On the complexity of solving linear congruences and computing nullspaces modulo a constant

We consider the problems of determining the feasibility of a linear congruence, producing a solution to a linear congruence, and finding a spanning set for the nullspace of an integer matrix, where each problem is considered modulo an arbitrary constant k>1. These problems are known to be complete for the logspace modular counting classes {Mod_k L} = {coMod_k L} in special case that k is prime (Buntrock et al, 1992). By considering variants of standard logspace function classes --- related to #L and functions computable by UL machines, but which only characterize the number of accepting paths modulo k --- we show that these problems of linear algebra are also complete for {coMod_k L} for any constant k>1. Our results are obtained by defining a class of functions FUL_k which are low for {Mod_k L} and {coMod_k L} for k>1, using ideas similar to those used in the case of k prime in (Buntrock et al, 1992) to show closure of Mod_k L under NC^1 reductions (including {Mod_k L} oracle reductions). In addition to the results above, we briefly consider the relationship of the class FUL_k for arbitrary moduli k to the class {F.coMod_k L} of functions whose output symbols are verifiable by {coMod_k L} algorithms; and consider what consequences such a comparison may have for oracle closure results of the form {Mod_k L}^{Mod_k L} = {Mod_k L} for composite k.Comment: 17 pages, one Appendix; minor corrections and revisions to presentation, new observations regarding the prospect of oracle closures. Comments welcom

Definability of linear equation systems over groups and rings

Motivated by the quest for a logic for PTIME and recent insights that the descriptive complexity of problems from linear algebra is a crucial aspect of this problem, we study the solvability of linear equation systems over finite groups and rings from the viewpoint of logical (inter-)definability. All problems that we consider are decidable in polynomial time, but not expressible in fixed-point logic with counting. They also provide natural candidates for a separation of polynomial time from rank logics, which extend fixed-point logics by operators for determining the rank of definable matrices and which are sufficient for solvability problems over fields. Based on the structure theory of finite rings, we establish logical reductions among various solvability problems. Our results indicate that all solvability problems for linear equation systems that separate fixed-point logic with counting from PTIME can be reduced to solvability over commutative rings. Moreover, we prove closure properties for classes of queries that reduce to solvability over rings, which provides normal forms for logics extended with solvability operators. We conclude by studying the extent to which fixed-point logic with counting can express problems in linear algebra over finite commutative rings, generalising known results on the logical definability of linear-algebraic problems over finite fields

Polygonal Complexity Counting Classes

In this work we introduce new counting classes defined by special sets of natural number like Triangular, Perfect Square, Pentagonal and generally K-gonal numbers. We shall see that  NP is a subclass of all  complements of K-gonal classes and all  K-gonal classes are subclasses of a class defined by only perfect square numbers. صفوف التعقيد العدّية المضلعة في هذا العمل تمّ تعريف صفوف تعقيد عدّية جديدة اعتمادا على مجموعات جزئية من مجموعة الأعداد الطبيعية كالأعداد المثلثية 3-gonal والمربعة 4-gonal والمخمسة5-gonalوعموما الأعداد المضلعة K-gonal numbers وسنبين أنّ الصف NP هو صف جزئي من مجموعة متممات الصف المولد بالمجموعةK-gonal وأنّ الصفوف التي تعرفها المجموعات K-gonal هي صفوف جزئية من الصف الذي تعرفه مجموعة الأعداد المربعة فق

Robustness for Space-Bounded Statistical Zero Knowledge

We show that the space-bounded Statistical Zero Knowledge classes SZK_L and NISZK_L are surprisingly robust, in that the power of the verifier and simulator can be strengthened or weakened without affecting the resulting class. Coupled with other recent characterizations of these classes [Eric Allender et al., 2023], this can be viewed as lending support to the conjecture that these classes may coincide with the non-space-bounded classes SZK and NISZK, respectively

Counting Classes Definedby Prime Numbers

ريف صفوف تعقيد جديدة لحاسبة تورينك اللاحتمية بزمن  P-NP-problem is the most important issue in computing theory and computational complexity,Through her study has been defined and studied the ranks of other complexity such ascoNP, PP, .. In this paper we have defined new complexity classes forpolynomial time nondeterministic Turing Machine using prime and composite numbersfor k-prime numbers  and we have proven that  is a subclass of it and the class is a subclass o

The Parallel Dynamic Complexity of the Abelian Cayley Group Membership Problem

Let $G$ be a finite group given as input by its multiplication table. For a subset $S$ of $G$ and an element $g\in G$ the Cayley Group Membership Problem (denoted CGM) is to check if $g$ belongs to the subgroup generated by $S$. While this problem is easily seen to be in polynomial time, pinpointing its parallel complexity has been of research interest over the years. In this paper we further explore the parallel complexity of the abelian CGM problem, with focus on the dynamic setting: the generating set $S$ changes with insertions and deletions and the goal is to maintain a data structure that supports efficient membership queries to the subgroup $\angle{S}$. We obtain the following results: 1. We first consider the more general problem of Monoid Membership. When $G$ is a commutative monoid we give a deterministic dynamic algorithm constant time parallel algorithm for membership testing that supports $O(1)$ insertions and deletions in each step. 2. Building on the previous result we show that there is a dynamic randomized constant-time parallel algorithm for abelian CGM that supports polylogarithmically many insertions/deletions to $S$ in each step. 3. If the number of insertions/deletions is at most $O(\log n/\log\log n)$ then we obtain a deterministic dynamic constant-time parallel algorithm for the problem. 4. We obtain analogous results for the dynamic abelian Group Isomorphism