# LambdaCalculus - cs3723 1 Language Semantics Lambda...

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Unformatted text preview: cs3723 1 Language Semantics Lambda Calculus: Variables and Functions cs3723 2 Describing Semantics  Informal definitions  Tutorials: learn by working examples  Reference Manuals  Natural language explanation for each syntax rule  Formal definitions (skip)  Attribute grammars  Associate attributes (values) with each grammar symbol  Associate semantics rules with each grammar rule  Operational semantics  Interpret the language on an abstract machine or using another language  Denotational semantics  Define language constructs as mathematical functions  Axiomatic semantics (proof rules)  Define properties (invariance) of language constructs  Goal: communication, automation and validation cs3723 3 Lambda Calculus  Mathematical system for functions  Computation with functions  Captures essence of variable binding  Function parameters and substitution  Can be extended with types, expressions, stores and side-effects  Introduced by Church in 1930s  Notation for function expressions  Proof system for equality of expressions  Calculation rules for function application (invocation) cs3723 4 Pure Lambda Calculus  Abstract syntax : M ::= x | λ x.M | M M  x represents variable names  Each expression in Lambda Calculus is called a lambda term or a lambda expression  Concrete syntax: how to resolve ambiguity?  ( M M ) has higher precedence than λ x.M ; i.e, λ x.M N <=> λ x. (M N)  M M is left associative; i.e. x y z <=> (x y) z  Add parentheses to allow alternative grouping of operations  Compare: concrete syntax in Lisp/Scheme  M ::= x | (lambda (x) M) | (M M) cs3723 5 The Applied Lambda Calculus  Can pure lambda calculi express all computation?  Yes, it is Turing complete. Other values/operations can be represented as function abstractions.  For example, booleans can be expressed as True = λ t. ( λ f. t) False = λ t. ( λ f. f)  But we are not going to be extreme....
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## This note was uploaded on 10/13/2009 for the course CS 3773 taught by Professor Staff during the Spring '08 term at The University of Texas at San Antonio- San Antonio.

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LambdaCalculus - cs3723 1 Language Semantics Lambda...

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