127 research outputs found
Tropical Geometry of Statistical Models
This paper presents a unified mathematical framework for inference in
graphical models, building on the observation that graphical models are
algebraic varieties.
From this geometric viewpoint, observations generated from a model are
coordinates of a point in the variety, and the sum-product algorithm is an
efficient tool for evaluating specific coordinates. The question addressed here
is how the solutions to various inference problems depend on the model
parameters. The proposed answer is expressed in terms of tropical algebraic
geometry. A key role is played by the Newton polytope of a statistical model.
Our results are applied to the hidden Markov model and to the general Markov
model on a binary tree.Comment: 14 pages, 3 figures. Major revision. Applications now in companion
paper, "Parametric Inference for Biological Sequence Analysis
Highly Automated Formal Verification of Arithmetic Circuits
This dissertation investigates the problems of two distinctive formal verification techniques for verifying large scale multiplier circuits and proposes two approaches to overcome some of these problems. The first technique is equivalence checking based on recurrence relations, while the second one is the symbolic computation technique which is based on the theory of Gröbner bases. This investigation demonstrates that approaches based on symbolic computation have better scalability and more robustness than state-of-the-art equivalence checking techniques for verification of arithmetic circuits. According to this conclusion, the thesis leverages the symbolic computation technique to verify floating-point designs. It proposes a new algebraic equivalence checking, in contrast to classical combinational equivalence checking, the proposed technique is capable of checking the equivalence of two circuits which have different architectures of arithmetic units as well as control logic parts, e.g., floating-point multipliers
Commutative Algebra of Statistical Ranking
A model for statistical ranking is a family of probability distributions
whose states are orderings of a fixed finite set of items. We represent the
orderings as maximal chains in a graded poset. The most widely used ranking
models are parameterized by rational function in the model parameters, so they
define algebraic varieties. We study these varieties from the perspective of
combinatorial commutative algebra. One of our models, the Plackett-Luce model,
is non-toric. Five others are toric: the Birkhoff model, the ascending model,
the Csiszar model, the inversion model, and the Bradley-Terry model. For these
models we examine the toric algebra, its lattice polytope, and its Markov
basis.Comment: 25 page
A Certified Polynomial-Based Decision Procedure for Propositional Logic
In this paper we present the formalization of a decision procedure for Propositional Logic based on polynomial normalization. This formalization is suitable for its automatic verification in an applicative logic like Acl2. This application of polynomials has been developed by reusing a previous work on polynomial rings [19], showing that a proper formalization leads to a high level of reusability. Two checkers are defined: the first for contradiction formulas and the second for tautology formulas. The main theorems state that both checkers are sound and complete. Moreover, functions for generating models and counterexamples of formulas are provided. This facility plays also an important role in the main proofs. Finally, it is shown that this allows for a highly automated proof development
A verified Common Lisp implementation of Buchberger's algorithm in ACL2
In this article, we present the formal verification of a Common
Lisp implementation of Buchberger's algorithm for computing
Gröbner bases of polynomial ideals. This work is carried out in
ACL2, a system which provides an integrated environment where
programming (in a pure functional subset of Common Lisp) and
formal verification of programs, with the assistance of a theorem
prover, are possible. Our implementation is written in a real
programming language and it is directly executable within the
ACL2 system or any compliant Common Lisp system. We provide
here snippets of real verified code, discuss the formalization details
in depth, and present quantitative data about the proof effort
Thomas Decomposition of Algebraic and Differential Systems
In this paper we consider disjoint decomposition of algebraic and non-linear
partial differential systems of equations and inequations into so-called simple
subsystems. We exploit Thomas decomposition ideas and develop them into a new
algorithm. For algebraic systems simplicity means triangularity, squarefreeness
and non-vanishing initials. For differential systems the algorithm provides not
only algebraic simplicity but also involutivity. The algorithm has been
implemented in Maple
Revealing the canalizing structure of Boolean functions: Algorithms and applications
Boolean functions can be represented in many ways including logical forms,
truth tables, and polynomials. Additionally, Boolean functions have different
canonical representations such as minimal disjunctive normal forms. Other
canonical representation is based on the polynomial representation of Boolean
functions where they can be written as a nested product of canalizing layers
and a polynomial that contains the noncanalizing variables. In this paper we
study the problem of identifying the canalizing layers format of Boolean
functions. First, we show that the problem of finding the canalizing layers is
NP-hard. Second, we present several algorithms for finding the canalizing
layers of a Boolean function, discuss their complexities, and compare their
performances. Third, we show applications where the computation of canalizing
layers can be used for finding a disjunctive normal form of a nested canalizing
function. Another application deals with the reverse engineering of Boolean
networks with a prescribed layering format. Finally, implementations of our
algorithms in Python and in the computer algebra system Macaulay2 are available
at https://github.com/ckadelka/BooleanCanalization.Comment: 13 pages, 1 figur
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