![]() ![]() Any quantitative analysis would be simply impossbible. Said differently, if $M \to_\beta N$ by performing a "real" $\beta$-step, then $M$ $\beta$-reduces to $N$ with an arbitrary number of $\beta$-steps. Where $N \to_\beta N$ because $\to_\beta$ is reflexive. ![]() M &\to_\beta N \to_\beta N \to_\beta N \to_\beta \dots\\ M &\to_\beta N \to_\beta N \to_\beta N \\ For instance, if $M \to_\beta N$ by performing a "real" $\beta$-step and $\to_\beta$ is reflexive by definition, then to estimate the evaluation length), it would be impossible to do that if $\beta$-reduction were reflexive. Moreover, if you are interested in counting the number of steps to reach the normal form (i.e. So, if the definition of $\beta$-reduction included reflexivity, it would be really clumsy to define the notion of normal form, which is the crucial notion corresponding to the idea of result of a computation. ![]() And saying that $M$ is normal if and only if $M \to_\beta N$ implies $M = N$ does not solve the problem at all, because according to this attempt of definition the term $(\lambda x.xx) \lambda x.xx$ would be normal but actually it $\beta$-reduces to itself by performing a "real" $\beta$-step, not thanks to reflexivity. What would be a normal form with such a definition? It would be really clumsy to define the notion of normal form, because for every term $M$, we have $M \to_\beta M$, which means that every term $M$ is not normal. It seems bothersome and inelegant to having to precede a beta-reduction with alpha-conversion, the operation of roughly the same complexity. Suppose that $\beta$-reduction were defined so as to be reflexive. On the other hand, alpha-conversion feels rather similar to beta-reduction: in either case, we traverse the body of a lambda-abstraction, looking for a particular variable and replacing it. a term in the $\lambda$-calculus where $\beta$-steps cannot apply) means to evaluate a program to its output, i.e. From the "computer scientist's point of view", a single step $\beta$-reduction roughly corresponds to an elementary step of computation. ![]()
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