Boolean Tree Research Articles

FR-CSG: Fast and Reliable Modeling for Constructive Solid Geometry.

Reconstructing CSG trees from CAD models is a critical subject in reverse engineering. While there have been notable advancements in CSG reconstruction, challenges persist in capturing geometric details and achieving efficiency. Additionally, since non-axis-aligned volumetric primitives cannot maintain coplanar characteristics due to discretization errors, existing Boolean operations often lead to zero-volume surfaces and suffer from topological errors during the CSG modeling process. To address these issues, we propose a novel workflow to achieve fast CSG reconstruction and reliable forward modeling. First, we employ feature removal and model subdivision techniques to decompose models into sub-components. This significantly expedites the reconstruction by simplifying the complexity of the models. Then, we introduce a more reasonable method for primitive generation and filtering, and utilize a size-related optimization approach to reconstruct CSG trees. By re-adding features as additional nodes in the CSG trees, our method not only preserves intricate details but also ensures the conciseness, semantic integrity, and editability of the resulting CSG tree. Finally, we develop a coplanar primitive discretization method that represents primitives as large planes and extracts the original triangles after intersection. We extend the classification of triangles and incorporate a coplanar-aware Boolean tree assessment technique, allowing us to achieve manifold and watertight modeling results without zero-volume surfaces, even in extreme degenerate cases. We demonstrate the superiority of our method over state-of-the-art approaches. Moreover, the reconstructed CSG trees generated by our method contain extensive semantic information, enabling diverse model editing tasks.

Implicit Conversion of Manifold B-Rep Solids by Neural Halfspace Representation

We present a novel implicit representation --- neural halfspace representation (NH-Rep), to convert manifold B-Rep solids to implicit representations. NH-Rep is a Boolean tree built on a set of implicit functions represented by the neural network, and the composite Boolean function is capable of representing solid geometry while preserving sharp features. We propose an efficient algorithm to extract the Boolean tree from a manifold B-Rep solid and devise a neural network-based optimization approach to compute the implicit functions. We demonstrate the high quality offered by our conversion algorithm on ten thousand manifold B-Rep CAD models that contain various curved patches including NURBS, and the superiority of our learning approach over other representative implicit conversion algorithms in terms of surface reconstruction, sharp feature preservation, signed distance field approximation, and robustness to various surface geometry, as well as a set of applications supported by NH-Rep.

Resolving Stanley's conjecture on $k$-fold acyclic complexes

In 1993 Stanley showed that if a simplicial complex is acyclic over some field, then its face poset can be decomposed into disjoint rank \(1\) boolean intervals whose minimal faces together form a subcomplex. Stanley further conjectured that complexes with a higher notion of acyclicity could be decomposed in a similar way using boolean intervals of higher rank. We provide an explicit counterexample to this conjecture. We also prove a version of the conjecture for boolean trees and show that the original conjecture holds when this notion of acyclicity is as high as possible.Mathematics Subject Classifications: 05E45, 55U10

Energy-Reliability Limits in Nanoscale Feedforward Neural Networks and Formulas

Due to energy-efficiency requirements, computational systems are now being implemented using noisy nanoscale semiconductor devices whose reliability depends on energy consumed. We study circuit-level energy-reliability limits for deep feedforward neural networks (multilayer perceptrons) built using such devices, and en route also establish the same limits for formulas (boolean tree-structured circuits). To obtain energy lower bounds, we extend Pippenger’s mutual information propagation technique for characterizing the complexity of noisy circuits, since small circuit complexity need not imply low energy. Many device technologies require all gates to have the same electrical operating point; in circuits of such uniform gates, we show that the minimum energy required to achieve any non-trivial reliability scales superlinearly with the number of inputs. Circuits implemented in emerging device technologies like spin electronics can, however, have gates operate at different electrical points; in circuits of such heterogeneous gates, we show energy scaling can be linear in the number of inputs. Building on our extended mutual information propagation technique and using crucial insights from convex optimization theory, we develop an algorithm to compute energy lower bounds for any given boolean tree under heterogeneous gates. This algorithm runs in linear time in number of gates, and is therefore practical for modern circuit design. As part of our development we find a simple procedure for energy allocation across circuit gates with different operating points and neural networks with differently-operating layers.

