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### Integer Vector Addition Systems with States [chapter]

Christoph Haase, Simon Halfon
2014 Lecture Notes in Computer Science
This paper studies reachability, coverability and inclusion problems for Integer Vector Addition Systems with States (ZVASS) and extensions and restrictions thereof.  ...  A ZVASS comprises a finite-state controller with a finite number of counters ranging over the integers.  ...  Z-VASS R ) if A = {λ 1 E 1 + • • • + λ d E d : λ i ∈ {0, 1}, i ∈ [d]}; -integer vector addition system with states (Z-VASS) if A = I d ; -integer vector addition system (Z-VAS) if A is a Z-VASS and |Q  ...

### Affine Extensions of Integer Vector Addition Systems with States [article]

Michael Blondin and Christoph Haase and Filip Mazowiecki and Mikhail Raskin
2021 arXiv   pre-print
We study the reachability problem for affine ℤ-VASS, which are integer vector addition systems with states in which transitions perform affine transformations on the counters.  ...  EXTENSIONS OF INTEGER VECTOR ADDITION SYSTEMS WITH STATES1:15 Vol. 17:3 M.  ...  RaskinVol. 17:3 1:10 AFFINE EXTENSIONS OF INTEGER VECTOR ADDITION SYSTEMS WITH STATES1:11 Vol. 17:3 AFFINE EXTENSIONS OF INTEGER VECTOR ADDITION SYSTEMS WITH STATES1:13 Vol. 17:3 AFFINE  ...

### Affine Extensions of Integer Vector Addition Systems with States

Michael Blondin, Christoph Haase, Filip Mazowiecki, Michael Wagner
2018 International Conference on Concurrency Theory
We study the reachability problem for affine Z-VASS, which are integer vector addition systems with states in which transitions perform affine transformations on the counters.  ...  hypothesis and by definition of M (σw k ). 2 0 1 8 14:10 Affine Extensions of Integer Vector Addition Systems with States Proof of Theorem 1.  ...  Affine Integer VASS An affine integer vector addition system with states (affine Z-VASS) is a tuple V = (d, Q, T ) where d ∈ N, Q is a finite set and T ⊆ Q × Z d×d × Z d × Q. Let us fix such a V.  ...

### Adding digit vectors [article]

Peter Hellekalek
2012 arXiv   pre-print
Our results show that there are at least 2^m-1 different additions of such vectors, while there exist only two types of addition that we may employ, addition with carry and addition without carry.  ...  In this paper, we study the different possibilities to add two vectors of digits of a given length m.  ...  in a hybrid function system (see  for this notion) cover all possible cases of function systems associated with additions of digit vectors.  ...

### Computer Arithmetic of Numbers, Vectors, Figures and Functions. Algorithms and Hardware

Solomon Khmelnik
2020 Zenodo
Multiplication of multidimensional Multiplication of integer codes of vectors by the base (1.31) \ 86 2.6. Multiplication of integer codes of vectors by the base (1.30) \ 86 2.7.  ...  Algebraic addition operations of C-codes \ 38 1. Unary operations with integer codes -Uni \ 38 2. Unary operations with non-integer codes -Unf \ 39 3.  ...

### Efficient Privacy-Preserving Fingerprint-Based Authentication System Using Fully Homomorphic Encryption

Taeyun Kim, Yongwoo Oh, Hyoungshick Kim
2020 Security and Communication Networks
The proposed system can perform the fingerprint matching process within about 166 seconds (±0.564 seconds) on average.  ...  To help smartphone users protect their phone, fingerprint-based authentication systems (e.g., Apple's Touch ID) have increasingly become popular in smartphones.  ...  Using this bit-level addition, we can expand it to the addition between two integers. e algorithm implements addition between two integers using the TFHE library, and it first initializes two carries,  ...

### Vector Arithmetic in the Triangular Grid

Khaled Abuhmaidan, Monther Aldwairi, Benedek Nagy
2021 Entropy
The triangular grid is also regular, but it is not a point lattice: it is not closed under vector-addition, which gives a challenge.  ...  This system is expanded to the plane using restrictions like, at least one of the coordinates is an integer and the sum of the three coordinates is in the interval [−1,1].  ...  Whenever the translation vector is also an integer vector (i.e., it belongs to ℤ 𝟐 ), again a simple vector addition gives the result.  ...

