469 research outputs found
Chain models, trees of singular cardinality and dynamic EF games
Let Îș be a singular cardinal. Karp's notion of a chain model of size ? is defined to be an ordinary model of size Îș along with a decomposition of it into an increasing union of length cf(Îș). With a notion of satisfaction and (chain)-isomorphism such models give an infinitary logic largely mimicking first order logic. In this paper we associate to this logic a notion of a dynamic EF-game which gauges when two chain models are chain-isomorphic. To this game is associated a tree which is a tree of size Îș with no Îș-branches (even no cf(Îș)-branches). The measure of how non-isomorphic the models are is reflected by a certain order on these trees, called reduction. We study the collection of trees of size Îș with no Îș-branches under this notion and prove that when cf(Îș) = Ï this collection is rather regular; in particular it has universality number exactly Îș+. Such trees are then used to develop a descriptive set theory of the space cf(Îș)Îș.The main result of the paper gives in the case of Îș strong limit singular an exact connection between the descriptive set-theoretic complexity of the chain isomorphism orbit of a model, the reduction order on the trees and winning strategies in the corresponding dynamic EF games. In particular we obtain a neat analog of the notion of Scott watershed from the Scott analysis of countable models
The singular world of singular cardinals
The article uses two examples to explore the statement that, contrary to the common wisdom, the properties of singular cardinals are actually more intuitive than those of the regular ones
Chain logic and Shelahâs infinitary logic
For a cardinal of the form kappa = (sic)(kappa), Shelah's logic L-kappa(1) has a characterisation as the maximal logic above boolean OR(lambda We then show that the chain logic gives a partial solution to Problem 1.4 from Shelah's [28], which asked whether for kappa singular of countable cofinality there was a logic strictly between L kappa+,omega and L kappa(+),kappa(+) having interpolation. We show that modulo accepting as the upper bound a model class of L-kappa,L-kappa, Karp's chain logic satisfies the required properties. In addition, we show that this chain logic is not kappa-compact, a question that we have asked on various occasions. We contribute to further development of chain logic by proving the Union Lemma and identifying the chainindependent fragment of the logic, showing that it still has considerable expressive power. In conclusion, we have shown that the simply defined chain logic emulates the logic L-kappa(1) in satisfying interpolation, undefinability of well-order and maximality with respect to it, and the Union Lemma. In addition it has a completeness theorem.Peer reviewe
On wide Aronszajn trees in the presence of MA
A wide Aronszajn tree is a tree of size and height with no uncountable branches. We prove that under there is no wide Aronszajn tree which is universal under weak embeddings. This solves an open question of Mekler and VÀÀnĂ€nen from 1994. We also prove that under, every wide Aronszajn tree weakly embeds in an Aronszajn tree, which combined with a result of TodorÄeviÄ from 2007, gives that under every wide Aronszajn tree embeds into a Lipschitz tree or a coherent tree. We also prove that under there is no wide Aronszajn tree which weakly embeds all Aronszajn trees, improving the result in the first paragraph as well as a result of TodorÄeviÄ from 2007 who proved that under there are no universal Aronszajn trees
Decision Problems for Partial Specifications: Empirical and Worst-Case Complexities
Partial specifications allow approximate models of systems such as Kripke structures, or labeled
transition systems to be created. Using the abstraction possible with these models, an avoidance
of the state-space explosion problem is possible, whilst still retaining a structure that can
have properties checked over it. A single partial specification abstracts a set of systems, whether
Kripke, labeled transition systems, or systems with both atomic propositions and named transitions.
This thesis deals in part with problems arising from a desire to efficiently evaluate
sentences of the modal Ό-calculus over a partial specification.
Partial specifications also allow a single system to be modeled by a number of partial specifications,
which abstract away different parts of the system. Alternatively, a number of partial
specifications may represent different requirements on a system. The thesis also addresses the
question of whether a set of partial specifications is consistent, that is to say, whether a single
system exists that is abstracted by each member of the set. The effect of nominals, special
atomic propositions true on only one state in a system, is also considered on the problem of the
consistency of many partial specifications. The thesis also addresses the question of whether
the systems a partial specification abstracts are all abstracted by a second partial specification,
the problem of inclusion.
