58 research outputs found
Encoding many-valued logic in {\lambda}-calculus
We extend the well-known Church encoding of two-valued Boolean Logic in
-calculus to encodings of -valued propositional logic (for ) in well-chosen infinitary extensions in -calculus. In case
of three-valued logic we use the infinitary extension of the finite
-calculus in which all terms have their B\"ohm tree as their unique
normal form. We refine this construction for . These -valued
logics are all variants of McCarthy's left-sequential, three-valued
propositional calculus. The four- and five-valued logic have been given
complete axiomatisations by Bergstra and Van de Pol. The encodings of these
-valued logics are of interest because they can be used to calculate the
truth values of infinitary propositions. With a novel application of McCarthy's
three-valued logic we can now resolve Russell's paradox. Since B\"ohm trees are
always finite in Church's original -calculus, we believe
their construction to be within the technical means of Church. Arguably he
could have found this encoding of three-valued logic and used it to resolve
Russell's paradox.Comment: 15 page
On Undefined and Meaningless in Lambda Definability
We distinguish between undefined terms as used in lambda definability
of partial recursive functions and meaningless terms as used in
infinite lambda calculus for the infinitary terms models that
generalise the Bohm model. While there are uncountable many known
sets of meaningless terms, there are four known sets of undefined
terms. Two of these four are sets of meaningless terms.
In this paper we first present set of sufficient conditions for a set
of lambda terms to serve as set of undefined terms in lambda
definability of partial functions. The four known sets of undefined
terms satisfy these conditions.
Next we locate the smallest set of meaningless terms satisfying these
conditions. This set sits very low in the lattice of all sets of
meaningless terms. Any larger set of meaningless terms than this
smallest set is a set of undefined terms. Thus we find uncountably
many new sets of undefined terms.
As an unexpected bonus of our careful analysis of lambda definability
we obtain a natural modification, strict lambda-definability, which
allows for a Barendregt style of proof in which the representation of
composition is truly the composition of representations
Nominal Coalgebraic Data Types with Applications to Lambda Calculus
We investigate final coalgebras in nominal sets. This allows us to define types of infinite data with binding for which all constructions automatically respect alpha equivalence. We give applications to the infinitary lambda calculus
Infinitary lambda calculus and discrimination of Berarducci trees
AbstractWe propose an extension of lambda calculus for which the Berarducci trees equality coincides with observational equivalence, when we observe rootstable or rootactive behavior of terms. In one direction the proof is an adaptation of the classical Böhm out technique. In the other direction the proof is based on confluence for strongly converging reductions in this extension
Transfinite reductions in orthogonal term rewriting systems
Strongly convergent reduction is the fundamental notion of reduction in infinitary orthogonal term rewriting systems (OTRSs). For these we prove the Transfinite Parallel Moves Lemma and the Compressing Lemma. Strongness is necessary as shown by counterexamples. Normal forms, which we allow to be infinite, are unique, in contrast to ω-normal forms. Strongly converging fair reductions result in normal forms. In general OTRSs the infinite Church-Rosser Property fails for strongly converging reductions. However for Böhm reduction (as in Lambda Calculus, subterms without head normal forms may be replaced by ⊥) the infinite Church-Rosser property does hold. The infinite Church-Rosser Property for non-unifiable OTRSs follows. The top-terminating OTRSs of Dershowitz c.s. are examples of non-unifiable OTRSs
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