3 research outputs found
From Quantifier Depth to Quantifier Number: Separating Structures with k Variables
Given two -element structures, and , which can
be distinguished by a sentence of -variable first-order logic
(), what is the minimum such that there is guaranteed to
be a sentence with at most quantifiers, such
that but ? We present
various results related to this question obtained by using the recently
introduced QVT games. In particular, we show that when we limit the number of
variables, there can be an exponential gap between the quantifier depth and the
quantifier number needed to separate two structures. Through the lens of this
question, we will highlight some difficulties that arise in analysing the QVT
game and some techniques which can help to overcome them. As a consequence, we
show that is exponentially more succinct than
. We also show, in the setting of the existential-positive
fragment, how to lift quantifier depth lower bounds to quantifier number lower
bounds. This leads to almost tight bounds.Comment: 53 pages, 8 figures; added new result on the relative succinctness of
finite variable logi
Structured d-DNNF Is Not Closed Under Negation
Both structured d-DNNF and SDD can be exponentially more succinct than OBDD.
Moreover, SDD is essentially as tractable as OBDD. But this has left two
important open questions. Firstly, does OBDD support more tractable
transformations than structured d-DNNF? And secondly, is structured d-DNNF more
succinct than SDD? In this paper, we answer both questions in the affirmative.
For the first question we show that, unlike OBDD, structured d-DNNF does not
support polytime negation, disjunction, or existential quantification
operations. As a corollary, we deduce that there are functions with an
equivalent polynomial-sized structured d-DNNF but with no such representation
as an SDD, thus answering the second question. We also lift this second result
to arithmetic circuits (AC) to show a succinctness gap between PSDD and the
monotone AC analogue to structured d-DNNF.Comment: 9 pages, 2 figure