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Dynamic Semantics
The dynamic semantics of a language defines what happens when you run a program. There are many approaches to defining the dynamic semantics of a language: I (^) Small-step operational semantics I (^) Big-step operational semantics I (^) Abstract machines I (^) Translation to another language I (^) Denotational semantics (translation to a mathematical system) I (^) Axiomatic semantics
In this class we will primarily be concerned with the small-step operational semantics, which is the most widely used approach.
Recall the language of Arithmetic Expressions
e ::= true | false | if e then e else e | 0 | succ e | pred e | iszero e
v ::= true | false | nv nv ::= 0 | succ nv
Definition of small-step semantics
We define a relation between two expressions that says when one expression evaluates to the other expression. We’ll call this the ReducesTo relation and define the relation inductively with the following rules. The first bunch of rules do some real computation
(1) (pred 0 , 0) ∈ ReducesTo (2) (pred succ nv , nv ) ∈ ReducesTo
(3) (iszero succ nv , false) ∈ ReducesTo
(4) (iszero 0 , true) ∈ ReducesTo
(5) (if true then e 2 else e 3 , e 2 ) ∈ ReducesTo
(6) (if false then e 2 else e 3 , e 3 ) ∈ ReducesTo
Some notation
Before defining the rest of the rules for ReducesTo, we introduce the following notation:
e 7 −→ e′^ ≡ (e, e′) ∈ ReducesTo
Inversion Lemmas
Suppose you know that for some e and e′, e 7 −→ e′. What further information can you glean from this based on the rules for ReducesTo? We can do case analysis on the rules. For example, suppose
if true then e 2 else e 3 7 −→ e 4
Then you know that e 2 = e 4 because the only rule that could have appeared as the final step in the derivation is rule (5). Why not rule (10)? Because then there would need to be a rule of the form true 7 −→ −, but there is no such rule.
A Proof by Rule Induction on ReducesTo
Theorem
If e 7 −→ e′^ and e 7 −→ e′′^ then e′^ = e′′. Proof by rule induction on e 7 −→ e′. Case (1) pred 0 7 −→ 0 : So e = pred 0 , e′^ = 0 and by assumption we have pred 0 7 −→ e′′. There are two rules that could have applied (1) and (8), but (8) can’t apply because there are no rules of the form 0 7 −→ −. Thus, e′′^ = 0 and we conclude that e′^ = e′′.
More Definitions
Definition
An expression e is in normal form if 6 ∃e′. e 7 −→ e′.
Definition
Multi-step reduction (written 7 −→∗) is the relation inductively defined by the following rules:
(1) e 7 −→∗^ e (2) e 1 7 −→ e 2 e 2 7 −→∗^ e 3 e 1 7 −→∗^ e 3
(This is an alternate but equivalent definition than the one in the textbook.)
The Untyped Lambda Calculus
Syntax: x Variables e ::= x | (λx.e) | (e e) Expressions v ::= λx.e Values
I (^) An expression of the form (λx. e) create an anonymous function. The x is the one parameter of the function and e is the function body. I (^) An expression of the form (e 1 e 2 ) is a function call. The expression e 1 should evaluate to a function, which is then called using the result of e 2 as the argument.
α-equivalence
The variable name associated with a λ doesn’t really matter. The meaning of the program remains unchanged if you change the variable name and consistently replace the occurrences of that variable in the λ.
λy .e = λz.[y 7 → z]e provided z is not in e
Substitution and Free Variables
We write [x 7 → e′]e to say that e′^ is substituted for the free occurrences of x in e. For example, in the following we substitute the expression (y z) for the variable x in the expression ((λx. x) x). [x 7 → (y z)]((λx. x) x) = ((λx. x) (y z)) A free occurrence of a variable x within an expression e is an occurrence of x in e that does not have a surrounding λ in e that binds x. The the following examples, free occurrences are enclosed in a box.
I (^) ( z x ) I (^) (λ y. y) I (^) (λ y. (y x )) I (^) ((λ x. x) x ) I (^) y
Capture-avoiding Substitution
I (^) We need to be careful when defining substitution ([x 7 → e′]e) to make sure that free variables in e′^ don’t get captured by λs in e. I (^) The following is an example of what happens if you don’t define substitution correctly:
(λy. (λx. λy. x) y ) −→ (λy. (λy. y ))
I (^) Here’s a correct definition of substitution:
[x 7 → e′]y = if x = y then e′^ else y [x 7 → e′](λy. e) = (λz.[x 7 → e′][y 7 → z]e) (where z is fresh) [x 7 → e′](e 1 e 2 ) = ([x 7 → e′]e 1 [x 7 → e′]e 2 )
Evaluation Contexts
Congruence rules are a verbose way to specify evaluation order. A more succinct way is to use evaluation contexts. Recall the small-step textual rewriting view of semantics: if iszero (pred (succ 0)) then 0 else succ 0 7 −→ if iszero 0 then 0 else succ 0 An evaluation context is everything not inside an evaluation box. We can use a grammar to define where the boxes are allowed and what the surrounding evaluation context looks like.
E ::= [] The evaluation box forms a hole in the context. (E e) (v E )
Small-step Semantics with Evaluation Contexts
We separate out the computational reduction rules into the relation −→. For the lambda calculus there is just one rule:
(β) ((λx.e 1 ) v 2 ) −→ [x 7 → v 2 ]e 1
We then define top-level reduction using evaluation contexts:
e −→ e′ E [e] 7 −→ E [e′]