The relation between tree size complexity and probability for Boolean functions generated by uniform random trees

An associative Boolean tree is a plane rooted tree whose internal nodes are labelled by and or or and whose leaves are labelled by literals taken from a fixed set of variables and their negations. We study the distribution induced on the set of Boolean functions by the uniform distribution on the set of associative trees of a large fixed size, where the size of a tree is defined as the number of its nodes. Using analytic combinatorics, we prove a relation between the probability of a given function and its tree size complexity.

Importance Measures for Epistatic Interactions in Case-Parent Trios

Ensemble methods (such as Bagging and Random Forests) take advantage of unstable base learners (such as decision trees) to improve predictions, and offer measures of variable importance useful for variable selection. LogicFS has been proposed as such an ensemble learner for case-control studies when interactions of single nucleotide polymorphisms (SNPs) are of particular interest. LogicFS uses bootstrap samples of the data and employs the Boolean trees derived in logic regression as base learners to create ensembles of models that allow for the quantification of the contributions of epistatic interactions to the disease risk. In this article, we propose an extension of logicFS suitable for case-parent trio data, and derive an additional importance measure that is much less influenced by linkage disequilibrium between SNPs than the measure originally used in logicFS. We illustrate the performance of the novel procedure in simulation studies and in a case study of 461 case-parent trios with autistic children.

A Non-Discrete Approach to the Evolution of Information Filtering Trees

Boolean keyword trees can be used for information filtering. When they are generated by an evolutionary algorithm they can adapt to changes in the user's interests. This paper presents a non-discrete tree model that has been designed to be easier to evolve than more commonly used Boolean trees. Each tree is encoded by a genome that contains binary data as well as real-valued data. The results of the experiments are promising and suggest that the proposed model indeed offers an improvement.

Learning read-once formulas with queries

A read-once formula is a Boolean formula in which each variable occurs, at most, once. Such formulas are also called μ-formulas or Boolean trees. This paper treats the problem of exactly identifying an unknown read-once formula using specific kinds of queries. The main results are a polynomial-time algorithm for exact identification of monotone read-once formulas using only membership queries, and a polynomial-time algorithm for exact identification of general read-once formulas using equivalence and membership queries (a protocol based on the notion of a minimally adequate teacher [1]). The results of the authors improve on Valiant's previous results for read-once formulas [26]. It is also shown, that no polynomial-time algorithm using only membership queries or only equivalence queries can exactly identify all read-once formulas.

Integration of solid modeling and finite element generation

Abstract Constructive solid geometry (CSG) appeals to the engineer as a method for the generation of complex geometries. CSG results in a ‘Boolean tree’ definition of the object, combining geometry, topology and logic. To use this definition as a starting point for the generation of a discretized finite element model, it is customary to first create a ‘boundary representation’ of the surface of the geometric model. A mesh is then created within the volume bound by the computed external surface. This paper describes an alternative approach in which the meshing is performed on the primitives, prior to the Boolean operations, and addresses the geometric definition as well as its discretized form simultaneously.

Near real-time CSG rendering using tree normalization and geometric pruning

A description is given of a set of algorithms for efficiently rendering an object defined by constructive solid geometry (CSG) directly onto a frame buffer without converting first to a boundary representation. This method requires only that the frame buffer contain sufficient memory to hold two color values, two depth values, and three one-bit flags. The algorithm first converts the CSG tree to a normalized form that is analogous to the sum-of-products form for Boolean switching functions. The following are developed: dynamic interleaving of Boolean tree normalization with bounding-box pruning, allowing efficient rendering for most CSG objects; a method for extending the technique to nonconvex primitives; and implementation of these ideas in an interactive CSG design system on the Pixel-planes 4 solid modeling system. In the design system the designer directly manipulates the CSG structure while continuously viewing the color rendering of the object being designed. >

