Obtaining representations for probabilities of voting outcomes with effectively unlimited precision integer arithmetic

2002 ◽  
Vol 19 (3) ◽  
pp. 503-512 ◽  
Author(s):  
William V. Gehrlein
Keyword(s):  
Integer Arithmetic

An Ada package for multi-precision integer arithmetic

1993 ◽  
Vol 19 (2) ◽  
pp. 17-24 ◽  
Author(s):  
D. G. Knight
Keyword(s):  
Integer Arithmetic

scholarly journals Computing the Exact Number of Similarity Classes in the Longest Edge Bisection of Tetrahedra

Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1447
Author(s):  
Jose P. Suárez ◽  
Agustín Trujillo ◽  
Tania Moreno
Keyword(s):  
Data Structure ◽  
Stability Condition ◽  
Open Problem ◽  
Integer Arithmetic ◽  
Exact Number ◽  
Similarity Class ◽  
The Stability ◽  
High Level ◽  
The Cost

Showing whether the longest-edge (LE) bisection of tetrahedra meshes degenerates the stability condition or not is still an open problem. Some reasons, in part, are due to the cost for achieving the computation of similarity classes of millions of tetrahedra. We prove the existence of tetrahedra where the LE bisection introduces, at most, 37 similarity classes. This family of new tetrahedra was roughly pointed out by Adler in 1983. However, as far as we know, there has been no evidence confirming its existence. We also introduce a new data structure and algorithm for computing the number of similarity tetrahedral classes based on integer arithmetic, storing only the square of edges. The algorithm lets us perform compact and efficient high-level similarity class computations with a cost that is only dependent on the number of similarity classes.

scholarly journals Automatic Synthesis of Generalized Winning Strategies of Impartial Combinatorial Games Using SMT Solvers

2020 ◽  
Author(s):  
Kaisheng Wu ◽  
Liangda Fang ◽  
Liping Xiong ◽  
Zhao-Rong Lai ◽  
Yong Qiao ◽  
...  
Keyword(s):  
Artificial Intelligence ◽  
Game Theory ◽  
Experimental Results ◽  
Combinatorial Games ◽  
Logical Framework ◽  
Automatic Synthesis ◽  
Integer Arithmetic ◽  
Smt Solvers ◽  
Winning Strategies

Strategy representation and reasoning has recently received much attention in artificial intelligence. Impartial combinatorial games (ICGs) are a type of elementary and fundamental games in game theory. One of the challenging problems of ICGs is to construct winning strategies, particularly, generalized winning strategies for possibly infinitely many instances of ICGs. In this paper, we investigate synthesizing generalized winning strategies for ICGs. To this end, we first propose a logical framework to formalize ICGs based on the linear integer arithmetic fragment of numeric part of PDDL. We then propose an approach to generating the winning formula that exactly captures the states in which the player can force to win. Furthermore, we compute winning strategies for ICGs based on the winning formula. Experimental results on several games demonstrate the effectiveness of our approach.

scholarly journals Verifying a Solver for Linear Mixed Integer Arithmetic in Isabelle/HOL

2020 ◽  
pp. 233-250
Author(s):  
Ralph Bottesch ◽  
Max W. Haslbeck ◽  
Alban Reynaud ◽  
René Thiemann
Keyword(s):  
Linear Programming ◽  
Branch And Bound ◽  
Decision Procedure ◽  
Simplex Algorithm ◽  
Upper Bounds ◽  
Mixed Integer ◽  
Small Solution ◽  
Code Generator ◽  
Integer Arithmetic ◽  
Termination Proofs

AbstractWe implement a decision procedure for linear mixed integer arithmetic and formally verify its soundness in Isabelle/HOL. We further integrate this procedure into one application, namely into , a formally verified certifier to check untrusted termination proofs. This checking involves assertions of unsatisfiability of linear integer inequalities; previously, only a sufficient criterion for such checks was supported. To verify the soundness of the decision procedure, we first formalize the proof that every satisfiable set of linear integer inequalities also has a small solution, and give explicit upper bounds. To this end we mechanize several important theorems on linear programming, including statements on integrality and bounds. The procedure itself is then implemented as a branch-and-bound algorithm, and is available in several languages via Isabelle’s code generator. It internally relies upon an adapted version of an existing verified incremental simplex algorithm.

scholarly journals Integer arithmetic approximation of the hog algorithm used for pedestrian detection

2017 ◽  
Vol 14 (2) ◽  
pp. 329-346 ◽  
Author(s):  
Srdjan Sladojevic ◽  
Andras Anderla ◽  
Dubravko Culibrk ◽  
Darko Stefanovic ◽  
Bojan Lalic
Keyword(s):  
Classification Accuracy ◽  
System Performance ◽  
Detection System ◽  
Pedestrian Detection ◽  
Lessons Learned ◽  
People Detection ◽  
Algorithm Performance ◽  
Integer Arithmetic ◽  
Point Arithmetic ◽  
Angle Calculation

This paper presents the results of a study of the effects of integer (fixed-point) arithmetic implementation on classification accuracy of a popular open-source people detection system based on Histogram of Oriented Gradients. It is investigated how the system performance deviates from the reference algorithm performance as integer arithmetic is introduced with different bit-width in several critical parts of the system. In performed experiments, the effects of different bit-width integer arithmetic implementation for four key operations were separately considered: HoG descriptor magnitude calculation, HoG descriptor angle calculation, normalization and SVM classification. It is found that a 13-bit representation of variables is more than sufficient to accurately implement this system in integer arithmetic. The experiments in the paper are conducted for pedestrian detection and the methodology and the lessons learned from this study allow generalization of conclusions to a broader class of applications.

Extending Multiple-Valued Clausal Forms with Linear Integer Arithmetic

2011 ◽  
Author(s):  
Carlos Ansótegui ◽  
Miquel Bofill ◽  
Felip Manyà ◽  
Mateu Villaret
Keyword(s):  
Integer Arithmetic
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