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sav08:untyped_lambda_calculus [2009/03/05 12:45] vkuncak |
sav08:untyped_lambda_calculus [2009/03/05 12:47] vkuncak |
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\end{array}\] | \end{array}\] | ||
That is, an expression is a variable, a lambda expression defining a function, or an application of one expression to another. | That is, an expression is a variable, a lambda expression defining a function, or an application of one expression to another. | ||
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===== Beta reduction ===== | ===== Beta reduction ===== | ||
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The main rule of lambda calculus is beta reduction, which formalizes function application: | The main rule of lambda calculus is beta reduction, which formalizes function application: | ||
\begin{equation*} | \begin{equation*} | ||
- | (\lambda x.\, s) t\ \to\limits^{\beta}\ s[x:=t] | + | (\lambda x.\, s) t\ \mathop{\to}\limits^{\beta}\ s[x:=t] |
\end{equation*} | \end{equation*} | ||
where we assume capture-avoiding substitution. In general, we identify terms that differ only by renaming of fresh bound variables. | where we assume capture-avoiding substitution. In general, we identify terms that differ only by renaming of fresh bound variables. | ||
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+ | Here $\mathop{\to}\limits^{\beta}$ is a binary 'reduction' relation on terms. We build larger relations on terms by | ||
+ | - performing reduction in any context $C$, so we have $C[(\lambda x.\, s) t]\ \mathop{\to}\limits^{\beta}\ C[s[x:=t]]$ | ||
+ | - taking its transitive closure: computes all terms reachable by reduction | ||
===== Non-termination and Difficulties with Semantics ===== | ===== Non-termination and Difficulties with Semantics ===== |