123,668 research outputs found
A Galois connection between Turing jumps and limits
Limit computable functions can be characterized by Turing jumps on the input
side or limits on the output side. As a monad of this pair of adjoint
operations we obtain a problem that characterizes the low functions and dually
to this another problem that characterizes the functions that are computable
relative to the halting problem. Correspondingly, these two classes are the
largest classes of functions that can be pre or post composed to limit
computable functions without leaving the class of limit computable functions.
We transfer these observations to the lattice of represented spaces where it
leads to a formal Galois connection. We also formulate a version of this result
for computable metric spaces. Limit computability and computability relative to
the halting problem are notions that coincide for points and sequences, but
even restricted to continuous functions the former class is strictly larger
than the latter. On computable metric spaces we can characterize the functions
that are computable relative to the halting problem as those functions that are
limit computable with a modulus of continuity that is computable relative to
the halting problem. As a consequence of this result we obtain, for instance,
that Lipschitz continuous functions that are limit computable are automatically
computable relative to the halting problem. We also discuss 1-generic points as
the canonical points of continuity of limit computable functions, and we prove
that restricted to these points limit computable functions are computable
relative to the halting problem. Finally, we demonstrate how these results can
be applied in computable analysis
Sheaf Logic, Quantum Set Theory and the Interpretation of Quantum Mechanics
Based on the Sheaf Logic approach to set theoretic forcing, a hierarchy of
Quantum Variable Sets is constructed which generalizes and simplifies the
analogous construction developed by Takeuti on boolean valued models of set
theory. Over this model two alternative proofs of Takeuti's correspondence,
between self adjoint operators and the real numbers of the model, are given.
This approach results to be more constructive showing a direct relation with
the Gelfand representation theorem, revealing also the importance of these
results with respect to the interpretation of Quantum Mechanics in close
connection with the Deutsch-Everett multiversal interpretation. Finally, it is
shown how in this context the notion of genericity and the corresponding
generic model theorem can help to explain the emergence of classicality also in
connection with the Deutsch- Everett perspective.Comment: 34 pages, 2 figure
On the Complexity of Temporal-Logic Path Checking
Given a formula in a temporal logic such as LTL or MTL, a fundamental problem
is the complexity of evaluating the formula on a given finite word. For LTL,
the complexity of this task was recently shown to be in NC. In this paper, we
present an NC algorithm for MTL, a quantitative (or metric) extension of LTL,
and give an NCC algorithm for UTL, the unary fragment of LTL. At the time of
writing, MTL is the most expressive logic with an NC path-checking algorithm,
and UTL is the most expressive fragment of LTL with a more efficient
path-checking algorithm than for full LTL (subject to standard
complexity-theoretic assumptions). We then establish a connection between LTL
path checking and planar circuits, which we exploit to show that any further
progress in determining the precise complexity of LTL path checking would
immediately entail more efficient evaluation algorithms than are known for a
certain class of planar circuits. The connection further implies that the
complexity of LTL path checking depends on the Boolean connectives allowed:
adding Boolean exclusive or yields a temporal logic with P-complete
path-checking problem
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