Ordering

Classical mereology is based on the idea that parthood is a reflexive, transitive, antisymmetric relation—a weak partial order. Proper parthood is irreflexive, transitive, and asymmetric—serving as the corresponding strict partial order. This chapter critically and carefully evaluates each of these conditions, and considers a range of alternate definitions for proper parthood and other basic notions.. Along the way, it discusses formal relationships between these definitions, and the logical implications of adopting them. The philosophical controversies surrounding these axioms are addressed: discussing self-parts, parthood loops, mutual parts and extensionality, and non-transitive notions such as immediate parthood, local transitivity, and more.

Hypernym Detection Using Strict Partial Order Networks

This paper introduces Strict Partial Order Networks (SPON), a novel neural network architecture designed to enforce asymmetry and transitive properties as soft constraints. We apply it to induce hypernymy relations by training with is-a pairs. We also present an augmented variant of SPON that can generalize type information learned for in-vocabulary terms to previously unseen ones. An extensive evaluation over eleven benchmarks across different tasks shows that SPON consistently either outperforms or attains the state of the art on all but one of these benchmarks.

Witnessing functions in bounded arithmetic and search problems

We investigate the possibility to characterize (multi)functions that are-definable with smalli(i= 1, 2, 3) in fragments of bounded arithmeticT2in terms of natural search problems defined over polynomial-time structures. We obtain the following results:(1) A reformulation of known characterizations of (multi)functions that areand-definable in the theoriesand.(2) New characterizations of (multi)functions that areand-definable in the theory.(3) A new non-conservation result: the theoryis not-conservative over the theory.To prove that the theoryis not-conservative over the theory, we present two examples of a-principle separating the two theories:(a) the weak pigeonhole principle WPHP(a2,f, g) formalizing that no functionfis a bijection betweena2andawith the inverseg,(b) the iteration principle Iter(a, R, f) formalizing that no functionfdefined on a strict partial order ({0,…, a},R) can have increasing iterates.

Time and Duration in Noninterleaving Concurrency

The purpose of this paper is to present a real-timed concurrency theory in the noninterleaving tradition. The theory is based on the occurrences of actions; each occurrence or event has a start and a finish. Causality is modelled by assigning a strict partial order to these starts and finishes, while timing is modelled by giving them reals. The theory is presented in some detail. All of the traditional notions found in concurrency theories (such as conflict, confusion, liveness, and so on) are found to be expressible. Four notions of causality arise naturally from the model, leading to notions of securing. Three of the notions give rise to underlying event structures, demonstrating that our model generalises Winskel’s. Infinite structures are then analysed: a poset of finite structures is defined and suitably completed to give one containing infinite structures. These infinite structures are characterised as just those arising as limits of finite ones. Our technique here, which relies on the structure of time, is of independent interest.