552 research outputs found
Complexity of equivalence relations and preorders from computability theory
We study the relative complexity of equivalence relations and preorders from
computability theory and complexity theory. Given binary relations , a
componentwise reducibility is defined by R\le S \iff \ex f \, \forall x, y \,
[xRy \lra f(x) Sf(y)]. Here is taken from a suitable class of effective
functions. For us the relations will be on natural numbers, and must be
computable. We show that there is a -complete equivalence relation, but
no -complete for .
We show that preorders arising naturally in the above-mentioned
areas are -complete. This includes polynomial time -reducibility
on exponential time sets, which is , almost inclusion on r.e.\ sets,
which is , and Turing reducibility on r.e.\ sets, which is .Comment: To appear in J. Symb. Logi
Eligibility and inscrutability
The philosophy of intentionality asks questions such as: in virtue of what
does a sentence, picture, or mental state represent that the world is a certain
way? The subquestion I focus upon here concerns the semantic properties
of language: in virtue of what does a name such as âLondonâ refer
to something or a predicate such as âis largeâ apply to some object?
This essay examines one kind of answer to this âmetasemanticâ1
question: interpretationism, instances of which have been proposed by
Donald Davidson, David Lewis, and others. I characterize the âtwostepâ
form common to such approaches and briefl y say how two versions
described by David Lewis fi t this pattern. Then I describe a fundamental
challenge to this approach: a âpermutation argumentâ that contends,
by interpretationist lights, there can be no fact of the matter about lexical
content (e.g., what individual words refer to). Such a thesis cannot be sustained,
so the argument threatens a reductio of interpretationism.
In the second part of the article, I will give what I take to be the
best interpretationist response to the inscrutability paradox: David Lewisâs
appeal to the differential âeligibilityâ of semantic theories. I contend that,
given an independently plausible formulation of interpretationism, the
eligibility response is an immediate consequence of Lewisâs general analysis
of the theoretical virtue of simplicity.
In the fi nal sections of the article, I examine the limitations of Lewisâs
response. By focusing on an alternative argument for the inscrutability
of reference, I am able to describe conditions under which the eligibility
result will deliver the wrong results. In particular, if the world is complex
enough and our language suffi ciently simple, then reference may
be determinately secured to the wrong things
Do Goedel's incompleteness theorems set absolute limits on the ability of the brain to express and communicate mental concepts verifiably?
Classical interpretations of Goedel's formal reasoning imply that the truth
of some arithmetical propositions of any formal mathematical language, under
any interpretation, is essentially unverifiable. However, a language of
general, scientific, discourse cannot allow its mathematical propositions to be
interpreted ambiguously. Such a language must, therefore, define mathematical
truth verifiably. We consider a constructive interpretation of classical,
Tarskian, truth, and of Goedel's reasoning, under which any formal system of
Peano Arithmetic is verifiably complete. We show how some paradoxical concepts
of Quantum mechanics can be expressed, and interpreted, naturally under a
constructive definition of mathematical truth.Comment: 73 pages; this is an updated version of the NQ essay; an HTML version
is available at http://alixcomsi.com/Do_Goedel_incompleteness_theorems.ht
Polyadic Algebras
This chapter surveys the development in the theory of polyadic algebras in the last decades
Uniform Interpolation in provability logics
We prove the uniform interpolation theorem in modal provability logics GL and
Grz by a proof-theoretical method, using analytical and terminating sequent
calculi for the logics. The calculus for G\"odel-L\"ob's logic GL is a variant
of the standard sequent calculus, in the case of Grzegorczyk's logic Grz, the
calculus implements an explicit loop-preventing mechanism inspired by work of
Heuerding
Factorization theory: From commutative to noncommutative settings
We study the non-uniqueness of factorizations of non zero-divisors into atoms
(irreducibles) in noncommutative rings. To do so, we extend concepts from the
commutative theory of non-unique factorizations to a noncommutative setting.
Several notions of factorizations as well as distances between them are
introduced. In addition, arithmetical invariants characterizing the
non-uniqueness of factorizations such as the catenary degree, the
-invariant, and the tame degree, are extended from commutative to
noncommutative settings. We introduce the concept of a cancellative semigroup
being permutably factorial, and characterize this property by means of
corresponding catenary and tame degrees. Also, we give necessary and sufficient
conditions for there to be a weak transfer homomorphism from a cancellative
semigroup to its reduced abelianization. Applying the abstract machinery we
develop, we determine various catenary degrees for classical maximal orders in
central simple algebras over global fields by using a natural transfer
homomorphism to a monoid of zero-sum sequences over a ray class group. We also
determine catenary degrees and the permutable tame degree for the semigroup of
non zero-divisors of the ring of upper triangular matrices over a
commutative domain using a weak transfer homomorphism to a commutative
semigroup.Comment: 45 page
On the Hilbert Property and the Fundamental Group of Algebraic Varieties
We review, under a perspective which appears different from previous ones,
the so-called Hilbert Property (HP) for an algebraic variety (over a number
field); this is linked to Hilbert's Irreducibility Theorem and has important
implications, for instance towards the Inverse Galois Problem.
We shall observe that the HP is in a sense `opposite' to the Chevalley-Weil
Theorem, which concerns unramified covers; this link shall immediately entail
the result that the HP can possibly hold only for simply connected varieties
(in the appropriate sense). In turn, this leads to new counterexamples to the
HP, involving Enriques surfaces. We also prove the HP for a K3 surface related
to the above Enriques surface, providing what appears to be the first example
of a non-rational variety for which the HP can be proved.
We also formulate some general conjectures relating the HP with the topology
of algebraic varieties.Comment: 24 page
Extremal properties of (epi)Sturmian sequences and distribution modulo 1
Starting from a study of Y. Bugeaud and A. Dubickas (2005) on a question in
distribution of real numbers modulo 1 via combinatorics on words, we survey
some combinatorial properties of (epi)Sturmian sequences and distribution
modulo 1 in connection to their work. In particular we focus on extremal
properties of (epi)Sturmian sequences, some of which have been rediscovered
several times
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