Dynamic Proof Research Articles

Two adaptive logics of norm-propositions

We present two defeasible logics of norm-propositions (statements about norms) that (i) consistently allow for the possibility of normative gaps and normative conflicts, and (ii) map each premise set to a sufficiently rich consequence set. In order to meet (i), we define the logic LNP, a conflict- and gap-tolerant logic of norm-propositions capable of formalizing both normative conflicts and normative gaps within the object language. Next, we strengthen LNP within the adaptive logic framework for non-monotonic reasoning in order to meet (ii). This results in the adaptive logics LNPr and LNPm, which interpret a given set of premises in such a way that normative conflicts and normative gaps are avoided ‘whenever possible’. LNPr and LNPm are equipped with a preferential semantics and a dynamic proof theory.

Polynomial Dynamics and a Proof of the Fermat Little Theorem

We present a dynamical proof of the Fermat little theorem, based on the Chebyshev polynomials and their extraordinary composition property.

A PROOF Analysis Framework

The analysis of the complex LHC data usually follows a standard path that aims at minimizing not only the amount of data but also the number of observables used. After a number of steps of slimming and skimming the data, the remaining few terabytes of ROOT files hold a selection of the events and a flat structure for the variables needed that can be more easily inspected and traversed in the final stages of the analysis. PROOF arises at this point as an efficient mechanism to distribute the analysis load by taking advantage of all the cores in modern CPUs through PROOF Lite, or by using PROOF Cluster or PROOF on Demand tools to build dynamic PROOF cluster on computing facilities with spare CPUs. However using PROOF at the level required for a serious analysis introduces some difficulties that may scare new adopters. We have developed the PROOF Analysis Framework (PAF) to facilitate the development of new analysis by uniformly exposing the PROOF related configurations across technologies and by taking care of the routine tasks as much as possible. We describe the details of the PAF implementation as well as how we succeeded in engaging a group of CMS physicists to use PAF as their daily analysis framework.

Non-Monotonic Set Theory as a Pragmatic Foundation of Mathematics

In this paper I propose a new approach to the foundation of mathematics: non-monotonic set theory. I present two completely different methods to develop set theories based on adaptive logics. For both theories there is a finitistic non-triviality proof and both theories contain (a subtle version of) the comprehension axiom schema. The first theory contains only a maximal selection of instances of the comprehension schema that do not lead to inconsistencies. The second allows for all the instances, also the inconsistent ones, but restricts the conclusions one can draw from them in order to avoid triviality. The theories have enough expressive power to form a justification/explication for most of the established results of classical mathematics. They are therefore not limited by Gödel’s incompleteness theorems. This remarkable result is possible because of the non-recursive character of the final proofs of theorems of non-monotonic theories. I shall argue that, precisely because of the computational complexity of these final proofs, we cannot claim that non-monotonic theories are ideal foundations for mathematics. Nevertheless, thanks to their strength, first order language and the recursive dynamic (defeasible) proofs of theorems of the theory, the non-monotonic theories form (what I call) interesting pragmatic foundations.

A Logic for prioritized normative reasoning

In this article we present the logic MP⊏ that explicates reasoning on the basis of prioritized obligations. Although formal criteria to handle prioritized obligations have been formulated in the literature, little attention has been paid to the actual (non-monotonic) reasoning that makes use of these criteria. The dynamic proof theory of MP⊏ fills this lacuna. This article focuses on premise sets consisting of possibly conflicting prima facie obligations that have a modular order. MP⊏ allows to derive—inter alia—the actual, all-things-considered obligations from such premise sets. It is defined in the format of lexicographic adaptive logics from [34], whence a rich meta-theory is immediately available (e.g. soundness and completeness, idempotence, reflexivity, etc.). In addition, we establish some meta-theoretic results that are specific to the context of prioritized obligations. With the aid of concrete examples, we illustrate properties of MP⊏ which improve on other existing criteria for prioritized obligations.

Dynamic PROOF clusters with PoD: architecture and user experience

PROOF on Demand (PoD) is a tool-set, which sets up a PROOF cluster on any resource management system. PoD is a user oriented product with an easy to use GUI and a command-line interface. It is fully automated. No administrative privileges or special knowledge is required to use it. PoD utilizes a plug-in system, to use different job submission front-ends. The current PoD distribution is shipped with LSF, Torque (PBS), Grid Engine, Condor, gLite, and SSH plug-ins. The product is to be extended. We therefore plan to implement a plug-in for AliEn Grid as well. Recently developed algorithms made it possible to efficiently maintain two types of connections: packet-forwarding and native PROOF connections. This helps to properly handle most kinds of workers, with and without firewalls. PoD maintains the PROOF environment automatically and, for example, prevents resource misusage in case when workers idle for too long. As PoD matures as a product and provides more plug-ins, it's used as a standard for setting up dynamic PROOF clusters in many different institutions. The GSI Analysis Facility (GSIAF) is in production since 2007. The static PROOF cluster has been phased out end of 2009. GSIAF is now completely based on PoD. Users create private dynamic PROOF clusters on the general purpose batch farm. This provides an easier resource sharing between interactive local batch and Grid usage. The main user communities are FAIR and ALICE.

