The Complexity of Logarithmic Space Bounded Counting Classes

TL;DR

The monograph develops a comprehensive theory of logarithmic-space bounded counting classes, tying together NL, #L, GapL, ModL, ModkL, and PL. It builds foundational models (Turing and circuit), introduces key tools (Immerman-Szelepcsenyi counting, the Isolating Lemma, and modular counting) and establishes complete problems and hierarchies that illuminate the structure of space-bounded counting. A central achievement is showing determinant-based completeness for GapL via Mahajan–Vinay constructions, and situating these classes within circuit complexity (L-uniform TC^1, AC^0) to reveal deep connections between space-bounded counting and uniform computation. The work also surveys closure properties, reductions, and hierarchies, offering both foundational results and a suite of techniques for analyzing logspace counting problems with broad implications for computational complexity. Overall, it provides a unified, rigorous framework for understanding how counting interacts with logarithmic space, with concrete complete problems and powerful tools such as the Isolating Lemma and determinant-based reductions.

Abstract

In this monograph, we study complexity classes that are defined using $O(\log n)$-space bounded non-deterministic Turing machines. We prove salient results of Computational Complexity in this topic such as the Immerman-Szelepcs$\rm\acute{e}$nyi Theorem, the Isolating Lemma, theorems of Mahajan-Vinay on the determinant and many consequences of these very important results. The manuscript is intended to be a comprehensive textbook on the topic of The Complexity of Logarithmic Space Bounded Counting Classes.

Figures (10)

• Figure 1.1: This figure is the computation tree of a non-deterministic Turing machine $M$, where $0$ and $1$ on the edges denote the two non-deterministic choices using which $M$ branches at every stage on its computation path. Leaf nodes are marked as "$a$" or "$r$" denoting that the computation path ends in an accepting configuration or in a rejecting configuration respectively. Also all the computation paths are of the same length.
• Figure 2.1: This figure is a directed graph $G$ which is an instance of $\hbox{\rm SLDAG}$. Assuming that vertices $s=v_{11}$ and $t=v_{44}$ we also infer that this instance $(G,s,t)$ is in $\hbox{\rm SLDAGSTCON}$.
• Figure 2.2: This figure explains the reduction from a directed graph $G$ with vertices $s_1=v_1$ to the vertex $t_1=v_4$ to a "yes" instance $G"$ of $\hbox{\rm SLDAGSTCON}$ where $s=v_{11}$ and $t=v_{44}$.
• Figure 2.3: This figure is the directed graph $G$ obtained in the logspace many-one reduction from $\overline{\hbox{\rm 2SAT}}$ to DSTCON. We are given the Boolean formula $\phi =(x\vee y)\wedge (x\vee \neg y)\wedge (\neg x\vee z)\wedge (\neg x\vee \neg z)$ as input and we obtain the directed graph $G$ in this figure as the output of the reduction. Clearly, there are $4$ clauses in $\phi$, $8$ vertices and $8$ edges in $G$. It is easy to see that $\phi$ is not satisfiable and there is a directed path in fact, from every literal to its negation in $G$.
• Figure 2.4: This figure is a directed graph $G$ in the reduction from DSTCON to $\overline{\hbox{\rm 2SAT}}$. We therefore obtain the Boolean formula $\phi =(\neg x\vee v_2)\wedge (\neg v_2\vee v_2)\wedge (\neg v_2\vee v_3)\wedge (\neg v_2\vee \neg x)\wedge (\neg v_3\vee \neg x)\wedge (x\vee y)\wedge (x\vee \neg y)$. Since there exists a directed path from $s$ to $t$ in $G$ it is easy to see that $\phi$ is not satisfiable.

Theorems & Definitions (288)

• Definition 1.1
• Definition 1.2
• Definition 1.3
• Definition 1.4
• Definition 1.5
• Definition 1.6
• Definition 1.7
• Definition 1.8
• Theorem 1.9
• Definition 1.10