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CS 252: Advanced Programming Language Principles
- Prof. Tom Austin
San José State University
Lambda Calculus Prof. Tom Austin San Jos State University Minimum - - PowerPoint PPT Presentation
Lambda Calculus Prof. Tom Austin San Jos State University Minimum - - PowerPoint PPT Presentation
CS 252: Advanced Programming Language Principles Lambda Calculus Prof. Tom Austin San Jos State University Minimum complete programming language? WARNING: I expect you to remember every construct of this language for exams Lambda
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WARNING: I expect you to remember every construct of this language for exams
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Lambda Calculus expressions
e ::= x | (λx.e) | e e expressions: variables lambda abstractions function application
We could have just said "function", but we want to sound cool
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Lambda Calculus values
v ::= (λx.e) values: lambda abstractions
When our program finishes running, it returns some complex function as its "value"
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Function application
Suppose we have a function: (λx.E) Where E is some complex expression. How do we evaluate: (λx.E) v à E[x->v]
v replaces x wherever it
- ccurs in E
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Small step semantics for λ-calculus
(in-class)
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Operational Semantics e1 -> e1' e1 e2 -> e1' e2 (λx.e) v -> e[x->v] e2 -> e2' (λx.e) e2 -> (λx.e) e2'
[Ctxt1] [Ctxt2] [Call]
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Example: Identity Function
(λx.x) (λa.λb.a) à x[x->(λa.λb.a)] à (λa.λb.a)
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When should we evaluate function arguments?
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Strict Evaluation Strategies
Evaluate function arguments first
- Call-by-value:
copy of the parameter is passed
- Call-by-reference:
implicit reference is passed
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Lazy Evaluation Strategies
Substitute arguments in function body
- Call-by-name:
re-evaluate the argument each time
- Call-by-need:
memoizes parameter value after use
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How powerful is this language?
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The lambda-calculus is Turing complete. You can also implement the λ-calculus w/ a Turing machine
In other words, the two are equal in power
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Translating λ-calc to Haskell
x (λx.e) e e x (\x -> e) e e
Lambda-calculus Haskell
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Extending the lambda calculus
(in class)
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