Modelling defeasible reasoning by means of adaptive logic games

In this article, I present a dynamic logic game for defeasible reasoning. I argue that, as far as defeasible reasoning is concerned, one should distinguish between practical and ideal rationality. Starting from the adaptive logic framework, I formalize both rationality notions by means of logic games. The presented adaptive logic games are based on (i) standard logic games on the one hand and (ii) dynamic proof procedures for adaptive logic on the other hand. The games are similar to standard logic games, but have the extra property that some moves are revisable. This is handled by means of a main control game, which starts different standard logic games. I argue that the adaptive logic games form intuitive reasoning models for rationality in defeasible reasoning contexts. Moreover, I will also demonstrate that the games give a good insight in the computational complexity of defeasible reasoning forms.

Universally measure-preserving homeomorphisms of cantor minimal systems

In 1995, T. Giordano, I. Putnam, and C. Skau [GPS] made a significant breakthrough in Cantor minimal system theory. They completely classified Cantor minimal systems in the sense of topological orbit equivalence by using C*-algebra and homological algebra techniques. Since then, a dynamical proof of their theorem has been conjectured. Such a proof is presented in this paper. We establish orbit equivalence theory based on a finite coordinate change relation arising from an ordered Bratteli diagram, which is known from [HK] in the finitary case of ergodic probability measure-preserving transformations. We obtain the Orbital Extension Theorem. This theorem is considered a topological version of the Copying Lemma of Y. Katznelson and B. Weiss [KW], which has played an important role in measured orbit equivalence theory.

A dynamic proof of mercury elimination from solution through a combined sorption–reduction process

Physico-chemical factors affecting mercury elimination from solution using fern as sorbent have been analysed. It was demonstrated that interaction of mercury with this biomass follows two processes, adsorption by the functional groups in the biopolymers of the cell wall and reduction by easily oxidized compounds of the biomass. Batch experiments have been done to analyse the effect of pH, ionic strength of the media or competition with other metals. Ionic strength did not show a significant influence in the process, but mercury speciation with the formation of negatively charge complexes represented an important drawback in metal elimination. Continuous flow conditions were also analysed since many industrial applications will require them. These studies allowed distinguishing the two processes: mercury sorption was observed but also reduction of this metal occurs. Reduction to mercury (I) and metallic mercury has been confirmed by scanning electron microscopy analysis of the column filling after the continuous flow experiments.

PROOF on Demand

PROOF on Demand (PoD) is a set of utilities, which allows starting a PROOF cluster at user request, on any resource management system. It provides a plug-in based system, which allows to use different job submission frontends, such as LSF or gLite WMS. Main components of PoD are the PROOFAgent and the PAConsole. PROOFAgent provides the communication layer between the PROOF master on the local machine and the PROOF workers on the remote resources, possibly behind a firewall. PAConsole provides a user-friendly GUI, which is used to setup, manage, and shutdown the dynamic PROOF cluster. Installation is simple and doesn't require administrator privileges, and all the processes run in user space. PoD gives users, who don't have a centrally-administrated static PROOF cluster at their institute, the possibility to enjoy the full power of interactive analysis with PROOF.

A dynamic proof of Thébault’s theorem

Thébault’s theorem, discovered in 1938 (see [5]), has the remarkable property that it required three decades to obtain a first proof, which then took 24 pages of calculations. Subsequent shorter proofs, but not easy ones, were published mainly in Elemente der Mathematik ([4] and [6]) and in Dutch language. We refer to [1], [2], and [7] for complete accounts of all these and recent proofs. In this note we want to emphasize the role of a certain parabola which comes out of a dynamic machine, and which allows an easy understanding.

Stepwise formation of P-cluster in nitrogenase MoFe protein

The P-cluster of nitrogenase is one of the most complex biological metallocenters known to date. Despite the recent advances in the chemical synthesis of P-cluster topologs, the biosynthetic mechanism of P-cluster has not been well defined. Here, we present a combined biochemical, electron paramagnetic resonance, and X-ray absorption spectroscopy/extended X-ray absorption fine-structure investigation of the maturation process of P-clusters in DeltanifH molybdenum-iron (MoFe) protein. Our data indicate that the previously identified, [Fe(4)S(4)]-like cluster pairs in DeltanifH MoFe protein are indeed the precursors to P-clusters, which can be reductively coupled into the mature [Fe(8)S(7)] structures in the presence of Fe protein, MgATP, and dithionite. Moreover, our observation of a biphasic maturation pattern of P-clusters in DeltanifH MoFe protein provides dynamic proof for the previously hypothesized, stepwise assembly mechanism of the two P-clusters in the alpha(2)beta(2)-tetrameric MoFe protein, i.e., one P-cluster is formed in one alphabeta dimer before the other in the second alphabeta dimer.

