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Register a new userA Functional Hitchhikers Guide to Hereditarily Finite Sets, Ackermann Encodings and Pairing Functions
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Added by
Paul Tarau
Publication date
2008
fields
Informatics Engineering
and research's language is
English
Authors
Paul Tarau
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The paper is organized as a self-contained literate Haskell program that implements elements of an executable finite set theory with focus on combinatorial generation and arithmetic encodings. The code, tested under GHC 6.6.1, is available at http://logic.csci.unt.edu/tarau/research/2008/fSET.zip . We introduce ranking and unranking functions generalizing Ackermanns encoding to the universe of Hereditarily Finite Sets with Urelements. Then we build a lazy enumerator for Hereditarily Finite Sets with Urelements that matches the unranking function provided by the inverse of Ackermanns encoding and we describe functors between them resulting in arithmetic encodings for powersets, hypergraphs, ordinals and choice functions. After implementing a digraph representation of Hereditarily Finite Sets we define {em decoration functions} that can recover well-founded sets from encodings of their associated acyclic digraphs. We conclude with an encoding of arbitrary digraphs and discuss a concept of duality induced by the set membership relation. Keywords: hereditarily finite sets, ranking and unranking functions, executable set theory, arithmetic encodings, Haskell data representations, functional programming and computational mathematics
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In many computer vision classification tasks, class priors at test time often differ from priors on the training set. In the case of such prior shift, classifiers must be adapted correspondingly to maintain close to optimal performance. This paper analyzes methods for adaptation of probabilistic classifiers to new priors and for estimating new priors on an unlabeled test set. We propose a novel method to address a known issue of prior estimation methods based on confusion matrices, where inconsistent estimates of decision probabilities and confusion matrices lead to negative values in the estimated priors. Experiments on fine-grained image classification datasets provide insight into the best practice of prior shift estimation and classifier adaptation and show that the proposed method achieves state-of-the-art results in prior adaptation. Applying the best practice to two tasks with naturally imbalanced priors, learning from web-crawled images and plant species classification, increased the recognition accuracy by 1.1% and 3.4% respectively.
Prologs ability to return multiple answers on backtracking provides an elegant mechanism to derive reversible encodings of combinatorial objects as Natural Numbers i.e. {em ranking} and {em unranking} functions. Starting from a generalization of Ackermans encoding of Hereditarily Finite Sets with Urelements and a novel tupling/untupling operation, we derive encodings for Finite Functions and use them as building blocks for an executable theory of {em Hereditarily Finite Functions}. The more difficult problem of {em ranking} and {em unranking} {em Hereditarily Finite Permutations} is then tackled using Lehmer codes and factoradics. The paper is organized as a self-contained literate Prolog program available at url{http://logic.csci.unt.edu/tarau/research/2008/pHFF.zip}
It is our view that the state of the art in constructing a large collection of graph algorithms in terms of linear algebraic operations is mature enough to support the emergence of a standard set of primitive building blocks. This paper is a position paper defining the problem and announcing our intention to launch an open effort to define this standard.
GlobalBioIm is an open-source MATLAB library for solving inverse problems. The library capitalizes on the strong commonalities between forward models to standardize the resolution of a wide range of imaging inverse problems. Endowed with an operator-algebra mechanism, GlobalBioIm allows one to easily solve inverse problems by combining elementary modules in a lego-like fashion. This user-friendly toolbox gives access to cutting-edge reconstruction algorithms, while its high modularity makes it easily extensible to new modalities and novel reconstruction methods. We expect GlobalBioIm to respond to the needs of imaging scientists looking for reliable and easy-to-use computational tools for solving their inverse problems. In this paper, we present in detail the structure and main features of the library. We also illustrate its flexibility with examples from multichannel deconvolution microscopy.
We construct a symmetric invertible binary pairing function $F(m,n)$ on the set of positive integers with a property of $F(m,n)=F(n,m)$. Then we provide a complete proof of its symmetry and bijectivity, from which the construction of symmetric invertible binary pairing functions on any custom set of integers could be seen.
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