CMSC 330: Organization of Programming Languages. Operational Semantics
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1 CMSC 330: Organization of Programming Languages Operational Semantics
2 Notes about Project 4, Parts 1 & 2 Still due today (7/2) Will not be graded until 7/11 (along with Part 3) You are strongly encouraged to finish Parts 1 & 2 ASAP and begin working on Part 3 There is one important function from Part 2 that you will need in Part 3: lookup CMSC 330 2
3 Review of CMSC 330 Syntax Regular expressions Finite automata Context-free grammars Semantics Operational semantics Lambda calculus Implementation Concurrency Generics Argument passing Scoping Exceptions CMSC 330 3
4 CMSC 330
5 CMSC 330
6 Operational Semantics There are several different kinds of semantics Denotational: A program is a mathematical function Axiomatic: Develop a logical proof of a program Give predicates that hold when a program (or part) is executed We will briefly look at operational semantics A program is defined by how you execute it on a mathematical model of a machine We will look at a subset of OCaml as an example CMSC 330 6
7 Evaluation We re going to define a relation E v This means expression E evaluates to v Sometimes simply read as E goes to v We ll use grammars to describe each of these One to describe program abstract syntax trees E One to describe result values v CMSC 330 7
8 OCaml Programs E ::= x n true false if E then E else E fun x = E E E x stands for any identifier n stands for any integer true and false stand for the two boolean values Using = in fun instead of -> to avoid confusion CMSC 330 8
9 Values v ::= n true false n is an integer (not a string corresp. to an integer) Same idea for true, false Important: Be sure to understand the difference between program text S and mathematical objects v. E.g., the text 3 evaluates to the mathematical number 3 To help, we ll use different colors and italics This is usually not done, and it s up to the reader to remember which is which CMSC 330 9
10 Grammars for Trees We re just using grammars to describe trees E ::= x n true false if E then E else E fun x = E E E v ::= n true false type expr = Id of string Num of int Bool of bool If of expr * expr * expr Fun of string * expr App of expr * expr Given a program, we have seen how to convert it to an AST(e.g., recursive descent parsing) type value = VNum of int VBool of bool Goal: For any expr, we want an operational rule describing how to obtain a value that represents its execution CMSC
11 Operational Semantics Rules n n true true false false Each basic entity evaluates to the corresponding value CMSC
12 Operational Semantics Rules (cont d) How about built-in functions? + n m n + m We re applying the + function Ignore currying for the moment, and pretend we have multiargument functions On the right-hand side, we re computing the mathematical sum; the left-hand side is source code But what about + (+ 3 4) 5? We need recursion CMSC
13 Recursive Rules To evaluate + E 1 E 2, we need to evaluate E 1, then evaluate E 2, then add the results This is call-by-value E 1 n E 2 m + E 1 E 2 n + m This is a natural deduction style rule It says that if the hypotheses above the line hold, then the conclusion below the line holds i.e., if E 1 executes to value n and if E 2 executes to value m, then + E 1 E 2 executes to value n+m CMSC
14 Trees of Semantic Rules When we apply rules to an expression, we actually get a tree Corresponds to the recursive evaluation procedure For example: + (+ 3 4 ) (+ 3 4) ( + 3 4) 5 12 CMSC
15 Rules for If E 1 true E 2 v if E 1 then E 2 else E 3 v E 1 false E 3 v if E 1 then E 2 else E 3 v Examples if false then 3 else 4 4 if true then 3 else 4 3 Notice that only one branch is evaluated CMSC
16 Rules for Identifiers Let s assume for now that the only identifiers are parameter names Ex. (fun x = + x 3) 4 When we see x in the body, we need to look it up So we need to keep some sort of environment This will be a map from identifiers to values CMSC
