A new coinductive confluence proof for infinitary lambda calculus

We present a new and formal coinductive proof of confluence and normalisation of B\"ohm reduction in infinitary lambda calculus. The proof is simpler than previous proofs of this result. The technique of the proof is new, i.e., it is not merely a coinductive reformulation of any earlier proofs. We formalised the proof in the Coq proof assistant.Comment: arXiv admin note: text overlap with arXiv:1501.0435

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 λ\lambda-Calculi from a Linear Perspective (Long Version)

We introduce a linear infinitary $\lambda$-calculus, called $\ell\Lambda_{\infty}$, in which two exponential modalities are available, the first one being the usual, finitary one, the other being the only construct interpreted coinductively. The obtained calculus embeds the infinitary applicative $\lambda$-calculus and is universal for computations over infinite strings. What is particularly interesting about $\ell\Lambda_{\infty}$, is that the refinement induced by linear logic allows to restrict both modalities so as to get calculi which are terminating inductively and productive coinductively. We exemplify this idea by analysing a fragment of $\ell\Lambda$ built around the principles of $\mathsf{SLL}$ and $\mathsf{4LL}$. Interestingly, it enjoys confluence, contrarily to what happens in ordinary infinitary $\lambda$-calculi

Encoding many-valued logic in {\lambda}-calculus

We extend the well-known Church encoding of two-valued Boolean Logic in $\lambda$-calculus to encodings of $n$-valued propositional logic (for $3\leq n\leq 5$) in well-chosen infinitary extensions in $\lambda$-calculus. In case of three-valued logic we use the infinitary extension of the finite $\lambda$-calculus in which all terms have their B\"ohm tree as their unique normal form. We refine this construction for $n\in\{4,5\}$. These $n$-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 $n$-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 $\lambda{\mathbf I}$-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

A Finite Semantics of Simply-Typed Lambda Terms for Infinite Runs of<br> Automata

Model checking properties are often described by means of finite automata. Any particular such automaton divides the set of infinite trees into finitely many classes, according to which state has an infinite run. Building the full type hierarchy upon this interpretation of the base type gives a finite semantics for simply-typed lambda-trees. A calculus based on this semantics is proven sound and complete. In particular, for regular infinite lambda-trees it is decidable whether a given automaton has a run or not. As regular lambda-trees are precisely recursion schemes, this decidability result holds for arbitrary recursion schemes of arbitrary level, without any syntactical restriction.Comment: 23 page