lec9.pdf - lec9 Lambda Calculus Semantics CS3100 Fall 2019 Review Last time Lambda Calculus Syntax Today Lambda Calculus Semantics Reductions

# lec9.pdf - lec9 Lambda Calculus Semantics CS3100 Fall 2019...

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14/08/2019 lec9 127.0.0.1:8888/notebooks/lec9/lec9.ipynb# 1/12 Lambda Calculus : Semantics CS3100 Fall 2019 Review Last time Lambda Calculus: Syntax Today Lambda Calculus: Semantics Reductions, Church-Rosser Theorem. β-reduction Lambda Calculus we have been looking so far is untyped . No static semantics, only dynamic semantics! A term of the form is called a β-redex . The act of evaluating lambda calculus terms is called β-reduction . β-reduction replaces with . A term without β-reduxes is said to be in β-normal form . β-reduction, formally
14/08/2019 lec9 127.0.0.1:8888/notebooks/lec9/lec9.ipynb# 2/12 Example Example Example Many steps of β-reduction Church-Rossser Theorem If and then there exists an such that and .
14/08/2019 lec9 127.0.0.1:8888/notebooks/lec9/lec9.ipynb# 3/12 β-normal form "β-normal form" "contains no reduxes" Theorem (Uniqueness of β-normal forms). If and and and are in β-normal form, then . Proof. By Church-Rosser, obtain an such that and . But and are in β-normal form. Hence, . β-equivalence iff there exists an such that and . Possible Non-termination Some terms do not have a normal form
14/08/2019 lec9 127.0.0.1:8888/notebooks/lec9/lec9.ipynb# 4/12 Such terms are said to diverge . Possible Non-termination Other terms may or may not terminate based on the redux chosen to reduce. Reduction Strategies Several different reduction strategies have been studied for lambda calculus. The β reduction we have seen so far is known as full β-reduction Any redex in the term can be reduced at any time.

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