746 research outputs found

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

    Get PDF
    We introduce a linear infinitary λ\lambda-calculus, called ℓΛ∞\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 λ\lambda-calculus and is universal for computations over infinite strings. What is particularly interesting about ℓΛ∞\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 ℓΛ\ell\Lambda built around the principles of SLL\mathsf{SLL} and 4LL\mathsf{4LL}. Interestingly, it enjoys confluence, contrarily to what happens in ordinary infinitary λ\lambda-calculi

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

    Full text link
    We extend the well-known Church encoding of two-valued Boolean Logic in λ\lambda-calculus to encodings of nn-valued propositional logic (for 3≀n≀53\leq n\leq 5) in well-chosen infinitary extensions in λ\lambda-calculus. In case of three-valued logic we use the infinitary extension of the finite λ\lambda-calculus in which all terms have their B\"ohm tree as their unique normal form. We refine this construction for n∈{4,5}n\in\{4,5\}. These nn-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 nn-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 λI\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

    Infinitary Combinatory Reduction Systems: Normalising Reduction Strategies

    Get PDF
    We study normalising reduction strategies for infinitary Combinatory Reduction Systems (iCRSs). We prove that all fair, outermost-fair, and needed-fair strategies are normalising for orthogonal, fully-extended iCRSs. These facts properly generalise a number of results on normalising strategies in first-order infinitary rewriting and provide the first examples of normalising strategies for infinitary lambda calculus

    Infinitary lambda calculus

    Get PDF
    In a previous paper we have established the theory of transfinite reduction for orthogonal term rewriting systems. In this paper we perform the same task for the lambda calculus. From the viewpoint of infinitary rewriting, the Böhm model of the lambda calculus can be seen as an infinitary term model. In contrast to term rewriting, there are several different possible notions of infinite term, which give rise to different Böhm-like models, which embody different notions of lazy or eager computation

    Strict Ideal Completions of the Lambda Calculus

    Get PDF
    The infinitary lambda calculi pioneered by Kennaway et al. extend the basic lambda calculus by metric completion to infinite terms and reductions. Depending on the chosen metric, the resulting infinitary calculi exhibit different notions of strictness. To obtain infinitary normalisation and infinitary confluence properties for these calculi, Kennaway et al. extend ÎČ\beta-reduction with infinitely many `⊄\bot-rules', which contract meaningless terms directly to ⊄\bot. Three of the resulting B\"ohm reduction calculi have unique infinitary normal forms corresponding to B\"ohm-like trees. In this paper we develop a corresponding theory of infinitary lambda calculi based on ideal completion instead of metric completion. We show that each of our calculi conservatively extends the corresponding metric-based calculus. Three of our calculi are infinitarily normalising and confluent; their unique infinitary normal forms are exactly the B\"ohm-like trees of the corresponding metric-based calculi. Our calculi dispense with the infinitely many ⊄\bot-rules of the metric-based calculi. The fully non-strict calculus (called 111111) consists of only ÎČ\beta-reduction, while the other two calculi (called 001001 and 101101) require two additional rules that precisely state their strictness properties: λx.⊄→⊄\lambda x.\bot \to \bot (for 001001) and ⊄ M→⊄\bot\,M \to \bot (for 001001 and 101101)

    A new coinductive confluence proof for infinitary lambda calculus

    Full text link
    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
    • 

    corecore