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comp_314_lambda_definitions_1_03 - Comp 314-calculus...

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Comp 314 λ -calculus definitions and notes Version 1.03 Chris Stephenson March 14, 2009 1 Grammar In this grammar for the λ -calculus, i represents one of an infinite number of names. You can think of i as representing a Scheme identifier. There are an infinite number of Scheme identifiers. But Scheme identifiers are formed from a finite alpahabet. We could embed the regular grammar for Scheme identifiers in the context free grammar of the λ -calculus to give us a grammar with a finite alphabet, but an infinite number of names. Expanded in this way, our λ -calculus definition below obeys the rules for a formal language by having a finite alphabet. The definition below with an apparently infinite alphabet is simply a convenient shorthand. Λ i Λ (Λ Λ) Λ ( λ i Λ) 2 Free Identifiers Definition of FI(Λ), the free identifiers of a λ -sentence. Λ = i FI (Λ) = { i } Λ = ( M N ) FI (Λ) = FI ( M ) FI ( N ) Λ = ( λ i M ) FI (Λ) = FI ( M ) \ { i } 1
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3 β -reduction A λ -sentence of the form (( λ i M ) N ), where M and N are λ
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