λ S : A Lambda Calculus with Side-effects

Transcription

1 L14-1 λ S : A Lambda Calculus with Side-effects delivered by Jacob Schwartz Laboratory for Computer Science M.I.T. Lecture 14 M-Structures and Barriers L14-2 Some problems cannot be expressed functionally Input / Output Gensym: Generate unique identifiers Gathering statistics Graph algorithms Non-deterministic algorithms Once side-effects are introduced, barriers are needed to control the execution of some operations The λ S calculus λ C + side-effects and barriers

2 The λ B Calculus : λ C + Barriers L14-3 Even adding barriers to a purely functional calculus (without side-effects) is significant Observability of Termination Using λ B as a stepping stone to λ S allows us to analyze the semantic effects of barriers separate from side-effects, simplifying the analysis λ S = λ B + side-effects Outline L14-4 Background The λ C calculus: λ + letrecs Observable values The λ B calculus: λ C + barriers Garbage collection The λ S calculus: λ B + side-effects

3 λ +Let : A way to model sharing L14-5 Instead of the normal β-rule (λx.e) e a e [e a /x] use the following β let rule (λx.e) e a { let t = e a in e[t/x] } where t is a new variable and only allow the substitution of values and variables to preserve sharing Previous work on Sharing L14-6 Differences are mainly regarding where variables can be instantiated the source language λ or λ + let or λ + letrec Graph reduction and lazy evaluation Wadsworth (71), Launchbury (POPL93) Environments and Explicit Substitution Abadi, Cardelli, Curien & Levy (POPL 92, JFP) Letrecs but no reductions inside λ-abstractions Ariola, Felleisen, Wadler,...(POPL 95) Letrecs Ariola et al. (96)

4 λ C Syntax L14-7 E ::= x λx.e E E { S in E } Cond (E, E, E) PF k (E 1,...,E k ) CN 0 CN k (E 1,...,E k ) CN k (SE 1,...,SE k ) PF 1 ::= negate not... Prj 1 Prj CN 0 ::= Number Boolean CN 2 ::= Cons... S ::= ε x = E S; S Not in initial expressions λ C Syntax L14-8 Values V ::= λx.e CN 0 CN k (SE 1,...,SE k ) Simple expressions SE ::= x V

5 Equivalence Rules L14-9 α-renaming λx.e λx.(e[x / x]) {x=e; S in e 0 } {x =e; S in e 0 }[x /x] Properties of ; ε ; S S S 1 ; S 2 S 2 ; S 1 S 1 ; (S 2 ; S 3 ) (S 1 ; S 2 ) ; S 3 λ let Instantiation Rules L14-10 a is a Simple Expression; [x] is a free occurrence of x in C[x] or SC[x] Instantiation Rule 1 { x = a ; S in C[x] } { x = a ; S in C [a] } Instantiation Rule 2 (x = a ; SC[x]) (x = a ; SC [a]) Instantiation Rule 3 x = C[x] x = C [C[x]] where C[x] is simple

6 λ C Rules L14-11 Cond-rules Cond (True, e 1, e 2 ) e 1 Cond (False, e 1, e 2 ) e 2 Constructors CN k (e 1,...,e k ) {t 1 = e 1 ;...; t k = e k in CN k (t 1,...,t k )} δ-rules PF k (v 1,...,v k ) pf k (v 1,...,v k ) Prj i (CN k (x 1,...,x i,...,x k )) x i Need for Lifting Rules L14-12 {f = { S 1 in λx.e 1 }; y = f a ; in ({ S 2 in λx.e 2 } e 3 ) } How do we juxtapose or (λx.e 1 ) a (λx.e 2 ) e3?

7 λ C Block Flattening and Lifting Rules L14-13 Block Flatten x = { S in e} (x = e ; S ) Lifting rules { S 1 in { S 2 in e } } { S 1 ; S 2 in e } { S in e} e 2 { S in e e 2 } Cond({ S in e}, e 1, e 2 ) { S in Cond (e, e 1, e 2 ) } PF k (e 1,...{ S in e },...e k ) { S in PF k (e 1,...e,...e k ) } { S in e } is the α-renaming of { S in e } to avoid name conflicts Non-confluence L14-14 odd = λn.cond(n=0, False, even (n-1)) ---- (M) even = λn.cond(n=0, True, odd (n -1)) substitute for even (n-1) in M odd = λn.cond(n=0, False, Cond(n-1 = 0, True, odd ((n-1)-1))) ---- (M 1 ) even = λn.cond(n=0, True, odd (n -1)) substitute for odd (n-1) in M odd = λn.cond(n=0, False, even (n-1)) ---- (M 2 ) even = λn.cond(n=0, True, Cond( n-1 = 0, False, even ((n-1)-1))) M 1 and M 2 cannot be reduced to the same expression! Ariola & Klop (LICS 94)

8 Printable Values L14-15 Printable values are trees and can be infinite We will compute the printable value of a term in 2 steps: Info: E --> T P (trees) Print: E --> {T P } (downward closed sets of trees) where T P ::= λ CN 0 CN k (T P1,...,T Pk ) t (bottom) t t (reflexive) CN k (v 1,...,v i,...,v k ) CN k (v 1,...,v i,...,v k ) if v i v i Info Procedure L14-16 Info : E --> T P Info [ { S in E } ] = Info [E] Info [λx.e] = λ Info [CN 0 ] = CN0 Info [CN k (a 1,...,a k )] = CNk (Info[a 1 ],...,Info[a k ]) Info [E] = Ω otherwise Proposition Reduction is monotonic wrt Info: If e ->> e 1 then Info[e] Info[e 1 ]. Proposition Confluence wrt Info: If e ->> e 1 and e ->> e 2 then e 3 s.t. e 1 ->> e 3 and Info[e 2 ] Info[e 3 ].

