Operational Semantics of Resolution and Productivity in Horn Clause Logic

This paper presents a study of operational and type-theoretic properties of different resolution strategies in Horn clause logic. We distinguish four different kinds of resolution: resolution by unification (SLD-resolution), resolution by term-matching, the recently introduced structural resolution, and partial (or lazy) resolution. We express them all uniformly as abstract reduction systems, which allows us to undertake a thorough comparative analysis of their properties. To match this small-step semantics, we propose to take Howard's System H as a type-theoretic semantic counterpart. Using System H, we interpret Horn formulas as types, and a derivation for a given formula as the proof term inhabiting the type given by the formula. We prove soundness of these abstract reduction systems relative to System H, and we show completeness of SLD-resolution and structural resolution relative to System H. We identify conditions under which structural resolution is operationally equivalent to SLD-resolution. We show correspondence between term-matching resolution for Horn clause programs without existential variables and term rewriting.Comment: Journal Formal Aspect of Computing, 201

The relationship between galaxy and dark matter halo size from z ∼ 3 to the present

We explore empirical constraints on the statistical relationship between the radial size of galaxies and the radius of their host dark matter haloes from z similar to 0.1-3 using the Galaxy And Mass Assembly (GAMA) and Cosmic Assembly Near Infrared Deep Extragalactic Legacy Survey (CANDELS) surveys. We map dark matter halo mass to galaxy stellar mass using relationships from abundance matching, applied to the Bolshoi-Planck dissipationless N-body simulation. We define SRHR equivalent to r(e)/R-h as the ratio of galaxy radius to halo virial radius, and SRHR lambda equivalent to r(e)/(lambda R-h) as the ratio of galaxy radius to halo spin parameter times halo radius. At z similar to 0.1, we find an average value of SRHR similar or equal to 0.018 and SRHR. similar or equal to 0.5 with very little dependence on stellar mass. Stellar radius-halo radius (SRHR) and SRHR lambda have a weak dependence on cosmic time since z similar to 3. SRHR shows a mild decrease over cosmic time for low-mass galaxies, but increases slightly or does not evolve formoremassive galaxies. We find hints that at high redshift (z similar to 2-3), SRHR. is lower for more massive galaxies, while it shows no significant dependence on stellar mass at z less than or similar to 0.5. We find that for both the GAMA and CANDELS samples, at all redshifts from z similar to 0.1-3, the observed conditional size distribution in stellar mass bins is remarkably similar to the conditional distribution of lambda R-h. We discuss the physical interpretation and implications of these results