53,936 research outputs found
Noncommutative Nullstellens\"atze and Perfect Games
The foundations of classical Algebraic Geometry and Real Algebraic Geometry
are the Nullstellensatz and Positivstellensatz. Over the last two decades the
basic analogous theorems for matrix and operator theory (noncommutative
variables) have emerged. This paper concerns commuting operator strategies for
nonlocal games, recalls NC Nullstellensatz which are helpful, extends these,
and applies them to a very broad collection of games. In the process it brings
together results spread over different literatures, hence rather than being
terse, our style is fairly expository.
The main results of this paper are two characterizations, based on
Nullstellensatz, which apply to games with perfect commuting operator
strategies. The first applies to all games and reduces the question of whether
or not a game has a perfect commuting operator strategy to a question involving
left ideals and sums of squares. Previously, Paulsen and others translated the
study of perfect synchronous games to problems entirely involving a
-algebra.The characterization we present is analogous, but works for all
games. The second characterization is based on a new Nullstellensatz we derive
in this paper. It applies to a class of games we call torically determined
games, special cases of which are XOR and linear system games. For these games
we show the question of whether or not a game has a perfect commuting operator
strategy reduces to instances of the subgroup membership problem and, for
linear systems games, we further show this subgroup membership characterization
is equivalent to the standard characterization of perfect commuting operator
strategies in terms of solution groups. Both the general and torically
determined games characterizations are amenable to computer algebra techniques,
which we also develop.Comment: 58 page
Curious properties of free hypergraph C*-algebras
A finite hypergraph consists of a finite set of vertices and a
collection of subsets which we consider as partition
of unity relations between projection operators. These partition of unity
relations freely generate a universal C*-algebra, which we call the "free
hypergraph C*-algebra" . General free hypergraph C*-algebras were first
studied in the context of quantum contextuality. As special cases, the class of
free hypergraph C*-algebras comprises quantum permutation groups, maximal group
C*-algebras of graph products of finite cyclic groups, and the C*-algebras
associated to quantum graph homomorphism, isomorphism, and colouring.
Here, we conduct the first systematic study of aspects of free hypergraph
C*-algebras. We show that they coincide with the class of finite colimits of
finite-dimensional commutative C*-algebras, and also with the class of
C*-algebras associated to synchronous nonlocal games. We had previously shown
that it is undecidable to determine whether is nonzero for given .
We now show that it is also undecidable to determine whether a given
is residually finite-dimensional, and similarly whether it only has
infinite-dimensional representations, and whether it has a tracial state. It
follows that for each one of these properties, there is such that the
question whether has this property is independent of the ZFC axioms,
assuming that these are consistent. We clarify some of the subtleties
associated with such independence results in an appendix.Comment: 19 pages. v2: minor clarifications. v3: terminology 'free hypergraph
C*-algebra', added Remark 2.2
A New Game Equivalence and its Modal Logic
We revisit the crucial issue of natural game equivalences, and semantics of
game logics based on these. We present reasons for investigating finer concepts
of game equivalence than equality of standard powers, though staying short of
modal bisimulation. Concretely, we propose a more finegrained notion of
equality of "basic powers" which record what players can force plus what they
leave to others to do, a crucial feature of interaction. This notion is closer
to game-theoretic strategic form, as we explain in detail, while remaining
amenable to logical analysis. We determine the properties of basic powers via a
new representation theorem, find a matching "instantial neighborhood game
logic", and show how our analysis can be extended to a new game algebra and
dynamic game logic.Comment: In Proceedings TARK 2017, arXiv:1707.0825
New Algorithms for Solving Tropical Linear Systems
The problem of solving tropical linear systems, a natural problem of tropical
mathematics, has already proven to be very interesting from the algorithmic
point of view: it is known to be in but no polynomial time
algorithm is known, although counterexamples for existing pseudopolynomial
algorithms are (and have to be) very complex.
In this work, we continue the study of algorithms for solving tropical linear
systems. First, we present a new reformulation of Grigoriev's algorithm that
brings it closer to the algorithm of Akian, Gaubert, and Guterman; this lets us
formulate a whole family of new algorithms, and we present algorithms from this
family for which no known superpolynomial counterexamples work. Second, we
present a family of algorithms for solving overdetermined tropical systems. We
show that for weakly overdetermined systems, there are polynomial algorithms in
this family. We also present a concrete algorithm from this family that can
solve a tropical linear system defined by an matrix with maximal
element in time , and this time matches the complexity of the best of
previously known algorithms for feasibility testing.Comment: 17 page
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