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Substitution & Evaluation Order cos 441 David Walker

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Reading Pierce Chapter 5: –5.1: intro, op. sem., evaluation order –5.2: encodings of booleans, pairs, numbers, recursion –5.3: substitution

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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]

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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}

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FV as an inductive definition FV(x) = {x} FV(e1 e2) = FV(e1) U FV(e2) FV(\x.e) = FV(e) – {x} FV(x) = {x} FV(e1) = S1 FV(e2) = S2 FV(e1 e2) = S1 U S2 FV(e) = S FV(\x.e) = S – {x} Previous slide: Equivalent definition:

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All Variables Vars(x) = {x} Vars(e1 e2) = Vars(e1) U Vars(e2) Vars(\x.e) = Vars(e) U {x}

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substitution examples (\x.\y.z z)[\w.w/z] = \x.\y.(\w.w) (\w.w)

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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

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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 ?

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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

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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)

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The Principle of Bound Variable Names Dont 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 doesnt conflict with the free variables of the expression. eg: let val global = 3 in global + global end bad

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Alpha-Equivalence in order to avoid variable clashes, it is very convenient to alpha-convert expressions so that bound variables dont 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

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capture-avoiding substitution defined inductively on the structure of exps: 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)

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Implicit Alpha-Conversion its 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 doesnt 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, well just assume that x is any variable we like other than one of the free variables in e.

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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). Well implicitly alpha-convert \y.e1 to \z.e1[[z/y]] for some z that doesnt appear in e1 whenever we need to satisfy the free variable side conditions)

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operational semantics again Is this the only possible operational semantics? e1 --> e1 e1 e2 --> e1 e2 e2 --> e2 v e2 --> v e2 (\x.e) v --> e [v/x]

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alternatives e1 --> e1 e1 e2 --> e1 e2 e2 --> e2 v e2 --> v e2 (\x.e) v --> e [v/x] e1 --> e1 e1 e2 --> e1 e2 (\x.e1) e2 --> e1 [e2/x] call-by-value call-by-name

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alternatives e1 --> e1 e1 e2 --> e1 e2 e2 --> e2 v e2 --> v e2 (\x.e) v --> e [v/x] e1 --> e1 e1 e2 --> e1 e2 (\x.e1) e2 --> e1 [e2/x] call-by-value full beta-reduction e2 --> e2 e1 e2 --> e1 e2 e --> e \x.e --> \x.e

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alternatives e1 --> e1 e1 e2 --> e1 e2 e2 --> e2 v e2 --> v e2 (\x.e) v --> e [v/x] call-by-value right-to-left call-by-value e1 --> e1 e1 v --> e1 v e2 --> e2 e1 e2 --> e1 e2 (\x.e) v --> e [v/x]

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Multi-step Op. Sem Given a single step op sem. relation: We extend it to a multi-step relation by taking its reflexive, transitive closure: e1 -->* e1 e1 --> e2 e2 -->* e3 e1 -->* e3 e1 --> e2 (reflexivity)(transitivity)

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Proving Theorems About O.S. Call-by-value o.s.: To prove property P of e1 --> e2, there are 3 cases: case: e1 --> e1 e1 e2 --> e1 e2 e2 --> e2 v e2 --> v e2 (\x.e) v --> e [v/x] e1 --> e1 e1 e2 --> e1 e2 e2 --> e2 v e2 --> v e2 IH = P(e1 --> e1) Must prove: P(e1 e2 --> e1 e2) IH = P(e2 --> e2) Must prove: P(v e2 --> v e2) Must prove: P((\x.e) v --> e [v/x]) ** Often requires a related property of substitution e [v/x]

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Proving Theorems About O.S. Call-by-value o.s.: To prove property P of e1 -->* e2, given youve already proven property P of e1 --> e2, there are 2 cases: case: IH = P(e2 -->* e3) Also available: P(e1 --> e2) Must prove: P(e1 -->* e3) e1 -->* e1 e1 --> e2 e2 -->* e3 e1 -->* e3 (reflexivity) (transitivity) e1 -->* e1 Must prove: P(e1 -->* e1) directly e1 --> e2 e2 -->* e3 e1 -->* e3

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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.

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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

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