MinML:an idealized programming

language

CS 510

David Walker

MinML

• Reading:– Pierce chapter 1, 9

• the meaning of program safety• the typed lambda calculus

– Harper 5-8• MinML

MinML

• MinML is an idealized programming language based on the lambda calculus

• the definition of MinML includes– a syntax for expressions involving recursive

functions, booleans and integers– a dynamic semantics similar to the call-by-

value semantics of lambda calculus– a set of typing rules to prevent programs from

incurring certain sorts of errors

syntax

• types:

t ::= int | bool | t1 -> t2

• expressions:

op ::= + | - | < | ...

e ::= x | n | b | e op e | if e then e else e

| fun f (x:t):t = e | e e

syntax

• A simple example:

fun fact (x : int ) : int =

if x <= 0 then 1

else x * fact (x-1)

dynamic semantics

• As we did for the lambda calculus, we will define the dynamic semantics as a relation:

e --> e’

• but we have some choices to make...

dynamic semantics

• there are at least two general approaches:– by translation: describe execution by

translating the language into some lower-level language

• we showed how to implement booleans, pairs in the lambda calculus by translating them into functions...

– language-based: describe execution entirely in terms of language itself

• we showed how to implement functions in the lambda calculus directly

translation-based approach

• Advantages– clear implementation strategy in terms of lower-level

concepts– sometimes simplifies complex language by translation

into a smaller core that can be more easily understood

– if translation to machine-level• facilitates low-level programming (writing device drivers;

interfacing with garbage collector)• supports low-level “hacks” that depend upon specific

machines

translation-based approach

• disadvantages– requires you understand how language is

compiled• problematic if compilation is complex

– can prohibit portability and optimization– run-time errors cannot necessarily be

understood in terms of the source program, only in terms of how it is compiled and executed

language-based models

• define execution entirely at the level of the language itself

• advantages– portable– (for the most part) do not need to introduce

new concepts to explain execution– simpler explanation of errors

language-based models

• disadvantages– cannot take advantage of machine-specific

details– does not to suggest a compilation strategy– can be more difficult to understand the time

and space complexity

back to MinML dynamic semantics

• For MinML, we’re going to give the dynamic semantics directly

• We will use a relation with the form

e1 e2

• values:v ::= n | b | fix f (x:t1) : t2 = e

MinML dynamic semantics

• Primitive instructions:

n1 + n2 n

(when, mathematically,the sum of n1 and n2 is n)

n = n true n = n’ false

(when n and n’ are different numbers)

if true then e1 else e2 e1

if false then e1 else e2 e2

MinML dynamic semantics

• Primitive instructions:

v1 v2 e [v1/f] [v2/x]

(when v1 = fix f (x:t1) : t2 = e)

MinML dynamic semantics

• By the way, normally:– I write the side conditions inside the body of

the rule– express them using mathematical notation

rather than English

• examples:

v1 v2 e [v1/f] [v2/x]

(v1 = (fix f (x:t1) : t2 = e)) (n = n1 + n2)n1 + n2 n

(n ≠ n’)n = n’ false

MinML dynamic semantics

• Search rules– recall, these rules specify the order of

evaluation– we’ll use left-to-right, call-by-value:

e1 e1’e1 op e2 e1’ op e2

e2 e2’v1 op e2 v1 op e2’

e1 e1’e1 e2 e1’ e2

e2 e2’v1 e2 v1 e2’

e1 e1’if e1 then e2 else e3 if e1’ then e2 else e3

MinML dynamic semantics

• once again, multi-step evaluation:

e * ee e’ e’ * e’’e * e’’

some properties

• proposition 1 (values are irreducible)– if v is a value then there is no e such that

v e.

Proof?

some properties

• proposition 1 (values are irreducible)– if v is a value then there is no e such that v e.

Proof? By inspection of the operational rules.There is no rule with the form:

(To call this proof “inductive” is technically correct, but misleading: we do not appeal to the inductive hypothesis. We could also call it a proof “by case analysis of the operational rules”)

...... premises....... v e

some properties

• proposition 2 (determinacy)– for every e there is at most one e’ such that

e e’.

Proof?

some properties

• proposition 2 (determinacy)– for every e there is at most one e’ such that

e e’.

Proof? By induction on the structure of e.

case (fix f (x : t1) : t2 = e). This is a value. By prop 1, there are no such e’.

case ...

some properties

• proposition 2 (determinacy)– for every e there is at most one e’ such that e e’.

Proof? By induction on the structure of e.case e1 e2: subcase e1, e2 both not values: ... subcase e1 value, e2 not value: ... subcase e1, e2 both values: ...

some properties

• proposition 3 (determinacy of values)– for every e there is at most one v such that

e * v.

Proof?

some properties

• proposition 3 (determinacy of values)– for every e there is at most one v such that

e * v.

Proof? By induction on the multi-step relation + it helps to be more formal about the induction hypothesis:

IH: If e * v and e * v’ then v = v’.

Proof uses prop 1 & 2.

stuck states

• not every irreducible expression is a value:– if 7 then 1 else 2– true + false– 0 (17)

• an expression that is not a value, but for which there exists no e’ such that e e’ is stuck

• but, all stuck expressions are ill-typed• if we type check programs in MinML before

running them, they will never get stuck

summary of syntax & dynamic semantics

• MinML is an idealized language that models some of the basic concepts in functional programming

• the dynamic semantics satisfies some nice properties: values are irreducible, evaluation is deterministic

• Next up: typing MinML