43,152 research outputs found
On finitely recursive programs
Disjunctive finitary programs are a class of logic programs admitting
function symbols and hence infinite domains. They have very good computational
properties, for example ground queries are decidable while in the general case
the stable model semantics is highly undecidable. In this paper we prove that a
larger class of programs, called finitely recursive programs, preserves most of
the good properties of finitary programs under the stable model semantics,
namely: (i) finitely recursive programs enjoy a compactness property; (ii)
inconsistency checking and skeptical reasoning are semidecidable; (iii)
skeptical resolution is complete for normal finitely recursive programs.
Moreover, we show how to check inconsistency and answer skeptical queries using
finite subsets of the ground program instantiation. We achieve this by
extending the splitting sequence theorem by Lifschitz and Turner: We prove that
if the input program P is finitely recursive, then the partial stable models
determined by any smooth splitting omega-sequence converge to a stable model of
P.Comment: 26 pages, Preliminary version in Proc. of ICLP 2007, Best paper awar
Finite Groebner bases in infinite dimensional polynomial rings and applications
We introduce the theory of monoidal Groebner bases, a concept which
generalizes the familiar notion in a polynomial ring and allows for a
description of Groebner bases of ideals that are stable under the action of a
monoid. The main motivation for developing this theory is to prove finiteness
theorems in commutative algebra and its applications. A major result of this
type is that ideals in infinitely many indeterminates stable under the action
of the symmetric group are finitely generated up to symmetry. We use this
machinery to give new proofs of some classical finiteness theorems in algebraic
statistics as well as a proof of the independent set conjecture of Hosten and
the second author.Comment: 24 pages, adds references to work of Cohen, adds more details in
Section
On 2-representation infinite algebras arising from dimer models
The Jacobian algebra arising from a consistent dimer model is a bimodule
-Calabi-Yau algebra, and its center is a -dimensional Gorenstein toric
singularity. A perfect matching of a dimer model gives the degree making the
Jacobian algebra -graded. It is known that if the degree zero part
of such an algebra is finite dimensional, then it is a -representation
infinite algebra which is a generalization of a representation infinite
hereditary algebra. In this paper, we show that internal perfect matchings,
which correspond to toric exceptional divisors on a crepant resolution of a
-dimensional Gorenstein toric singularity, characterize the property that
the degree zero part of the Jacobian algebra is finite dimensional. Moreover,
combining this result with the theorems due to Amiot-Iyama-Reiten, we show that
the stable category of graded maximal Cohen-Macaulay modules admits a tilting
object for any -dimensional Gorenstein toric isolated singularity. We then
show that all internal perfect matchings corresponding to the same toric
exceptional divisor are transformed into each other using the mutations of
perfect matchings, and this induces derived equivalences of -representation
infinite algebras.Comment: 28 pages, v2: improved some proof
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