### Creative Mathematics Education

Edgar E. Escultura
2012 Creative Education
that results in the construction of the contradiction-free new real number system and the complex vector plane.  ...  Critical thinking is consolidated in the critique of current mathematics and its foundations, creativity in the construction of a mathematical space or system.  ...  Vector addition, translation and dot product as well as inversion are introduced here. At all times, both systems will be related to experience.  ...

### MOP: Matrix operations program

Alan B. Zonderman
1979 Behavior Research Methods
The left-hand side may also be an operand for addition and subtraction. However, it is a fatal error when the result is also an operand in a matrix multiplication. Integer Variable System.  ...  Subsequently, additional memory is requested from the operating system in minimum 1,000-word segments.Installation Notes.  ...

### DIV, FLOOR, CEIL, MOD and STEP Functions in Nested Loop Programs and Linearly Bounded Lattices [chapter]

P.C. Held, A.C.J. Kienhuis
1995 Algorithms and Parallel VLSI Architectures III
The nested loop programs may contain the integer operators: integer division, floor, ceil, and modulo, in expressions and the stride, or step size, of for loops may be greater than one.  ...  We will show the relation between the integer division operators in the SAP and linearly bounded lattices in the corresponding DG.  ...  This decomposition leads to matrices ½ and ¾ , with corresponding variable vectors Ã and Ã , respectively. Matrix ¾ defines the lattice vectors with the variables of Ã as free variables.  ...

### Reducibilities Among Decision Problems for HNN Groups, Vector Addition Systems and Subsystems of Peano Arithmetic

Michael Anshel, Kenneth McAloon
1983 Proceedings of the American Mathematical Society
Our purpose is to exhibit reducibilities among decision problems for conjugate powers in HNN groups, reachability sets of vector addition systems and sentences in subsystems of Peano arithmetic, and show  ...  By a vector addition system (of dimension n) we mean a pair (d, W), d in A/", IV a finite subset of Z" (where N and Z denote the nonnegative integers and integers respectively).  ...  Let VAS denote the class of vector addition systems and BVAS denote the bounded vector addition systems (i.e. (d, W) is in BVAS iff R(d,W) is finite).  ...

### Reducibilities among decision problems for HNN groups, vector addition systems and subsystems of Peano arithmetic

Michael Anshel, Kenneth McAloon
1983 Proceedings of the American Mathematical Society
Our purpose is to exhibit reducibilities among decision problems for conjugate powers in HNN groups, reachability sets of vector addition systems and sentences in subsystems of Peano arithmetic, and show  ...  By a vector addition system (of dimension n) we mean a pair (d, W), d in A/", IV a finite subset of Z" (where N and Z denote the nonnegative integers and integers respectively).  ...  Let VAS denote the class of vector addition systems and BVAS denote the bounded vector addition systems (i.e. (d, W) is in BVAS iff R(d,W) is finite).  ...

### Typing the Numeric Tower [chapter]

Vincent St-Amour, Sam Tobin-Hochstadt, Matthew Flatt, Matthias Felleisen
2012 Lecture Notes in Computer Science
In addition, the type system provides information to the compiler so that it can perform standard numeric optimizations.  ...  In this paper, we demonstrate how the type system of Typed Racket accommodates both a textbook programming style and expressive static checking.  ...  In addition, bounds on range types are static; bounds cannot depend on values, unlike in dependently-typed systems.  ...

### Testing additive integrality gaps [chapter]

Friedrich Eisenbrand, Nicolai Hähnle, Dömötör Pálvölgyi, Gennady Shmonin
2010 Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms
However, if, in addition, the objective is the all-one vector, then one can test in polynomial time whether the additive gap is bounded by a constant.  ...  We consider the problem of testing whether the maximum additive integrality gap of a family of integer programs in standard form is bounded by a given constant.  ...  function vector into account if one wants to have a similar notion for additive integrality gaps.  ...

### Testing additive integrality gaps

Friedrich Eisenbrand, Nicolai Hähnle, Dömötör Pálvölgyi, Gennady Shmonin
2012 Mathematical programming
However, if, in addition, the objective is the all-one vector, then one can test in polynomial time whether the additive gap is bounded by a constant.  ...  We consider the problem of testing whether the maximum additive integrality gap of a family of integer programs in standard form is bounded by a given constant.  ...  function vector into account if one wants to have a similar notion for additive integrality gaps.  ...
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