9 Print Procedure L14-17 Print : E --> {T P } Print[e] = { i i Info[e 1 ] and e i >> e 1 } i > is simple instantiation: let x = v ; S in C[x] i > let x = v ; S in C[v] Unwind the value as much as possible Keep track of all the unwindings Terms with infinite unwindings lead to infinite sets. Print*: Maximum Printable Info L14-18 Print*[e] = { Ui Print[s i ] s ε PRS(e) } where Definition: Reduction Sequence RS(e) = { s s 0 = e, s i-1 -> s i, 0 < i < s } Definition: Progressive Reduction Sequence PRS(e) = { s s ε RS(e), and i j > i. s j ->> t k. Print[t] Print[s k ] } Proposition: if e ->> e 1 then Print*[e] = Print*[e 1 ]. Print*[e] has precisely one element.

10 λ B Syntax L14-19 E ::= x λx.e E E { S in E } Cond (E, E, E) PF k (E 1,...,E k ) CN 0 CN k (E 1,...,E k ) CN k (x 1,...,x k ) PF 1 ::= negate not... Prj 1 Prj CN 0 ::= Number Boolean S ::= ε x = E S; S S >>> S Not in initial expressions Barriers L14-20 { ( y = 1+7 >>> z = 3 ) in z } { ( y = 8 >>> z = 3 ) in z } { y = 8 ; ( z = 3 ) in z } { y = 8 ; z = 3 in 3 } Barriers discharge when all the bindings in the pre-region terminate, i.e., all expressions become values.

11 Stability and Termination L14-21 Definition: Expression e is said to be stable if when e ->> e 1, Print[e] = Print[e 1 ] In general, an expression cannot be tested for stability. Terminated Terms E T ::= V {H in SE} H ::= x = V H; H Proposition: All terminated terms are stable. Values and Heap Terms L14-22 Values V ::= λx.e CN 0 CN k (x 1,...,x k ) Simple expressions SE ::= x V Terminated Terms E T ::= V {H in SE} H ::= x = V H; H

12 Barrier Rules L14-23 Barrier discharge (ε >>> S) S Barrier equivalence ((H ; S 1 ) >>> S 2 ) (H ; (S 1 >>> S 2 )) (H >>> S) (H ; S) (derivable) L14-24 λ C Versus λ B In λ B termination of a term is observable. Thus, 5 {x = in 5} Consider the context: { (y = [ ] >>> z = 3) in z } equality in λ C does not imply equality in λ B However, barriers can only make a term less defined.

13 L14-25 Properties of λ B Proposition Barriers are associative: S1 >>> (S2 >>> S3) = (S1 >>> S2) >>> S3 in all contexts. Proposition Barriers reduce results: Every reduction in C[S1 >>> S2] can be modeled by a reduction in C[S1 ; S2]. Proposition Postregions can be postponed: If C1[S1 >>> S2] ->> C3[S3 >>> S4] where the barrier is the same in both terms, there is a C2 such that: C1[S1 >>> S2] ->> C2[S3 >>> S2] ->> C3[S3 >>> S4] Garbage Collection L14-26 A Garbage collection rule erases part of a term. Definition: A garbage collection rule, GC, is said to be correct if for all e, Print*(e) = Print*(GC(e))

14 λ B Garbage Collection Rule L14-27 GC 0 -rule { S G ; S in e } { S in e } if forall x, x (FV(e) U FV(S)) then x BV(S G ) GC v -rule { H ; S in e } { S in e } if forall x, x (FV(e) U FV(S)) then x BV(H) While both GC 0 and GC v rules are correct for λ let, only the GC v -rule is correct for λ B. λ S Syntax L14-28 E ::= x λx.e E E { S in E } Cond (E, E, E) PF k (E 1,...,E k ) CN 0 CN k (E 1,...,E k ) CN k (x 1,...,x k ) allocate() o i object descriptors PF 1 ::= negate not... Prj 1 Prj 2... ifetch mfetch... CN 0 ::= Number Boolean () Not in initial expressions S ::= ε x = E S; S S >>> S sstore(e,e) allocator empty(o i ) full(o i,e) error(o i )

15 Values and Heap Terms L14-29 Values V ::= λx.e CN 0 CN k (x 1,...,x k ) o i Simple expressions SE ::= x V Heap Terms H ::= x = V H; H allocator empty(o i ) full(o i,v) Terminal Expressions E T ::= V let H in SE Side-effect Rules L14-30 Allocation rule (allocator; x=allocate()) allocator; x = o; empty(o)) where o is a new object descriptor Fetch and Take rules (x=ifetch(o) ; full(o,v)) (x=mfetch(o) ; full(o,v)) Store rules (sstore(o,v) ; empty(o)) (sstore(o,v) ; full(o,v )) (x=v ; full(o,v)) (x=v ; empty(o)) full(o,v) (error(o); full(o,v )) Lifting rules sstore({ S in e}, e 2 ) ( S ; sstore(e,e 2 )) sstore(e 1, { S in e}) ( S ; sstore(e 1,e))

16 Nondeterministic Choice L14-31 choose = choose 100 λx. { m = allocate(); sstore(m, True); ( y = mfetch(m) >>> sstore(m, False) ); ( z = mfetch(m) >>> sstore(m,true) ) in z }?