Defining ETH, NEXP, and EXP The Exponential Time Hypothesis (ETH) is a conjecture in computational complexity theory that states that 3-SAT, the satisfiability problem for Boolean formulas in 3-conjunctive normal form where each clause has at most 3 literals, cannot be solved in subexponential time by a deterministic Turing machine. More formally, ETH claims that…
The Role of Randomness in Computational Complexity Randomness plays an intriguing role within the field of computational complexity theory. At the heart of this field lies the P vs NP problem – one of the central open questions in computer science and mathematics. This problem centers around the relationship between two complexity classes: P –…
Defining Counting Problems Counting problems involve determining the number of solutions for a particular computational problem. For example, considering the Boolean satisfiability problem (SAT), instead of asking whether a satisfiable assignment exists, a counting version asks “how many satisfying assignments are there?” This is the #SAT problem. More formally, counting problems take the decision version…
Formalizing Concurrency and Distribution The development of mathematical models to analyze concurrent and distributed systems relies heavily on formal methods from computer science and discrete mathematics. Process calculi provide powerful algebraic languages to specify interacting processes and reason about their behaviors. Popular process calculi like the pi-calculus, CCS, and CSP capture the semantics of concurrency,…
As more systems and devices connect to the internet, the risk of malicious cyber attacks and data breaches grows exponentially. Recent years have seen several high-profile data breaches impacting billions of users’ personal information. In 2022 alone, over 4 billion records were exposed in data breaches according to cybersecurity company Varonis. The increasing prevalence of…
Defining Positive Topological Ordering A positive topological ordering or topology sort is a linear ordering of the vertices in a directed acyclic graph (DAG) such that for every edge (u, v), u comes before v in the ordering. Intuitively, it is an ordering that does not violate the dependencies between vertices. Formally, given a DAG…
The Core Issue: Lack of Understanding Complexity Classes Below Polynomial Space A glaring gap in our comprehension of computational complexity theory manifests in the shortage of natural problems known to reside in sub-polynomial space classes beneath polynomial space (PSPACE). While space classes exceeding polynomial space host multifarious problems of practical relevance, few natural candidates seem…
Constructing Small World Graphs with SAT Formulas Generating random 3-SAT formulas with planted solutions is a common technique to construct small world graphs. By tuning the clause density and variable connectivity, small world properties emerge in the variable constraint networks. Analyzing the graph structure reveals high local clustering along with short average path lengths. Specifically,…
Harnessing the Power of Quantum Superposition Quantum superposition is a fundamental principle of quantum mechanics in which a quantum system can exist in multiple states simultaneously. This contrasts with classical systems that can only exist in a single, definite state. Quantum superposition enables exponentially greater information density and parallelism compared to classical systems. A key…
Defining the P and NP Complexity Classes The complexity classes P and NP are formal definitions used in computational complexity theory to categorize mathematical problems according to the computational resources required to solve them. The class P consists of all decision problems that can be solved in polynomial time by a deterministic Turing machine. This…