11 research outputs found
Generalizations of Kochen and Specker's Theorem and the Effectiveness of Gleason's Theorem
Kochen and Specker's theorem can be seen as a consequence of Gleason's
theorem and logical compactness. Similar compactness arguments lead to stronger
results about finite sets of rays in Hilbert space, which we also prove by a
direct construction. Finally, we demonstrate that Gleason's theorem itself has
a constructive proof, based on a generic, finite, effectively generated set of
rays, on which every quantum state can be approximated.Comment: 14 pages, 6 figures, read at the Robert Clifton memorial conferenc
Quantifier elimination for the reals with a predicate for the powers of two
In 1985, van den Dries showed that the theory of the reals with a predicate
for the integer powers of two admits quantifier elimination in an expanded
language, and is hence decidable. He gave a model-theoretic argument, which
provides no apparent bounds on the complexity of a decision procedure. We
provide a syntactic argument that yields a procedure that is primitive
recursive, although not elementary. In particular, we show that it is possible
to eliminate a single block of existential quantifiers in time ,
where is the length of the input formula and denotes -fold
iterated exponentiation
Algorithms for Game Metrics
Simulation and bisimulation metrics for stochastic systems provide a
quantitative generalization of the classical simulation and bisimulation
relations. These metrics capture the similarity of states with respect to
quantitative specifications written in the quantitative {\mu}-calculus and
related probabilistic logics. We first show that the metrics provide a bound
for the difference in long-run average and discounted average behavior across
states, indicating that the metrics can be used both in system verification,
and in performance evaluation. For turn-based games and MDPs, we provide a
polynomial-time algorithm for the computation of the one-step metric distance
between states. The algorithm is based on linear programming; it improves on
the previous known exponential-time algorithm based on a reduction to the
theory of reals. We then present PSPACE algorithms for both the decision
problem and the problem of approximating the metric distance between two
states, matching the best known algorithms for Markov chains. For the
bisimulation kernel of the metric our algorithm works in time O(n^4) for both
turn-based games and MDPs; improving the previously best known O(n^9\cdot
log(n)) time algorithm for MDPs. For a concurrent game G, we show that
computing the exact distance between states is at least as hard as computing
the value of concurrent reachability games and the square-root-sum problem in
computational geometry. We show that checking whether the metric distance is
bounded by a rational r, can be done via a reduction to the theory of real
closed fields, involving a formula with three quantifier alternations, yielding
O(|G|^O(|G|^5)) time complexity, improving the previously known reduction,
which yielded O(|G|^O(|G|^7)) time complexity. These algorithms can be iterated
to approximate the metrics using binary search.Comment: 27 pages. Full version of the paper accepted at FSTTCS 200
Combining decision procedures for the reals
We address the general problem of determining the validity of boolean
combinations of equalities and inequalities between real-valued expressions. In
particular, we consider methods of establishing such assertions using only
restricted forms of distributivity. At the same time, we explore ways in which
"local" decision or heuristic procedures for fragments of the theory of the
reals can be amalgamated into global ones. Let Tadd[Q] be the
first-order theory of the real numbers in the language of ordered groups, with
negation, a constant 1, and function symbols for multiplication by
rational constants. Let Tmult[Q] be the analogous theory for the
multiplicative structure, and let T[Q] be the union of the two. We
show that although T[Q] is undecidable, the universal fragment of
T[Q] is decidable. We also show that terms of T[Q]can
fruitfully be put in a normal form. We prove analogous results for theories in
which Q is replaced, more generally, by suitable subfields F
of the reals. Finally, we consider practical methods of establishing
quantifier-free validities that approximate our (impractical) decidability
results.Comment: Will appear in Logical Methods in Computer Scienc
Algorithmic Semi-algebraic Geometry and Topology -- Recent Progress and Open Problems
We give a survey of algorithms for computing topological invariants of
semi-algebraic sets with special emphasis on the more recent developments in
designing algorithms for computing the Betti numbers of semi-algebraic sets.
Aside from describing these results, we discuss briefly the background as well
as the importance of these problems, and also describe the main tools from
algorithmic semi-algebraic geometry, as well as algebraic topology, which make
these advances possible. We end with a list of open problems.Comment: Survey article, 74 pages, 15 figures. Final revision. This version
will appear in the AMS Contemporary Math. Series: Proceedings of the Summer
Research Conference on Discrete and Computational Geometry, Snowbird, Utah
(June, 2006). J.E. Goodman, J. Pach, R. Pollack Ed