Some more thoughts on

 let Arvind Computer Science and Artificial Intelligence Laboratory M.I.T.

September 21, 2006 September 21, 2006 http://www.csg.csail.mit.edu/6.827

L04Ext-1

Why

 let

Calculus ?

Programming without (recursive) let blocks is tedious Recursive let blocks can be translated into the  -calculus with constants and Y combinator but the translation is • is complicated (not simple syntactical substitutions) ; • is not intuitive or illustrative • does not match any implementation  extend the  -calculus with recursive let

blocks.

September 21, 2006 http://www.csg.csail.mit.edu/6.827

L04Ext-2



-calculus with Constants & Letrec

E ::= x | | PF k  x.E | E E | Cond (E, E, E) (E 1 ,...,E k ) | CN 0 | CN | k (E 1 ,...,E let S in E k ) | CN k (SE 1 ,...,SE k ) PF 1 PF 2 CN 0 CN 2 ::= negate | not | ... | Prj 1 | Prj 2 ::= + | ... ::= Number | Boolean ::= cons | ...

| ...

not in initial terms Statements

S ::=  | x = E | S; S

Variables on the LHS in a let expression must be pairwise distinct

September 21, 2006 http://www.csg.csail.mit.edu/6.827

L04Ext-3

Issues in giving semantics for lets- 1

1. Creating redexes ((let S in  x.e

1 ) e 2 )

How do we juxtapose

(  x.e

1 ) e 2 ?

Solution: Lifting rules September 21, 2006 http://www.csg.csail.mit.edu/6.827

L04Ext-4

Issues in giving semantics for lets- 2

2. How to refer to a variable binding

let

f =  x.e

1

in

y = e 2 e (f y) + y 3 Solution: Instantiation rules

How and when f and y refer to their definitions

( (  x.e

1 ) y) + y ? September 21, 2006 http://www.csg.csail.mit.edu/6.827

L04Ext-5

How to define the operational semantics of let blocks: Environments Eval [[e]]  environment An environment-based interpreter.

– An environment where all the (variable name, value) bindings are kept and is passed around for expression evaluation – When a let expression is encountered the environment is extended with all the let-bindings. Very complicated if the environment contains unevaluated expressions – Not abstract enough – too many concrete data structures and associated functions for proper execution September 21, 2006 http://www.csg.csail.mit.edu/6.827

L04Ext-6

How to define the operational semantics of let blocks: graphs A let simply represents a wiring diagram, i.e., a graph

let

f =  x.e

1 y = e 2 e 3

in

(f y) + y ap + root  x f e 1 e 2 ap y e 3 September 21, 2006 – Quite complicated to explain  -substitution in a graph based interpreter http://www.csg.csail.mit.edu/6.827

L04Ext-7

How to define the operational semantics of let blocks: via a calculus • Rewrite rules – Lifting rules – Instantiation rules (need some new way of writing rules): contexts, ... • Reduction Strategy • Normal forms? Equivalences?

let x = 5 in x September 21, 2006 let x = 5 in 5 the  let calculus http://www.csg.csail.mit.edu/6.827

5 let x = 5 y = 6 in x L04Ext-8

Lifting Rules

(let S’ in e’) is the 

-renamed

(let S in e) avoid name conflicts in the following rules: to x = let S in e  x = e’; S’ let S 1 in (let S in e) (let S in e) e 1   let S 1 ; S’ in e’ let S’ in e’ e 1 Cond((let S in e), e 1 , e 2 )  let S’ in Cond(e’, e 1 , e 2 ) PF k (e 1 ,...(let S in e),...e

k )  let S’ in PF k (e 1 ,...e’,...e

k ) September 21, 2006 http://www.csg.csail.mit.edu/6.827

L04Ext-9

 let

Instantiation Rules

A free variable in an expression can be instantiated by a simple expression Instantiation rule 1 (let x = a ; S in C[x])  (let x = a ; S in C’[a]) simple expression free occurrence of x in some context C renamed C[ ] to avoid free variable capture Instantiation rule 2 (x = a ; SC[x])  Instantiation rule 3 x = a  where a = C[x] (x = a ; SC’[a]) x = C’[C[x]] September 21, 2006 http://www.csg.csail.mit.edu/6.827

L04Ext-10

Once we have lets we use them elsewhere too ...

The normal  -rule (  x.e) e a  e [e a /x] is replaced the following  -rule (  x.e) e a  let t = e a where t in e[t/x] is a new variable and the Instantiation rules which are used to refer to the value of a variable September 21, 2006 http://www.csg.csail.mit.edu/6.827

L04Ext-11