Quantum phases due to cosmic gravity and the spin-statistics connection

Recently, it was proposed that the spin-statistics connection arises due to a quantum dynamical phase involving cosmic gravity (Unnikrishnan C S, gr-qc/0406043). There it was assumed that the gravitational quantum phase accumulated on the wavefunctions of each of two identical particles undergoing scattering in the presence of cosmic matter with the direction of their momentum changing by some angle is equivalent to the particles forward scattering with the entire universe rotating oppositely by the same angle. However, if one considers the gravitational phase accumulated by an electron in orbit to be analogous to that acquired by a scattering particle, the accumulated phase inferred from the fine-structure splitting in an atom is too small to explain the spin statistics connection. Though, the desirability of a dynamical proof of the spin-statistics connection remains.

Equilibrium Solution to the Inelastic Boltzmann Equation Driven by a Particle Bath

We show the existence of smooth stationary solutions for the inelastic Boltzmann equation under the thermalization induced by a host medium with a fixed distribution. This is achieved by controlling the L p -norms, the moments and the regularity of the solutions to the Cauchy problem together with arguments related to a dynamical proof for the existence of stationary states.

Multivalued maps as a tool in modeling and rigorous numerics

Applications of the fixed point theory of multivalued maps can be classified into several areas: (1) Game theory and mathematical economics; (2) Discontinuous differential equations, differential inclusions, and optimal control; (3) Computing homology of maps; (4) Computer assisted proofs in dynamics; (5) Digital imaging. We give an overview of the most classical and well developed areas of applications (1) and (2), where a multivalued map is used as a generalization of a single-valued continuous map, and we survey more recent applications (3), (4), and (5), where multivalued maps play the role of a numerical tool.

Liapunov Stability and the ring of P-adic integers

It is well-known that stable Cantor sets are topologically conjugate to adding machines. In this work we show are also conjugate to an algebraic object, the ring of P−adic integers with respect to group tramnslation. This ring is closely related to the field of p-adic numbers; connections and distintions are explored. The inverse limit construction provides a purely dynamical proof of an algebraic result: the classification of adding machines, or P−adic integers, up to group isomorphism.

Yes fellows, most human reasoning is complex

This paper answers the philosophical contentions defended in Horsten and Welch (2007, Synthese, 158, 41-60). It contains a description of the standard format of adaptive logics, analyses the notion of dynamic proof required by those logics, discusses the means to turn such proofs into demonstrations, and argues that, notwithstanding their formal complexity, adaptive logics are important because they explicate an abundance of reasoning forms that occur frequently, both in scientific contexts and in common sense contexts.

The AProS Project: Strategic Thinking & Computational Logic

The paper discusses tools for teaching logic used in Logic & Proofs, a web-based introduction to modern logic that has been taken by more than 1,300 students since the fall of 2003. The tools include a wide array of interactive learning environments or cognitive mini-tutors; most important among them is the Carnegie Proof Lab. The Proof Lab is a sophisticated interface for constructing natural deduction proofs and is central, as strategically guided discovery of proofs is the distinctive focus of the course. My discussion makes explicit the broader intellectual context, but also the pursuit of pedagogical goals and their experimental examination. The intellectual context includes i) the theoretical background for the proof search algorithm AProS and its use for a dynamic Proof Tutor, and ii) the programmatic expansion of the course to Computational Logic. (I recommend that the reader enter the virtual classroom of Logic & Proofs: the interactive components just cannot be properly reflected in a narrative. It is also very easy to download AProS and observe its ways of searching for proofs.)1

A Dynamical Proof of Pisot's Theorem

AbstractWe give a geometric proof of classical results that characterize Pisot numbers as algebraic λ > 1 for which there is x ≠ 0 with λnx → 0 (mod 1) and identify such x as members of ℤ[λ–1]· ℤ[λ]* where ℤ[λ]* is the dual module of ℤ[λ].

A Formal Logic for Abductive Reasoning

This paper presents and illustrates a formal logic for the abduction of singular hypotheses. The logic has a semantics and a dynamic proof theory that is sound and complete with respect to the semantics. The logic presupposes that, with respect to a specific application, the set of explananda and the set of possible explanantia are disjoint (but not necessarily exhaustive). Where an explanandum can be explained by different explanantia, the logic allows only for the abduction of their disjunction.