552 research outputs found

    Complexity of equivalence relations and preorders from computability theory

    Full text link
    We study the relative complexity of equivalence relations and preorders from computability theory and complexity theory. Given binary relations R,SR, S, a componentwise reducibility is defined by R\le S \iff \ex f \, \forall x, y \, [xRy \lra f(x) Sf(y)]. Here ff is taken from a suitable class of effective functions. For us the relations will be on natural numbers, and ff must be computable. We show that there is a Π1\Pi_1-complete equivalence relation, but no Πk\Pi k-complete for k≄2k \ge 2. We show that ÎŁk\Sigma k preorders arising naturally in the above-mentioned areas are ÎŁk\Sigma k-complete. This includes polynomial time mm-reducibility on exponential time sets, which is ÎŁ2\Sigma 2, almost inclusion on r.e.\ sets, which is ÎŁ3\Sigma 3, and Turing reducibility on r.e.\ sets, which is ÎŁ4\Sigma 4.Comment: To appear in J. Symb. Logi

    Eligibility and inscrutability

    Get PDF
    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 ‘London’ refer to something or a predicate such as ‘is large’ apply to some object? This essay examines one kind of answer to this “metasemantic”1 question: interpretationism, instances of which have been proposed by Donald Davidson, David Lewis, and others. I characterize the “twostep” 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 “permutation argument” 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 Lewis’s appeal to the differential “eligibility” of semantic theories. I contend that, given an independently plausible formulation of interpretationism, the eligibility response is an immediate consequence of Lewis’s general analysis of the theoretical virtue of simplicity. In the fi nal sections of the article, I examine the limitations of Lewis’s 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?

    Full text link
    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

    Get PDF
    This chapter surveys the development in the theory of polyadic algebras in the last decades

    Uniform Interpolation in provability logics

    Full text link
    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

    Full text link
    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 ω\omega-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 n×nn \times n 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

    Full text link
    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

    Get PDF
    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
    • 

    corecore