Approximately multiplicative maps from weighted semilattice algebras

We investigate which weighted convolution algebras $\ell^1_\omega(S)$, where $S$ is a semilattice, are AMNM in the sense of Johnson (JLMS, 1986). We give an explicit example where this is not the case. We show that the unweighted examples are all AMNM, as are all $\ell^1_\omega(S)$ where $S$ has either finite width or finite height. Some of these finite-width examples are isomorphic to function algebras studied by Feinstein (IJMMS, 1999). We also investigate when $(\ell^1_\omega(S),{\bf M}_2)$ is an AMNM pair in the sense of Johnson (JLMS, 1988), where ${\bf M}_2$ denotes the algebra of 2-by-2 complex matrices. In particular, we obtain the following two contrasting results: (i) for many non-trivial weights on the totally ordered semilattice ${\bf N}_{\min}$, the pair $(\ell^1_\omega({\bf N}_{\min}),{\bf M}_2)$ is not AMNM; (ii) for any semilattice $S$, the pair $(\ell^1(S),{\bf M}_2)$ is AMNM. The latter result requires a detailed analysis of approximately commuting, approximately idempotent $2\times 2$ matrices.Comment: AMS-LaTeX. v3: 31 pages, additional minor corrections to v2. Final version, to appear in J. Austral. Math. Soc. v4: small correction of mis-statement at start of Section 4 (this should also be fixed in the journal version

A Generic Undo Support for State-Based CRDTs

CRDTs (Conflict-free Replicated Data Types) have properties desirable for large-scale distributed systems with variable network latency or transient partitions. With CRDT, data are always available for local updates and data states converge when the replicas have incorporated the same updates. Undo is useful for correcting human mistakes and for restoring system-wide invariant violated due to long delays or network partitions. There is currently no generally applicable undo support for CRDTs. There are at least two reasons for this. First, there is currently no abstraction that we can practically use to capture the relations between undo and normal operations with respect to concurrency and causality. Second, using inverse operations as the existing partial solutions, the CRDT designer has to hard-code certain rules and design a new CRDT for almost every operation that needs undo support. In this paper, we present an approach to generic support of undo for CRDTs. The approach consists of two major parts. We first work out an abstraction that captures the semantics of concurrent undo and redo operations through equivalence classes. The abstraction is a natural extension of undo and redo in sequential applications and is straightforward to implement in practice. By using this abstraction, we then device a mechanism to augment existing CRDTs. The mechanism provides an "out of the box" support for undo without the involvement of the CRDT designers. We also present a practical application of the approach in collaborative editing

Tabling with Sound Answer Subsumption

Tabling is a powerful resolution mechanism for logic programs that captures their least fixed point semantics more faithfully than plain Prolog. In many tabling applications, we are not interested in the set of all answers to a goal, but only require an aggregation of those answers. Several works have studied efficient techniques, such as lattice-based answer subsumption and mode-directed tabling, to do so for various forms of aggregation. While much attention has been paid to expressivity and efficient implementation of the different approaches, soundness has not been considered. This paper shows that the different implementations indeed fail to produce least fixed points for some programs. As a remedy, we provide a formal framework that generalises the existing approaches and we establish a soundness criterion that explains for which programs the approach is sound. This article is under consideration for acceptance in TPLP.Comment: Paper presented at the 32nd International Conference on Logic Programming (ICLP 2016), New York City, USA, 16-21 October 2016, 15 pages, LaTeX, 0 PDF figure

Categoricity and multidimensional diagrams

We study multidimensional diagrams in independent amalgamation in the framework of abstract elementary classes (AECs). We use them to prove the eventual categoricity conjecture for AECs, assuming a large cardinal axiom. More precisely, we show assuming the existence of a proper class of strongly compact cardinals that an AEC which has a single model of some high-enough cardinality will have a single model in any high-enough cardinal. Assuming a weak version of the generalized continuum hypothesis, we also establish the eventual categoricity conjecture for AECs with amalgamation.Comment: 63 page