not a ‘residual’ of a redex to the left of ∆ j . We write M − → N if there is a s Separability 03 PL Workshop - 3

standard reduction σ : M − → N . • Example. λa. ( λb. ( λc.c ) bb ) d − → λa. ( λb.bb ) d − → λa.dd. not standard λa. ( λb. ( λc.c ) bb ) d − → λa. ( λc.c ) dd − → λa.dd. standard → ∗ N , then M − ∗ N such that • Standardization Theorem If M − → s → ∗ → ∗ ∃ Z M − Z − N i h Separability 03 PL Workshop - 4

B¨ ohm Trees → • BT ( M ), the B¨ ohm tree of M ≡ λ x.yM 1 · · · M k – if M is undefined, BT ( M ) = ⊥ → – if M ≡ λ x.y � � BT ( M 1 ) · · · BT ( M k ) • Example – BT ( S ) = λabc.a � � c · · · b | c – BT ( Sa Ω) = λc.a λc.ac (Ω c ) � � c · · · ⊥ Separability 03 PL Workshop - 5

– BT ( Y ) = λf.f | f | . . . ≡ λf.ω f ω f ω f ≡ λx.f ( xx ) ω f ω f = f ( ω f ω f ) Y Separability 03 PL Workshop - 6

Separability of the λ -calculus • Separability. Let T = { M 1 , · · · , M p } be a set of λ -terms. 1. If T ⊆ Λ 0 , then T is called separable if ∀ N 1 · · · N p ∈ Λ ∃ F ∈ Λ FM 1 = N 1 = FM 2 N 2 · · · · · · = FM p N p → → → 2. If T ⊆ Λ, then T is separable if its closure λ x. T = { λ x.M 1 , · · · , λ x.M p } is separable. • Distinct Terms. T is distinct if T consists of one element or some α ∈ Seq is ‘useful’ for T and the ∼ α equivalence classes of elements of T are all distinct. • Let M ≡ λx 1 · · · x n .yM 1 · · · M m and N ≡ λx 1 · · · x n ′ .y ′ M 1 · · · M m ′ . Then M ∼ N iff y ≡ y ′ and n − m = n ′ − m ′ Separability 03 PL Workshop - 7

�∼ ∼ λx.y λy.y λx.x λyz.y | | | � � M M M M N • Separability Theorem . T is separable ⇔ T is distinct Being given a constructive proof, we can find a λ -term F in the separable equations. Separability 03 PL Workshop - 8

Sequentiality of the λ -calculus • The λ -calculus is sequential . “parallel-or” functions cannot be definable in the λ -calculus (by Plotkin 1997). = por x T T = por T x T • The computation of a definable λ -calculus function is sequential rather than parallel (by Berry 1978.) • No F ∈ Λ such that FMN = I if M or N is solvable = unsolvable else Such an F is clearly parallel computable (simultaneously try to find the hnf of M and N ; if you find one, then give output I , else give no output). Separability 03 PL Workshop - 9

B¨ ohm-out Transformation Primitive functions • <M 0 , . . . , M n > ≡ λz.zM 0 · · · M n . • U n i ≡ λx 1 . . . x n .x i (selection). • P n ≡ λx 1 . . . x n . <x 1 , . . . , x n > = λx 1 . . . x n x n +1 .x n +1 x 1 . . . x n (permutation). Algorithm of B¨ ohm-out π α T : a set of λ -terms , α ∈ BT ( M ) 1. π f – transform T into λ -free form. ( ) π f = ( ) x 1 · · · x n . Separability 03 PL Workshop - 10

π f transformation λx 1 . . . x n .y y ⇒ = 1 m 1 m x k x n π f . . . . . . Figure 1: Transformation π f 2. π o – transform T π f into original form. ( ) π o = ( ) a 1 · · · a p +1 [ y := P p ], (a) The head variable at the root Separability 03 PL Workshop - 11

π o transformation (a) The head variable at the root a q +1 y P q ⇒ = = m a m +1 a q +1 1 m a m +1 a q 1 m 1 π o . . . . . . . . . (b) Internal node having a head variable λ� z.y λ� z. P q λ� z.x p +1 . . . x q +1 .x q +1 ⇒ = = 1 1 1 p p p x p +1 x q π o . . . . . . * * . . . * Figure 2: Transformation π o Separability 03 PL Workshop - 12

