12,032 research outputs found
A complex analogue of Toda's Theorem
Toda \cite{Toda} proved in 1989 that the (discrete) polynomial time
hierarchy, , is contained in the class \mathbf{P}^{#\mathbf{P}},
namely the class of languages that can be decided by a Turing machine in
polynomial time given access to an oracle with the power to compute a function
in the counting complexity class #\mathbf{P}. This result, which illustrates
the power of counting is considered to be a seminal result in computational
complexity theory. An analogous result (with a compactness hypothesis) in the
complexity theory over the reals (in the sense of Blum-Shub-Smale real machines
\cite{BSS89}) was proved in \cite{BZ09}. Unlike Toda's proof in the discrete
case, which relied on sophisticated combinatorial arguments, the proof in
\cite{BZ09} is topological in nature in which the properties of the topological
join is used in a fundamental way. However, the constructions used in
\cite{BZ09} were semi-algebraic -- they used real inequalities in an essential
way and as such do not extend to the complex case. In this paper, we extend the
techniques developed in \cite{BZ09} to the complex projective case. A key role
is played by the complex join of quasi-projective complex varieties. As a
consequence we obtain a complex analogue of Toda's theorem. The results
contained in this paper, taken together with those contained in \cite{BZ09},
illustrate the central role of the Poincar\'e polynomial in algorithmic
algebraic geometry, as well as, in computational complexity theory over the
complex and real numbers -- namely, the ability to compute it efficiently
enables one to decide in polynomial time all languages in the (compact)
polynomial hierarchy over the appropriate field.Comment: 31 pages. Final version to appear in Foundations of Computational
Mathematic
Languages of Dot-depth One over Infinite Words
Over finite words, languages of dot-depth one are expressively complete for
alternation-free first-order logic. This fragment is also known as the Boolean
closure of existential first-order logic. Here, the atomic formulas comprise
order, successor, minimum, and maximum predicates. Knast (1983) has shown that
it is decidable whether a language has dot-depth one. We extend Knast's result
to infinite words. In particular, we describe the class of languages definable
in alternation-free first-order logic over infinite words, and we give an
effective characterization of this fragment. This characterization has two
components. The first component is identical to Knast's algebraic property for
finite words and the second component is a topological property, namely being a
Boolean combination of Cantor sets.
As an intermediate step we consider finite and infinite words simultaneously.
We then obtain the results for infinite words as well as for finite words as
special cases. In particular, we give a new proof of Knast's Theorem on
languages of dot-depth one over finite words.Comment: Presented at LICS 201
Quantum finite automata and linear context-free languages: a decidable problem
We consider the so-called measure once finite quantum automata model introduced by Moore and Crutchfield in 2000. We show that given a language recognized by such a device and a linear context-free language, it is recursively decidable whether or not they have a nonempty intersection. This extends a result of Blondel et al. which can be interpreted as solving the problem with the free monoid in place of the family of linear context-free languages. © 2013 Springer-Verlag
Modal Ω-Logic: Automata, Neo-Logicism, and Set-Theoretic Realism
This essay examines the philosophical significance of -logic in Zermelo-Fraenkel set theory with choice (ZFC). The duality between coalgebra and algebra permits Boolean-valued algebraic models of ZFC to be interpreted as coalgebras. The modal profile of -logical validity can then be countenanced within a coalgebraic logic, and -logical validity can be defined via deterministic automata. I argue that the philosophical significance of the foregoing is two-fold. First, because the epistemic and modal profiles of -logical validity correspond to those of second-order logical consequence, -logical validity is genuinely logical, and thus vindicates a neo-logicist conception of mathematical truth in the set-theoretic multiverse. Second, the foregoing provides a modal-computational account of the interpretation of mathematical vocabulary, adducing in favor of a realist conception of the cumulative hierarchy of sets
A Categorical View on Algebraic Lattices in Formal Concept Analysis
Formal concept analysis has grown from a new branch of the mathematical field
of lattice theory to a widely recognized tool in Computer Science and
elsewhere. In order to fully benefit from this theory, we believe that it can
be enriched with notions such as approximation by computation or
representability. The latter are commonly studied in denotational semantics and
domain theory and captured most prominently by the notion of algebraicity, e.g.
of lattices. In this paper, we explore the notion of algebraicity in formal
concept analysis from a category-theoretical perspective. To this end, we build
on the the notion of approximable concept with a suitable category and show
that the latter is equivalent to the category of algebraic lattices. At the
same time, the paper provides a relatively comprehensive account of the
representation theory of algebraic lattices in the framework of Stone duality,
relating well-known structures such as Scott information systems with further
formalisms from logic, topology, domains and lattice theory.Comment: 36 page
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