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The Omega Rule is $\mathbf{\Pi_{1}^{1}}$-Complete in the $\lambda\beta$-Calculus

Benedetto Intrigila ; Richard Statman.

In a functional calculus, the so called \Omega-rule states that if two terms P and Q applied to any closed term <i>N</i> return the same value (i.e. PN = QN), then they are equal (i.e. P = Q holds). As it is well known, in the \lambda\beta-calculus the \Omega-rule does not hold, even when the&nbsp;[&hellip;]

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Solution of a Problem of Barendregt on Sensible lambda-Theories

Benedetto Intrigila ; Richard Statman.

<i>H</i> is the theory extending &#946;-conversion by identifying all closed unsolvables. <i>H</i>&#969; is the closure of this theory under the &#969;-rule (and &#946;-conversion). A long-standing conjecture of H. Barendregt states that the provable equations of <i>H</i>&#969; form&nbsp;[&hellip;]

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Degrees of extensionality in the theory of B\"ohm trees and Sall\'e's conjecture

Benedetto Intrigila ; Giulio Manzonetto ; Andrew Polonsky.

The main observational equivalences of the untyped lambda-calculus have been characterized in terms of extensional equalities between B\"ohm trees. It is well known that the lambda-theory H*, arising by taking as observables the head normal forms, equates two lambda-terms whenever their B\"ohm trees&nbsp;[&hellip;]

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Addressing Machines as models of lambda-calculus

Giuseppe Della Penna ; Benedetto Intrigila ; Giulio Manzonetto.

Turing machines and register machines have been used for decades in theoretical computer science as abstract models of computation. Also the $\lambda$-calculus has played a central role in this domain as it allows to focus on the notion of functional computation, based on the substitution mechanism,&nbsp;[&hellip;]

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