10a-lambda-calc

10a-lambda-calc - The Other Model: Calculus So far we have...

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The Other Model: λ Calculus So far we have only looked at Turing machines as our model. Question: Where does the Church in the Church-Turing thesis come from? Answer: Alonzo Church invented the λ calculus to model computing with functions. Turns out the λ calculus is as powerful as the Turing machine, we are just not used to thinking in terms of universal computing using just functions. –p
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The Other Model: λ Calculus The next couple of lectures will show that the λ calculus and the Turing machine are equivalent in terms of computational power. 1. Introduce the λ calculus. 2. Introduce the primitive- and µ -recursive. 3. Show that the λ calculus implements the µ -recursive functions. 4. Show that a Turing machine implements the µ -recursive functions. 5. Show that a Turing machine implements the λ -calculus. 6. Show that µ -recursive functions can implement Turing machines. 7. Because of 3 we can conclude that λ calculus can implement Turing machines. 8. From 5 and 7 we conclude that the λ -calculus and Turing machines are computationally equivalent. One way to look at the λ calculus is as a term rewriting system. –p
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λ Calculus 101 λ -calculus aims to model computation with functions. At the core of this calculus are λ -expressions of the form λx. E denoting functions with a parameter x and a function body E . Here is the variable x is assumed to be free in E (i.e. the variable is assumed not bound by a λ -operator). The syntax for the calculus can be summarized by the following context-free grammar, <function> ::= λ <var> . <expression> <expression> ::= <var> | <function> | <application> <application> ::= <expression><expression> Example: ( λx. x )( λf. ( λy. f y )) –p
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λ Calculus 101 Rules: The calculus is very simple, it essentially consists of only three rules:
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10a-lambda-calc - The Other Model: Calculus So far we have...

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