MinML: an idealized programming language

1. MinML:an idealized programming language CS 510 David Walker

2. MinML • Reading: • Pierce chapter 1, 9 • the meaning of program safety • the typed lambda calculus • Harper 5-8 • MinML

3. 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

4. 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

5. syntax • A simple example: fun fact (x : int ) : int = if x <= 0 then 1 else x * fact (x-1)

6. 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...

7. 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

8. 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

9. 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

10. 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

11. 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

12. 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

13. MinML dynamic semantics • Primitive instructions: n = n’  false n1 + n2  n n = n  true (when n and n’ are different numbers) (when, mathematically, the sum of n1 and n2 is n) if true then e1 else e2  e1 if false then e1 else e2  e2

14. MinML dynamic semantics • Primitive instructions: v1 v2  e [v1/f] [v2/x] (when v1 = fix f (x:t1) : t2 = e)

15. 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 = (fix f (x:t1) : t2 = e)) (n = n1 + n2) n1 + n2  n (n ≠ n’) n = n’  false v1 v2  e [v1/f] [v2/x]

16. 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 e1  e1’ e1 e2  e1’ e2 e2  e2’ v1 op e2  v1 op e2’ e2  e2’ v1 e2  v1 e2’ e1  e1’ if e1 then e2 else e3  if e1’ then e2 else e3

17. MinML dynamic semantics • once again, multi-step evaluation: e * e e  e’ e’ * e’’ e * e’’

18. some properties • proposition 1 (values are irreducible) • if v is a value then there is no e such that v  e. Proof?

19. 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

20. some properties • proposition 2 (determinacy) • for every e there is at most one e’ such that e  e’. Proof?

21. 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 ...

22. 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: ...

23. some properties • proposition 3 (determinacy of values) • for every e there is at most one v such that e * v. Proof?

24. 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.

25. 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

26. 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