Highly Undecidable Problems For Infinite Computations

We show that many classical decision problems about 1-counter omega-languages, context free omega-languages, or infinitary rational relations, are $\Pi_2^1$-complete, hence located at the second level of the analytical hierarchy, and "highly undecidable". In particular, the universality problem, the inclusion problem, the equivalence problem, the determinizability problem, the complementability problem, and the unambiguity problem are all $\Pi_2^1$-complete for context-free omega-languages or for infinitary rational relations. Topological and arithmetical properties of 1-counter omega-languages, context free omega-languages, or infinitary rational relations, are also highly undecidable. These very surprising results provide the first examples of highly undecidable problems about the behaviour of very simple finite machines like 1-counter automata or 2-tape automata.Comment: to appear in RAIRO-Theoretical Informatics and Application

BPS States in Omega Background and Integrability

We reconsider string and domain wall central charges in N=2 supersymmetric gauge theories in four dimensions in presence of the Omega background in the Nekrasov-Shatashvili (NS) limit. Existence of these charges entails presence of the corresponding topological defects in the theory - vortices and domain walls. In spirit of the 4d/2d duality we discuss the worldsheet low energy effective theory living on the BPS vortex in N=2 Supersymmetric Quantum Chromodynamics (SQCD). We discuss some aspects of the brane realization of the dualities between various quantum integrable models. A chain of such dualities enables us to check the AGT correspondence in the NS limit.Comment: 48 pages, 10 figures, minor changes, references added, typos correcte

The Apokatastasis Essays in Context: Leibniz and Thomas Burnet on the Kingdom of Grace and the Stoic/Platonic Revolutions

One of Leibniz’s more unusual philosophical projects is his presentation (in a series of unpublished drafts) of an argument for the conclusion that a time will necessarily come when “nothing would happen that had not happened before." Leibniz’s presentations of the argument for such a cyclical cosmology are all too brief, and his discussion of its implications is obscure. Moreover, the conclusion itself seems to be at odds with the main thrust of Leibniz’s own metaphysics. Despite this, we can discern a serious and important point to Leibniz’s consideration of the doctrine, namely in what it suggests about the proper boundary between metaphysics and theology, on the one hand, and ordinary history (whether human or natural), on the other. And we can get a better sense of Leibniz purpose in the essays by considering them in the context of Leibniz's response to Thomas Burnet's "Telluris theoria sacra" (1681-89). Leibniz praises Burnet's history of earth for presenting a harmony between the principles of nature and grace, a harmony absent in the cosmogonies of Descartes and the Newtonians. But Leibniz also complains that Burnet misconceives the boundary between natural explanation and reflections on divine wisdom. And Leibniz's essays on cyclical cosmology suggest the alternative to Burnet's account: a natural history of the earth and its inhabitants should be radically autonomous from, even if ultimately harmonious with, theological principles

Migrating agile methods to standardized development practice

Situated process and quality frame-works offer a way to resolve the tensions that arise when introducing agile methods into standardized software development engineering. For these to be successful, however, organizations must grasp the opportunity to reintegrate software development management, theory, and practice

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

Computing with rational symmetric functions and applications to invariant theory and PI-algebras

Let the formal power series f in d variables with coefficients in an arbitrary field be a symmetric function decomposed as a series of Schur functions, and let f be a rational function whose denominator is a product of binomials of the form (1 - monomial). We use a classical combinatorial method of Elliott of 1903 further developed in the Partition Analysis of MacMahon in 1916 to compute the generating function of the multiplicities (i.e., the coefficients) of the Schur functions in the expression of f. It is a rational function with denominator of a similar form as f. We apply the method to several problems on symmetric algebras, as well as problems in classical invariant theory, algebras with polynomial identities, and noncommutative invariant theory.Comment: 37 page