QUADRIL

Most man-made objects can be closely approximated by bodies whose surfaces are composed of portions of second-order (quadric) surfaces. These surfaces include elliptic, hyperbolic, and parabolic cylinders, as well as quadric cones, paraboloids, hyperboloids, ellipsoids, and pairs of planes. Simple planes (first-order surfaces) may be included as degenerate quadric surfaces. Because these quadric-surface bodies are so useful for modelling man-made objects, it is important that any Computer-Aided Design (CAD) system be able to work with such bodies. The “QUADRIL” language described here was designed to accept descriptions of quadric-surface bodies in character-string form. QUADRIL has a mixture of English-like and algebraic syntax. It may be used to specify quadric-surface bodies and then to display them on various media. QUADRIL will accept descriptions of quadric-surface bodies either as “volumetric” combinations of basic bodies, or as boolean functions of bounding surfaces. English-like syntax is used for specifying what surfaces and basic bodies are used, while algebraic syntax is used to transform the canonical forms of the surfaces or bodies into the shape, position, and orientation that the user desires. Volumetric combination of bodies involves the operations of union (+), intersection (*), and subtraction (-). Boolean specification of volumes is in terms of a boolean tree with the bounding surfaces as leaf nodes. The tree is expressed as a character string. QUADRIL permits using user-created “STRUCTURES” as component bodies (“OBJECTS”) in greater STRUCTURES. The display of the quadric-surface bodies may also be specified in QUADRIL. The user is considered fixed in space, while the body is transformed to give the desired view.

Fault tree analysis using bit manipulation : Dean B. Wheeler, Jason S. Hsuan, Ralph R. Duersch and Glenn M. Roe. IEEE Trans. Reliab.R-26, (2) 95 (June 1977)

numeric identification ofgates andprimary events, representa- tionoftheBoolean relationship (AND, OR)present ateach KeyWords-Fault tree, Bitmanipulation, Computerized fault tree gate, andtheordering ofthese appropriate tothesubsequent analysis, Boolean tree reduction. analysis (inthis case abottom-up sequence solvable bysequen- tial substitution). Toobtain theminimal cutsets, three types Reader Aids- Purpose: Efficient technique forcomputerized fault tree analysis oflogical redundancy mustthenbeidentified andeliminated: Special mathneeded forexplanation &results: Boolean algebra Results useful to:Engineers, Reliability andrisk analysts, Computer Redundant Factors (AA-A), (1) scientists Summary andConclusions-This paper describes anefficient tech- Subset Redundancy (A+AB A),

J. Paul Roth. Two logical minimization problems. Summaries of talks presented at the Summer Institute for Symbolic Logic, Cornell University, 1957, 2nd edn., Communications Research Division, Institute for Defense Analyses, Princeton, N.J., 1960, pp. 396–401. - J. Paul Roth. Algebraic topological methods for the synthesis of switching systems. II. Proceedings of the International Symposium on the Theory of Switching, Harvard

J. Paul Roth. Two logical minimization problems. Summaries of talks presented at the Summer Institute for Symbolic Logic, Cornell University, 1957, 2nd edn., Communications Research Division, Institute for Defense Analyses, Princeton, N.J., 1960, pp. 396–401. - J. Paul Roth. Algebraic topological methods for the synthesis of switching systems. II. Proceedings of the International Symposium on the Theory of Switching, Harvard University, April 2, 1957, The annals of the computation laboratory, vol. 29 (1959), pp. 57–73. - J. Paul Roth and E. G. Wagner. Algebraic topological methods for the synthesis of switching systems. III: Minimization of nonsingtdar Boolean trees. IBM journal of research and development, vol. 3 (1959), pp. 326–344. - Volume 25 Issue 4

Minimization over Boolean Trees

An algorithm is provided for what might be termed the general problem of logical design of circuits with one output and no feedback. Given a set B of logical building blocks, each with a positive cost, each with one output, and given a Boolean function f the problem is to prescribe a Boolean tree constructed from the available set of building blocks which realizes f and which has a minimum cost. Actually a more general problem involving don't care conditions is treated. The cost of a Boolean tree shall be the sum of the costs of the building blocks of which it is composed. A special case of this problem is the classical logical problem of finding a functional expression for a given logical function which uses a minimum number of conjunctions, disjunctions and negations. Programmed on an IBM 704 computer, the algorithm is believed to be efficient on problems with eight or less variables.

Algebraic Topological Methods for the Synthesis of Switching Systems—Part III: Minimization of Nonsingular Boolean Trees

An algorithm is given for solving a general problem in combinational switching-circuit minimization theory. The circuits considered consist of a disjunction (OR-ing together) of trees of any set of logical elements, with the restriction that in any given tree no input appears more than once. To each logical element is attached a positive cost. A method is presented for designing a minimum-cost circuit of this variety for any given logical function. Two parallel treatments are given, one viewing it as an abstract mathematical problem, the other considering it as an engineering problem.