2,406 research outputs found
The case of equality in the Livingstone-Wagner Theorem
Let G be a permutation group acting on a set Ω of size n∈ℕ and let 1≤k<(n−1)/2. Livingstone and Wagner proved that the number of orbits of G on k-subsets of Ω is less than or equal to the number of orbits on (k+1)-subsets. We investigate the cases when equality occurs
The Use of Proof Planning for Cooperative Theorem Proving
AbstractWe describebarnacle: a co-operative interface to theclaminductive theorem proving system. For the foreseeable future, there will be theorems which cannot be proved completely automatically, so the ability to allow human intervention is desirable; for this intervention to be productive the problem of orienting the user in the proof attempt must be overcome. There are many semi-automatic theorem provers: we call our style of theorem provingco-operative, in that the skills of both human and automaton are used each to their best advantage, and used together may find a proof where other methods fail. The co-operative nature of thebarnacleinterface is made possible by the proof planning technique underpinningclam. Our claim is that proof planning makes new kinds of user interaction possible.Proof planning is a technique for guiding the search for a proof in automatic theorem proving. Common patterns of reasoning in proofs are identified and represented computationally as proof plans, which can then be used to guide the search for proofs of new conjectures. We have harnessed the explanatory power of proof planning to enable the user to understand where the automatic prover got to and why it is stuck. A user can analyse the failed proof in terms ofclam's specification language, and hence override the prover to force or prevent the application of a tactic, or discover a proof patch. This patch might be to apply further rules or tactics to bridge the gap between the effects of previous tactics and the preconditions needed by a currently inapplicable tactic
Middle-Out Reasoning for Logic Program Synthesis
We propose a novel approach to automating the synthesis of logic programs: Logic programs are synthesized as a by-product of the planning of a verification proof. The approach is a two-level one: At the object level, we prove program verification conjectures in a sorted, first-order theory. The conjectures are of the form 8args \Gamma\Gamma\Gamma\Gamma! : prog(args \Gamma\Gamma\Gamma\Gamma! ) $ spec(args \Gamma\Gamma\Gamma\Gamma! ). At the meta-level, we plan the object-level verification with an unspecified program definition. The definition is represented with a (second-order) meta-level variable, which becomes instantiated in the course of the planning
A note on commuting graphs for symmetric groups
The commuting graph C(G;X) , where G is a group and X a subset of G, has X as its vertex set with two distinct elements of X joined by an edge when they commute in G. Here the diameter and disc structure of C(G;X) is investigated when G is the symmetric group and X a conjugacy class of
G
Do We Expect Most AGN to Live in Disks?
Recent observations have indicated that a large fraction of the low to
intermediate luminosity AGN population lives in disk-dominated hosts, while the
more luminous quasars live in bulge-dominated hosts, in conflict with some
previous model predictions. We therefore build and compare a semi-empirical
model for AGN fueling which accounts for both merger and non-merger
'triggering.' In particular, we show that the 'stochastic accretion' model - in
which fueling in disk galaxies is essentially a random process arising whenever
dense gas clouds reach the nucleus - provides a good match to the present
observations at low/intermediate luminosities. However it falls short of the
high-luminosity population. We combine this with models for major
merger-induced AGN fueling, which lead to rarer but more luminous events, and
predict the resulting abundance of disk-dominated and bulge-dominated AGN host
galaxies as a function of luminosity and redshift. We compile and compare
observational constraints from z~0-2. The models and observations generically
show a transition from disk to bulge dominance in hosts near the Seyfert-quasar
transition, at all redshifts. 'Stochastic' fueling dominates AGN by number
(dominant at low luminosity), and dominates BH growth below the knee in the
present-day BH mass function (<10^7 M_sun). However it accounts for just ~10%
of BH mass growth at masses >10^8 M_sun. In total, fueling in disky hosts
accounts for ~30% of the total AGN luminosity density/BH mass density. The
combined model also accurately predicts the AGN luminosity function and
clustering/bias as a function of luminosity and redshift; however, we argue
that these are not sensitive probes of BH fueling mechanisms.Comment: 13 pages, 5 figures, PDF updated to match published versio
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