12,656 research outputs found
Finite Orbits of Language Operations
We consider a set of natural operations on languages, and prove that the
orbit of any language L under the monoid generated by this set is finite and
bounded, independently of L. This generalizes previous results about
complement, Kleene closure, and positive closure
Finite Orbits of Language Operations
We consider a set of natural operations on languages, and prove that the orbit of any language L under the monoid generated by this set is finite and bounded, independently of L. This generalizes previous results about complement, Kleene closure, and positive closure
SMT Solving for Functional Programming over Infinite Structures
We develop a simple functional programming language aimed at manipulating
infinite, but first-order definable structures, such as the countably infinite
clique graph or the set of all intervals with rational endpoints. Internally,
such sets are represented by logical formulas that define them, and an external
satisfiability modulo theories (SMT) solver is regularly run by the interpreter
to check their basic properties.
The language is implemented as a Haskell module.Comment: In Proceedings MSFP 2016, arXiv:1604.0038
Timed pushdown automata revisited
This paper contains two results on timed extensions of pushdown automata
(PDA). As our first result we prove that the model of dense-timed PDA of
Abdulla et al. collapses: it is expressively equivalent to dense-timed PDA with
timeless stack. Motivated by this result, we advocate the framework of
first-order definable PDA, a specialization of PDA in sets with atoms, as the
right setting to define and investigate timed extensions of PDA. The general
model obtained in this way is Turing complete. As our second result we prove
NEXPTIME upper complexity bound for the non-emptiness problem for an expressive
subclass. As a byproduct, we obtain a tight EXPTIME complexity bound for a more
restrictive subclass of PDA with timeless stack, thus subsuming the complexity
bound known for dense-timed PDA.Comment: full technical report of LICS'15 pape
The algebraic dichotomy conjecture for infinite domain Constraint Satisfaction Problems
We prove that an -categorical core structure primitively positively
interprets all finite structures with parameters if and only if some stabilizer
of its polymorphism clone has a homomorphism to the clone of projections, and
that this happens if and only if its polymorphism clone does not contain
operations , , satisfying the identity .
This establishes an algebraic criterion equivalent to the conjectured
borderline between P and NP-complete CSPs over reducts of finitely bounded
homogenous structures, and accomplishes one of the steps of a proposed strategy
for reducing the infinite domain CSP dichotomy conjecture to the finite case.
Our theorem is also of independent mathematical interest, characterizing a
topological property of any -categorical core structure (the existence
of a continuous homomorphism of a stabilizer of its polymorphism clone to the
projections) in purely algebraic terms (the failure of an identity as above).Comment: 15 page
Order 3 Symmetry in the Clifford Hierarchy
We investigate the action of the first three levels of the Clifford hierarchy
on sets of mutually unbiased bases comprising the Ivanovic MUB and the Alltop
MUBs. Vectors in the Alltop MUBs exhibit additional symmetries when the
dimension is a prime number equal to 1 modulo 3 and thus the set of all Alltop
vectors splits into three Clifford orbits. These vectors form configurations
with so-called Zauner subspaces, eigenspaces of order 3 elements of the
Clifford group highly relevant to the SIC problem. We identify Alltop vectors
as the magic states that appear in the context of fault-tolerant universal
quantum computing, wherein the appearance of distinct Clifford orbits implies a
surprising inequivalence between some magic states.Comment: 20 pages, 2 figures. Published versio
Schaefer's theorem for graphs
Schaefer's theorem is a complexity classification result for so-called
Boolean constraint satisfaction problems: it states that every Boolean
constraint satisfaction problem is either contained in one out of six classes
and can be solved in polynomial time, or is NP-complete.
We present an analog of this dichotomy result for the propositional logic of
graphs instead of Boolean logic. In this generalization of Schaefer's result,
the input consists of a set W of variables and a conjunction \Phi\ of
statements ("constraints") about these variables in the language of graphs,
where each statement is taken from a fixed finite set \Psi\ of allowed
quantifier-free first-order formulas; the question is whether \Phi\ is
satisfiable in a graph.
We prove that either \Psi\ is contained in one out of 17 classes of graph
formulas and the corresponding problem can be solved in polynomial time, or the
problem is NP-complete. This is achieved by a universal-algebraic approach,
which in turn allows us to use structural Ramsey theory. To apply the
universal-algebraic approach, we formulate the computational problems under
consideration as constraint satisfaction problems (CSPs) whose templates are
first-order definable in the countably infinite random graph. Our method to
classify the computational complexity of those CSPs is based on a
Ramsey-theoretic analysis of functions acting on the random graph, and we
develop general tools suitable for such an analysis which are of independent
mathematical interest.Comment: 54 page
Local integrability results in harmonic analysis on reductive groups in large positive characteristic
Let be a connected reductive algebraic group over a non-Archimedean local
field , and let be its Lie algebra. By a theorem of Harish-Chandra, if
has characteristic zero, the Fourier transforms of orbital integrals are
represented on the set of regular elements in by locally constant
functions, which, extended by zero to all of , are locally integrable. In
this paper, we prove that these functions are in fact specializations of
constructible motivic exponential functions. Combining this with the Transfer
Principle for integrability [R. Cluckers, J. Gordon, I. Halupczok, "Transfer
principles for integrability and boundedness conditions for motivic exponential
functions", preprint arXiv:1111.4405], we obtain that Harish-Chandra's theorem
holds also when is a non-Archimedean local field of sufficiently large
positive characteristic. Under the hypothesis on the existence of the mock
exponential map, this also implies local integrability of Harish-Chandra
characters of admissible representations of , where is an
equicharacteristic field of sufficiently large (depending on the root datum of
) characteristic.Comment: Compared to v2/v3: some proofs simplified, the main statement
generalized; slightly reorganized. Regarding the automatically generated text
overlap note: it overlaps with the Appendix B (which is part of
arXiv:1208.1945) written by us; the appendix and this article cross-reference
each other, and since the set-up is very similar, some overlap is unavoidabl
- …