On the Computational Complexity of Finite Operations

2016 ◽  
Vol 27 (01) ◽  
pp. 15-38 ◽  
Author(s):  
Slavcho Shtrakov ◽  
Ivo Damyanov
Keyword(s):  
Computational Complexity ◽  
Combinatorial Problems ◽  
Affine Group ◽  
Equivalence Relations ◽  
Transformation Groups ◽  
Finite Function ◽  
Important Measure ◽  
Discrete Functions

The essential variables in a finite function f are defined as variables which occur in f and weigh with the values of that function. The number of essential variables is an important measure of complexity for discrete functions. When replacing some variables in a function with constants the resulting functions are called subfunctions, and when replacing all essential variables in a function with constants we obtain an implementation of this function. Such an implementation corresponds with a path in an ordered decision diagram (ODD) of the function which connects the root with a leaf of the diagram. The sets of essential variables in subfunctions of f are called separable in f. In this paper we study several properties of separable sets of variables in functions which directly affect the number of implementations and subfunctions in these functions. We define equivalence relations which classify the functions of k-valued logic into classes with the same number of: (i) implementations; (ii) subfunctions; and (iii) separable sets. These relations induce three transformation groups which are compared with the lattice of all subgroups of restricted affine group (RAG). This allows us to solve several important computational and combinatorial problems.

IMBEDDING OF TRANSFORMATION GROUPS IN AFFINE ONES AND RENORMALIZATION OF TWO-DIMENSIONAL HOMOGENEOUS σ-MODELS

1993 ◽  
Vol 08 (31) ◽  
pp. 2937-2942
Author(s):  
A. V. BRATCHIKOV
Keyword(s):  
Homogeneous Space ◽  
Transformation Group ◽  
Affine Group ◽  
Coupling Constants ◽  
Transformation Groups ◽  
Two Dimensional

The BLZ method for the analysis of renormalizability of the O(N)/O(N − 1) model is extended to the σ-model built on an arbitrary homogeneous space G/H and in arbitrary coordinates. For deriving Ward-Takahashi (WT) identities an imbedding of the transformation group G in an affine group is used. The structure of the renormalized action is found. All the infinities can be absorbed in a coupling constants renormalization and in a renormalization of auxiliary constants which are related to the imbedding.

scholarly journals The universality of polynomial time Turing equivalence

2016 ◽  
Vol 28 (3) ◽  
pp. 448-456 ◽  
Author(s):  
ANDREW MARKS
Keyword(s):  
Computational Complexity ◽  
Large Class ◽  
Complexity Theory ◽  
Polynomial Time ◽  
Equivalence Relations ◽  
Turing Degrees ◽  
Computational Complexity Theory ◽  
Borel Sets ◽  
Borel Equivalence Relations

We show that polynomial time Turing equivalence and a large class of other equivalence relations from computational complexity theory are universal countable Borel equivalence relations. We then discuss ultrafilters on the invariant Borel sets of these equivalence relations which are related to Martin's ultrafilter on the Turing degrees.

scholarly journals On the Families of Stable Multivariate Transformations of Large Order and Their Cryptographical Applications

2017 ◽  
Vol 70 (1) ◽  
pp. 107-117 ◽  
Author(s):  
Vasyl Ustimenko
Keyword(s):  
Computational Complexity ◽  
Commutative Ring ◽  
Finite Fields ◽  
Transformation Groups ◽  
Affine Spaces ◽  
Exponential Order ◽  
Cyclic Groups ◽  
Key Exchange Protocols ◽  
Polynomial Transformations ◽  
High Degree

Abstract Families of stable cyclic groups of nonlinear polynomial transformations of affine spaces Kn over general commutative ring K of with n increasing order can be used in the key exchange protocols and El Gamal multivariate cryptosystems related to them. We suggest to use high degree of noncommutativity of affine Cremona group and modify multivariate El Gamal algorithm via conjugations of two polynomials of kind gk and g−1 given by key holder (Alice) or giving them as elements of different transformation groups. Recent results on the existence of families of stable transformations of prescribed degree and density and exponential order over finite fields can be used for the implementation of schemes as above with feasible computational complexity.

Computational Complexity and Statistical Physics

2005 ◽  
Keyword(s):  
Phase Transitions ◽  
Computational Complexity ◽  
Computer Science ◽  
Statistical Physics ◽  
Combinatorial Problems ◽  
Complete Problems ◽  
Computer Scientists ◽  
Background Material ◽  
Np Complete ◽  
Exciting Area

Computer science and physics have been closely linked since the birth of modern computing. In recent years, an interdisciplinary area has blossomed at the junction of these fields, connecting insights from statistical physics with basic computational challenges. Researchers have successfully applied techniques from the study of phase transitions to analyze NP-complete problems such as satisfiability and graph coloring. This is leading to a new understanding of the structure of these problems, and of how algorithms perform on them. Computational Complexity and Statistical Physics will serve as a standard reference and pedagogical aid to statistical physics methods in computer science, with a particular focus on phase transitions in combinatorial problems. Addressed to a broad range of readers, the book includes substantial background material along with current research by leading computer scientists, mathematicians, and physicists. It will prepare students and researchers from all of these fields to contribute to this exciting area.

scholarly journals Minor complexity of discrete functions

2020 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Slavcho Shtrakov
Keyword(s):  
Data Structure ◽  
Combinatorial Problems ◽  
Complexity Measures ◽  
Essential Arity ◽  
Discrete Functions

In this paper we study a class of complexity measures, induced by a new data structure for representing k-valued functions (operations), called minor decision diagram. When assigning values to some variables in a function the resulting functions are called subfunctions, and when identifying some variables the resulting functions are called minors. The sets of essential variables in subfunctions of f are called separable in f.We examine the maximal separable subsets of variables and their conjugates, introduced in the paper, proving that each such set has at least one conjugate. The essential arity gap gap(f) of the function f is the minimal number of essential variables in f which become fictive when identifying distinct essential variables in f. We also investigate separable sets of variables in functions with non-trivial arity gap. This allows us to solve several important algebraic, computational and combinatorial problems about the finite-valued functions.

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