Substitution
& Evaluation Order
cos 441
David Walker
Reading
• Pierce Chapter 5:
– 5.1: intro, op. sem., evaluation order
– 5.2: encodings of booleans, pairs, numbers,
recursion
– 5.3: substitution
Substitution
• In order to be precise about the
operational semantics of the lambda
calculus, we need to define substitution
properly
• For the call-by-value operational
semantics, we need to define:
– e1 [v/x] where v contains no free variables
• For other operational semantics, we need:
– e1 [e2/x]
Free Variables
FV : Given an expression, compute its free
variables
FV : lambda expression variable set
FV(x) = {x}
FV(e1 e2) = FV(e1) U FV(e2)
FV(\x.e) = FV(e) – {x}
FV as an inductive definition
FV(x) = {x}
Previous slide: FV(e1 e2) = FV(e1) U FV(e2)
FV(\x.e) = FV(e) – {x}
FV(x) = {x}
FV(e1) = S1 FV(e2) = S2
Equivalent definition: FV(e1 e2) = S1 U S2
FV(e) = S
FV(\x.e) = S – {x}
All Variables
Vars(x) = {x}
Vars(e1 e2) = Vars(e1) U Vars(e2)
Vars(\x.e) = Vars(e) U {x}
substitution examples
(\x.\y.z z)[\w.w/z] = \x.\y.(\w.w) (\w.w)
examples
(\x.\y.z z)[\w.w/z] = \x.\y.(\w.w) (\w.w)
(\x.\z.z z)[\w.w/z] = \x.\z.z z
examples
(\x.\y.z z)[\w.w/z] = \x.\y.(\w.w) (\w.w)
(\x.\z.z z)[\w.w/z] = \x.\z.z z
(\x.x z)[x/z] = \x.x x ?
examples
(\x.\y.z z)[\w.w/z] = \x.\y.(\w.w) (\w.w)
(\x.\z.z z)[\w.w/z] = \x.\z.z z
(\x.x z)[x/z] = \x.x x
(\x.x z)[x/z] = (\y.y z)[x/z] = \y.y x
alpha-equivalent expressions = the same except for consistent renaming of variables
“special” substitution
(ignoring capture issues)
definition of e1 [[e/x]] assuming FV(e) Vars(e1) = { }:
x [[e/x]] =e
y [[e/x]] =y (if y ≠ x)
e1 e2 [[e/x]] = (e1 [[e/x]]) (e2 [[e/x]])
(\x.e1) [[e/x]] = \x.e1
(\y.e1) [[e/x]] = \y.(e1 [[e/x]]) (if y ≠ x)
The Principle of
“Bound Variable Names Don’t Matter”
when you write
“let val x = 3 in x + global end”
you assume you can change the declaration of x to a
declaration of y (or other name) provided you
systematically change the uses of x. eg:
“let val y = 3 in y + global end”
provided that the name you pick doesn’t conflict with the free
variables of the expression. eg:
“let val global = 3 in global + global end” bad
Alpha-Equivalence
in order to avoid variable clashes, it is very convenient to
alpha-convert expressions so that bound variables
don’t get in the way.
eg: to alpha-convert \x.e we:
1. pick z such that z not in Vars(\x.e)
2. return \z.(e[[z/x]])
we just defined this form of substitution e[[z/x]] so it is a
total function when z is not in Vars(\x.e)
terminology: Expressions e1 and e2 are called alpha-
equivalent when they are the same after alpha-
converting some of their bound variables
capture-avoiding substitution
defined inductively on the structure of exp’s:
x [e/x] =e
y [e/x] = y (if y ≠ x)
e1 e2 [e/x] = (e1 [e/x]) (e2 [e/x])
(\x.e1) [e/x] = \x.e1
(\y.e1) [e/x] = \y.(e1 [e/x]) (if y ≠ x and y FV(e))
(\y.e1) [e/x] = \z.((e1[[z/y]]) [e/x]) (if y ≠ x and y FV(e))
for some z such that z FV(e) U Vars(e1)
Implicit Alpha-Conversion
it’s irritating to explicitly alpha-convert all the time in our
definitions. ie: to explicitly write down that before
doing something like substitution (or type checking)
that we are going to pick some new variable z that
doesn’t interfere with any other variables in the current
context and alpha-convert the given term.