The thesis demonstrates how commonly used âspecification patternsâ â useful properties specified
in the modal Ό-calculus, can be efficiently evaluated over partial specifications, and gives
upper and lower complexity bounds on the problems related to sets of partial specifications
The Universality Problem
The theme of this thesis is to explore the universality problem in set theory in connection to model theory, to present some methods for finding universality results, to analyse how these methods were applied, to mention some results and to emphasise some philosophical interrogations that these aspects entail.
A fundamental aspect of the universality problem is to find what determines the existence of universal objects. That means that we have to take into consideration and examine the methods that we use in proving their existence or nonexistence, the role of cardinal arithmetic, combinatorics etc. The proof methods used in the mathematical part will be mostly set-theoretic, but some methods from model theory and category theory will also be present.
A graph might be the simplest, but it is also one of the most useful notions in mathematics. We show that there is a faithful functor F from the category L of linear orders to the category G of graphs that preserves model theoretic-related universality results (classes of objects having universal models in exactly the same cardinals, and also having the same universality spectrum).
Trees constitute combinatorial objects and have a central role in set theory. The universality of trees is connected to the universality of linear orders, but it also seems to present more challenges, which we survey and present some results. We show that there is no embedding between an â”2-Souslin tree and a non-special wide â”2 tree T with no cofinal branches. Furthermore, using the notion of ascent path, we prove that the class of non-special â”2-Souslin tree with an Ï-ascent path a has maximal complexity number, 2â”2 = â”3.
Within the general framework of the universality problem in set theory and model theory, while emphasising their approaches and their connections with regard to this topic, we examine the possibility of drawing some philosophical conclusions connected to, among others, the notions of mathematical knowledge, mathematical object and proof
Lazy Evaluation: From natural semantics to a machine-checked compiler transformation
In order to solve a long-standing problem with list fusion, a new compiler transformation, \u27Call Arity\u27 is developed and implemented in the Haskell compiler GHC. It is formally proven to not degrade program performance; the proof is machine-checked using the interactive theorem prover Isabelle. To that end, a formalization of Launchbury`s Natural Semantics for Lazy Evaluation is modelled in Isabelle, including a correctness and adequacy proof
Structure from Articulated Motion: Accurate and Stable Monocular 3D Reconstruction without Training Data
Recovery of articulated 3D structure from 2D observations is a challenging
computer vision problem with many applications. Current learning-based
approaches achieve state-of-the-art accuracy on public benchmarks but are
restricted to specific types of objects and motions covered by the training
datasets. Model-based approaches do not rely on training data but show lower
accuracy on these datasets. In this paper, we introduce a model-based method
called Structure from Articulated Motion (SfAM), which can recover multiple
object and motion types without training on extensive data collections. At the
same time, it performs on par with learning-based state-of-the-art approaches
on public benchmarks and outperforms previous non-rigid structure from motion
(NRSfM) methods. SfAM is built upon a general-purpose NRSfM technique while
integrating a soft spatio-temporal constraint on the bone lengths. We use
alternating optimization strategy to recover optimal geometry (i.e., bone
proportions) together with 3D joint positions by enforcing the bone lengths
consistency over a series of frames. SfAM is highly robust to noisy 2D
annotations, generalizes to arbitrary objects and does not rely on training
data, which is shown in extensive experiments on public benchmarks and real
video sequences. We believe that it brings a new perspective on the domain of
monocular 3D recovery of articulated structures, including human motion
capture.Comment: 21 pages, 8 figures, 2 table
Succinctness and Formula Size Games
TÀmÀ vÀitöskirja tutkii erilaisten logiikoiden tiiviyttÀ kaavan pituuspelien avulla. Logiikan tiiviys viittaa ominaisuuksien ilmaisemiseen tarvittavien kaavojen kokoon. Kaavan pituuspelit ovat hyvÀksi todettu menetelmÀ tiiviystulosten todistamiseen. VÀitöskirjan kontribuutio on kaksiosainen. EnsinnÀkin vÀitöskirjassa mÀÀritellÀÀn kaavan pituuspeli useille logiikoille ja tarjotaan nÀin uusia menetelmiÀ tulevaan tutkimukseen. Toiseksi nÀitÀ pelejÀ ja muita menetelmiÀ kÀytetÀÀn tiiviystulosten todistamiseen tutkituille logiikoille.