3. π s – select one of following success terms. ( ) π s = ( )[ z := U n j ], where α = j · α ′ . π s transformation - select a subterm at j U q z j = ⇒ = π s 1 · · · j 1 · · · j · · · m a m +1 a q m a m +1 a q * * * * Figure 3: Transformation π s 4. Repeat the above procedures with ((( T ) π f ) π o ) π s until α ′ becomes empty. Separability 03 PL Workshop - 13

Lambda Definability of Term Rewriting Systems • Question [RTA‘90] “Which rewrite systems can be directly defined in the λ -calculus? One has to find λ -terms representing rewrite system operators such that a rewrite step in TRS translates to a reduction in the λ -calculus.” • Main Idea 1. Define a separable system : f ( a 11 , a 12 , . . . , a 1 n ) → b 1 f ( a 21 , a 22 , . . . , a 2 n ) → b 2 · · · · · · f ( a m 1 , a m 2 , . . . , a mn ) → b m Let P f = { ( a 11 , a 12 , . . . , a 1 n ), ( a 21 , a 22 , . . . , a 2 n ), · · · , ( a m 1 , a m 2 , . . . , a mn ) } . The f -rules are separable if P f is ‘distinct’. Each element of P f is in a B¨ ohm tree. Separability 03 PL Workshop - 14

2. Define a homogeneous function φ ( P f ) such that φ ( P f ) is a set of distinct λ -terms. Then, by B¨ ohm’s separability theorem, φ ( f ) is obtained such that s → t implies φ ( s ) → ∗ φ ( t ). Separability 03 PL Workshop - 15

Separable Systems E.g. F ( x, A, B, C, S ( D )) → 1 F ( B, x, A, C, S ( D )) → 2 F ( A, B, x, C, S ( E )) → 3 is separable. Each separable system has a separation tree. Separability 03 PL Workshop - 16

5 5.1 4 4 3 1 2 2 2 Separation Tree Figure 4: Separation Tree Separability 03 PL Workshop - 17

Encoding of Separable Systems Example A (0 , x ) → x A ( S ( x ) , y ) → S ( A ( x, y )) A separation tree U T = 1 Assume φ c (0) = < 0 > , φ c ( S ) = λx. < 2 , x> . Then, φ p (0 , y ) = < < 0 >, � > , φ p ( S ( x ) , y ) = < < 2 , � >, � > , and α = 1 · 1. The whole encoding π = π 1 · π α · α ′ . • π α · α ′ : B¨ ohm-out terms at α · α ′ • π 1 : pass terms matches with variables at LHS to ones at RHS. Then, φ ( A ) = Y ( λax.x U 2 1 P 2 U 3 1 U 2 1 ( λb. < 2 , ( a < ( x U 2 1 U 2 2 ) , ( x U 2 2 ) > ) > ) ( x U 2 2 )) Separability 03 PL Workshop - 18

Reductions A ( S (0) , 0) → S ( A (0 , 0)) → S (0) are simulated as follows. φ ( A ( S (0) , 0)) = φ ( A ) φ p ( S (0) , 0) = φ ( A ) <φ ( S (0)) , φ (0) > = φ ( A ) < < 2 , < 0 > >, < 0 > > ≡ Y ( λax.x U 2 1 P 2 U 3 1 U 2 1 ( λb. < 2 , ( a < ( x U 2 1 U 2 2 ) , ( x U 2 ) ( x U 2 2 ) > ) > 2 )) < < 2 , < 0 > >, < 0 > > → ∗ < 2 , ( φ ( A ) < < 0 >, < 0 > > ) > → ∗ < 2 , < 0 > (= φ ( S (0)) ) > Separability 03 PL Workshop - 19

Correctness 1. s → ∗ t = ⇒ φ ( s ) → ∗ φ ( t ). 2. Let s include no operator normal term. Then, φ ( s ) → ∗ φ ( t ) = ⇒ s → ∗ t . 3. If s has a normal form, then φ ( s ) has a β -normal form. Separability 03 PL Workshop - 20