Consequently, we are going to take a short-cut: implicit
alpha-conversion. When dealing with a bound variable
as in \x.e, we’ll just assume that x is any variable we
like other than one of the free variables in e.
capture-avoiding substitution
(the short-cut definition)
x [e/x] =e
y [e/x] = y (if y ≠ x)
e1 e2 [e/x] = (e1 [e/x]) (e2 [e/x])
(\x.e1) [e/x] = \x.e1
(\y.e1) [e/x] = \y.(e1 [e/x]) (if y ≠ x and y FV(e))
(note, we left out the case for \y.e1 [e/x] when y ≠ x
and y FV(e). We’ll implicitly alpha-convert \y.e1 to
\z.e1[[z/y]] for some z that doesn’t appear in e1 whenever
we need to satisfy the free variable side conditions)
operational semantics again
(\x.e) v --> e [v/x]
e1 --> e1’ e2 --> e2’
e1 e2 --> e1’ e2 v e2 --> v e2’
• Is this the only possible operational
semantics?
alternatives
(\x.e) v --> e [v/x] (\x.e1) e2 --> e1 [e2/x]
e1 --> e1’ e1 --> e1’
e1 e2 --> e1’ e2 e1 e2 --> e1’ e2
e2 --> e2’
v e2 --> v e2’
call-by-value call-by-name
alternatives
(\x.e) v --> e [v/x] (\x.e1) e2 --> e1 [e2/x]
e1 --> e1’
e1 e2 --> e1’ e2
e1 --> e1’
e1 e2 --> e1’ e2
e2 --> e2’
e1 e2 --> e1 e2’
e2 --> e2’
v e2 --> v e2’ e --> e’
\x.e --> \x.e’
call-by-value full beta-reduction
alternatives
(\x.e) v --> e [v/x] (\x.e) v --> e [v/x]
e1 --> e1’ e1 --> e1’
e1 e2 --> e1’ e2 e1 v --> e1’ v
e2 --> e2’ e2 --> e2’
v e2 --> v e2’ e1 e2 --> e1 e2’
call-by-value right-to-left call-by-value
Multi-step Op. Sem
• Given a single step op sem. relation:
e1 --> e2
• We extend it to a multi-step relation by
taking its “reflexive, transitive closure:”
(reflexivity) e1 --> e2 e2 -->* e3 (transitivity)
e1 -->* e1 e1 -->* e3
Proving Theorems About O.S.
Call-by-value o.s.:
e1 --> e1’ e2 --> e2’
(\x.e) v --> e [v/x] e1 e2 --> e1’ e2 v e2 --> v e2’
To prove property P of e1 --> e2, there are 3 cases:
case:
(\x.e) v --> e [v/x] Must prove: P((\x.e) v --> e [v/x])
** Often requires a related property of
substitution e [v/x]
case: e1 --> e1’ IH = P(e1 --> e1’)
e1 e2 --> e1’ e2 Must prove: P(e1 e2 --> e1’ e2)
case: e2 --> e2’ IH = P(e2 --> e2’)
v e2 --> v e2’ Must prove: P(v e2 --> v e2’)
Proving Theorems About O.S.
Call-by-value o.s.:
e1 --> e2 e2 -->* e3 (transitivity)
(reflexivity)
e1 -->* e1 e1 -->* e3
To prove property P of e1 -->* e2, given you’ve already
proven property P’ of e1 --> e2, there are 2 cases:
case:
e1 -->* e1 Must prove: P(e1 -->* e1) directly
case:
IH = P(e2 -->* e3)
e1 --> e2 e2 -->* e3 Also available: P’(e1 --> e2)
e1 -->* e3 Must prove: P(e1 -->* e3)
Example
Definition: An expression e is closed
if FV(e) = { }.
Theorem:
If e1 is closed and e1 -->* e2 then e2 is
closed.
Proof: by induction on derivation of e1 -->* e2.
summary
• the operational semantics
– primary rule: beta-reduction
– depends upon careful definition of substitution
– many evaluation strategies
• definitions/terminology to remember:
– free variable
– bound variable
– closed expression
– capture-avoiding substitution
– alpha-equivalence; alpha-conversion
– call-by-value, call-by-name, full beta reduction