Termination of rewrite relations on λ\lambda-terms based on Girard's notion of reducibility

In this paper, we show how to extend the notion of reducibility introduced by Girard for proving the termination of $\beta$-reduction in the polymorphic $\lambda$-calculus, to prove the termination of various kinds of rewrite relations on $\lambda$-terms, including rewriting modulo some equational theory and rewriting with matching modulo $\beta$$\eta$, by using the notion of computability closure. This provides a powerful termination criterion for various higher-order rewriting frameworks, including Klop's Combinatory Reductions Systems with simple types and Nipkow's Higher-order Rewrite Systems

Fusion in Fractional Level sl^(2)-Theories with k=-1/2

The fusion rules of conformal field theories admitting an sl^(2)-symmetry at level k=-1/2 are studied. It is shown that the fusion closes on the set of irreducible highest weight modules and their images under spectral flow, but not when "highest weight" is replaced with "relaxed highest weight". The fusion of the relaxed modules, necessary for a well-defined u^(1)-coset, gives two families of indecomposable modules on which the Virasoro zero-mode acts non-diagonalisably. This confirms the logarithmic nature of the associated theories. The structures of the indecomposable modules are completely determined as staggered modules and it is shown that there are no logarithmic couplings (beta-invariants). The relation to the fusion ring of the c=-2 triplet model and the implications for the beta gamma ghost system are briefly discussed.Comment: 33 pages, 8 figures; v2 - added a ref and deleted a paragraph from the conclusion

Exploring the Boundaries of Monad Tensorability on Set

We study a composition operation on monads, equivalently presented as large equational theories. Specifically, we discuss the existence of tensors, which are combinations of theories that impose mutual commutation of the operations from the component theories. As such, they extend the sum of two theories, which is just their unrestrained combination. Tensors of theories arise in several contexts; in particular, in the semantics of programming languages, the monad transformer for global state is given by a tensor. We present two main results: we show that the tensor of two monads need not in general exist by presenting two counterexamples, one of them involving finite powerset (i.e. the theory of join semilattices); this solves a somewhat long-standing open problem, and contrasts with recent results that had ruled out previously expected counterexamples. On the other hand, we show that tensors with bounded powerset monads do exist from countable powerset upwards

The graph rewriting calculus: confluence and expressiveness

Introduced at the end of the nineties, the Rewriting Calculus (rho-calculus, for short) is a simple calculus that uniformly integrates term-rewriting and lambda-calculus. The Rhog has been recently introduced as an extension of the rho-calculus, handling structures with cycles and sharing. The calculus over terms is naturally generalized by using unification constraints in addition to the standard rho-calculus matching constraints. This leads to a term-graph representation in an equational style where terms consist of unordered lists of equations. In this paper we show that the (linear) Rhog is confluent. The proof of this result is quite elaborated, due to the non-termination of the system and to the fact that we work on equivalence classes of terms. We also show that the Rhog can be seen as a generalization of first-order term-graph rewriting, in the sense that for any term-graph rewrite step a corresponding sequence of rewritings can be found in the Rhog

Degenerate Viterbi decoding

We present a decoding algorithm for quantum convolutional codes that finds the class of degenerate errors with the largest probability conditioned on a given error syndrome. The algorithm runs in time linear with the number of qubits. Previous decoding algorithms for quantum convolutional codes optimized the probability over individual errors instead of classes of degenerate errors. Using Monte Carlo simulations, we show that this modification to the decoding algorithm results in a significantly lower block error rate