17 Semantics with Environments Extend rules to the form A E v Means in environment A, the program text E evaluates to v Notation: We write for the empty environment We write A(x) for the value that x maps to in A We write A, x v for the same environment as A, except x is now v might be a new (replacement) mapping for x We write A, A' for the environment with the bindings of A' added to and overriding the bindings of A CMSC
18 Rules for Identifiers and Application A x A(x) no hypothesis means in all cases A E 2 v A, x v E 1 v' A ( (fun x = E 1 ) E 2 ) v' To evaluate a user-defined function applied to an argument: Evaluate the argument (call-by-value) Evaluate the function body in an environment in which the formal parameter is bound to the actual argument Return the result CMSC
19 Operational Semantics Rules SInt A n n SVar x v A A x v SAdd A e 1 n A e 2 m A + e 1 e 2 n + m SFun A e 2 v A, x v e 1 v' A (fun x = e 1 ) e 2 v' CMSC
20 Example: (fun x = + x 3) 4? SInt SVar x 4 x 4 x x 4 + x 3 7 (fun x = + x 3) 4 7 SFun SAdd SInt CMSC
21 Exercises A + 3 m A: m 5 (fun x = x) 7 (fun b = (fun a = a + b) 8) 9 CMSC
22 Error Cases Recall the rule for built-in addition: E 1 n E 2 m Because we wrote n, m in the hypothesis, we mean that they must be integers But what if E 1 and E 2 aren t integers? E.g., what if we write + false true? It can be parsed, but we can t execute it We will have no rule that covers such a case Runtime error! + E 1 E 2 n + m CMSC
23 Why Did We Do This? Operational semantics are useful for Describing languages Not just OCaml! It s pretty hard to describe a big language like C or Java, but we can at least describe the core components of the language Giving a precise specification of how they work Look in any language standard they tend to be vague in many places and leave things undefined Reasoning about programs We can actually prove that programs do something or don t do something, because we have a precise definition of how they work CMSC
24 Why Did We REALLY Do This? P4, Part 3 is all about operational semantics You are implementing an interpreter for the Rube language, which means understanding and implementing its operational semantics AST definition ( expr ) on page 2 Value definition ( value ) on page 5 Rule definitions on page 6 Other concepts: Classes and methods (page 2) Heaps and environments (page 5) Built-in methods (page 8) CMSC
25 Conclusion READ THE SPEC Print it Study it Ask questions about it Focus on the definitions Pay attention to details For instance: H H H 0 H 1 (etc.) The better you understand the project description, the easier you will find Part 3 to be CMSC
26 Aside: Nested Functions All this works for cases of nested functions...as long as they are fully applied But what about the true higher-order cases? Passing functions as arguments, and returning functions as results We need closures to handle this case...and a closure was just a function and an environment We already have notation around for writing both parts CMSC
27 Closures Formally, we add closures (A, λx.e) to values A is the environment in which the closure was created x is the parameter name E is the source code for the body λx will be discussed next time. Means a binding of x in E. v ::= n true false [] v::v (A, λx.e) CMSC
28 Revised Rule for Lambda SLambda A fun x = E (A, λx.e) To evaluate a function definition, create a closure when the function is created Notice that we don t look inside the function body CMSC
29 Revised Rule for Application SApp A E 1 (A', λx.e) A E 2 v A, A', x v E v' A (E 1 E 2 ) v' To apply something to an argument: Evaluate it to produce a closure Evaluate the argument (call-by-value) Evaluate the body of the closure, in The current environment, extended with the closure s environment, extended with the binding for the parameter CMSC
30 Exercise (fun x = + x 3) 7 CMSC
31 Example (fun x = (fun y = + x y)) (, λx.(fun y = + x y)) x 3 (fun y = + x y) 3 3 (x 3, λy.(+ x y)) (fun x = (fun y = + x y)) 3 (x 3, λy.(+ x y)) Let <previous> = (fun x = (fun y = + x y)) 3 CMSC
32 Example (cont d) <previous> (x 3, λy.(+ x y)) 4 4 x 3,y 4 (+ x y) 7 ( <previous> 4 ) 7 CMSC
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