Tarkemmin sanottuna vĂ€itöskirjassa mÀÀritellÀÀn uudet parametrisoidut kaavan pituuspelit perusmodaalilogiikalle, modaaliselle ÎŒ-kalkyylille, tiimilauselogiikalle ja yleistetyille sÀÀnnöllisille lausekkeille. Yleistettyjen sÀÀnnöllisten lausekkeiden pelistĂ€ esitellÀÀn myös variantit, jotka vastaavat sÀÀnnöllisiĂ€ lausekkeita ja uusia âRE over star-freeâ -lausekkeita, joissa tĂ€htiĂ€ ei esiinny komplementtien sisĂ€llĂ€.
PelejĂ€ kĂ€ytetÀÀn useiden tiiviystulosten todistamiseen. Predikaattilogiikan nĂ€ytetÀÀn olevan epĂ€elementaarisesti tiiviimpi kuin perusmodaalilogiikka ja modaalinen ÎŒ-kalkyyli. Tiimilauselogiikassa tutkitaan systemaattisesti yleisten riippuvuuksia ilmaisevien atomien mÀÀrittelemisen tiiviyttĂ€. Klassinen epĂ€elementaarinen tiiviysero predikaattilogiikan ja sÀÀnnöllisten lausekkeiden vĂ€lillĂ€ osoitetaan uudelleen yksinkertaisemmalla tavalla ja saadaan tĂ€htien lukumÀÀrĂ€lle âRE over star-freeâ -lausekkeissa hierarkia ilmaisuvoiman suhteen.
Monissa yllĂ€mainituista tuloksista hyödynnetÀÀn eksplisiittisiĂ€ kaavoja peliargumenttien lisĂ€ksi. TĂ€llaisia kaavoja ja tyyppien laskemista hyödyntĂ€en saadaan epĂ€elementaarisia ala- ja ylĂ€rajoja yksittĂ€isten sanojen mÀÀrittelemisen tiiviydelle predikaattilogiikassa ja monadisessa toisen kertaluvun logiikassa.This thesis studies the succinctness of various logics using formula size games. The succinctness of a logic refers to the size of formulas required to express properties. Formula size games are some of the most successful methods of proof for results on succinctness. The contribution of the thesis is twofold. Firstly, we deïŹne formula size games for several logics, providing methods for future research. Secondly, we use these games and other methods to prove results on the succinctness of the studied logics.
More precisely, we develop new parameterized formula size games for basic modal logic, modal ÎŒ-calculus, propositional team logic and generalized regular expressions. For the generalized regular expression game we introduce variants that correspond to regular expressions and the newly deïŹned RE over star-free expressions, where stars do not occur inside complements.
We use the games to prove a number of succinctness results. We show that ïŹrst-order logic is non-elementarily more succinct than both basic modal logic and modal ÎŒ-calculus. We conduct a systematic study of the succinctness of deïŹning common atoms of dependency in propositional team logic. We reprove a classic non-elementary succinctness gap between ïŹrst-order logic and regular expressions in a much simpler way and establish a hierarchy of expressive power for the number of stars in RE over star-free expressions.
Many of the above results utilize explicit formulas in addition to game arguments. We use such formulas and a type counting argument to obtain non-elementary lower and upper bounds for the succinctness of deïŹning single words in ïŹrst-order logic and monadic second-order logic
- âŠ