33,008 research outputs found
A decidable quantified fragment of set theory with ordered pairs and some undecidable extensions
In this paper we address the decision problem for a fragment of set theory
with restricted quantification which extends the language studied in [4] with
pair related quantifiers and constructs, in view of possible applications in
the field of knowledge representation. We will also show that the decision
problem for our language has a non-deterministic exponential time complexity.
However, for the restricted case of formulae whose quantifier prefixes have
length bounded by a constant, the decision problem becomes NP-complete. We also
observe that in spite of such restriction, several useful set-theoretic
constructs, mostly related to maps, are expressible. Finally, we present some
undecidable extensions of our language, involving any of the operators domain,
range, image, and map composition.
[4] Michael Breban, Alfredo Ferro, Eugenio G. Omodeo and Jacob T. Schwartz
(1981): Decision procedures for elementary sublanguages of set theory. II.
Formulas involving restricted quantifiers, together with ordinal, integer, map,
and domain notions. Communications on Pure and Applied Mathematics 34, pp.
177-195Comment: In Proceedings GandALF 2012, arXiv:1210.202
Type-elimination-based reasoning for the description logic SHIQbs using decision diagrams and disjunctive datalog
We propose a novel, type-elimination-based method for reasoning in the
description logic SHIQbs including DL-safe rules. To this end, we first
establish a knowledge compilation method converting the terminological part of
an ALCIb knowledge base into an ordered binary decision diagram (OBDD) which
represents a canonical model. This OBDD can in turn be transformed into
disjunctive Datalog and merged with the assertional part of the knowledge base
in order to perform combined reasoning. In order to leverage our technique for
full SHIQbs, we provide a stepwise reduction from SHIQbs to ALCIb that
preserves satisfiability and entailment of positive and negative ground facts.
The proposed technique is shown to be worst case optimal w.r.t. combined and
data complexity and easily admits extensions with ground conjunctive queries.Comment: 38 pages, 3 figures, camera ready version of paper accepted for
publication in Logical Methods in Computer Scienc
Who witnesses The Witness? Finding witnesses in The Witness is hard and sometimes impossible
We analyze the computational complexity of the many types of
pencil-and-paper-style puzzles featured in the 2016 puzzle video game The
Witness. In all puzzles, the goal is to draw a simple path in a rectangular
grid graph from a start vertex to a destination vertex. The different puzzle
types place different constraints on the path: preventing some edges from being
visited (broken edges); forcing some edges or vertices to be visited
(hexagons); forcing some cells to have certain numbers of incident path edges
(triangles); or forcing the regions formed by the path to be partially
monochromatic (squares), have exactly two special cells (stars), or be singly
covered by given shapes (polyominoes) and/or negatively counting shapes
(antipolyominoes). We show that any one of these clue types (except the first)
is enough to make path finding NP-complete ("witnesses exist but are hard to
find"), even for rectangular boards. Furthermore, we show that a final clue
type (antibody), which necessarily "cancels" the effect of another clue in the
same region, makes path finding -complete ("witnesses do not exist"),
even with a single antibody (combined with many anti/polyominoes), and the
problem gets no harder with many antibodies. On the positive side, we give a
polynomial-time algorithm for monomino clues, by reducing to hexagon clues on
the boundary of the puzzle, even in the presence of broken edges, and solving
"subset Hamiltonian path" for terminals on the boundary of an embedded planar
graph in polynomial time.Comment: 72 pages, 59 figures. Revised proof of Lemma 3.5. A short version of
this paper appeared at the 9th International Conference on Fun with
Algorithms (FUN 2018
On some varieties associated with trees
This article considers some affine algebraic varieties attached to finite
trees and closely related to cluster algebras. Their definition involves a
canonical coloring of vertices of trees into three colors. These varieties are
proved to be smooth and to admit sometimes free actions of algebraic tori. Some
results are obtained on their number of points over finite fields and on their
cohomology.Comment: 37 pages, 7 figure
Tiling Problems on Baumslag-Solitar groups
We exhibit a weakly aperiodic tile set for Baumslag-Solitar groups, and prove
that the domino problem is undecidable on these groups. A consequence of our
construction is the existence of an arecursive tile set on Baumslag-Solitar
groups.Comment: In Proceedings MCU 2013, arXiv:1309.104
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