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 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  * e e  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