Computability and Complexity

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,…

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…

Reductions are fundamental tools in computational complexity theory that establish relationships between computational problems. By transforming one problem into another, reductions allow us to transfer qualities like computability and complexity from one problem to another. As such, reductions give us insights into the structure and boundaries of complexity classes – sets of problems with related…

Defining the CLIQUE Problem The CLIQUE problem is a classic NP-complete computational problem in graph theory and computer science. Given an undirected graph G = (V,E) and a positive integer k, the CLIQUE problem asks whether G contains a complete subgraph (clique) of size at least k. More formally: CLIQUE Input: An undirected graph G…

The Promise of Quantum Computing Quantum computing utilizes quantum bits or qubits which can represent a superposition of both 0 and 1 simultaneously, enabling massively parallel processing. This quantum parallelism theoretically allows quantum computers to solve certain problems exponentially faster than classical computers. Potential applications include breaking current encryption schemes, complex optimizations, and simulating quantum…

The Core Problem of Resource Bounds Defining efficiency in the context of computation requires an understanding of resource bounds. Computational efficiency refers to the amount of computational resources – namely time and space – needed to execute an algorithm or program. An algorithm can be considered efficient if it achieves its objective while minimizing the…

The Power of Toda’s Theorem Toda’s theorem, proved by Seinosuke Toda in 1991, establishes that the entire polynomial hierarchy (PH) is contained in the class P#P. This theorem demonstrates the immense power of the counting complexity class #P in its relationship to the polynomial hierarchy. By showing that access to a #P oracle would allow…

Demystifying the Quantum-Classical Divide There have long been perceived separations between quantum and classical proofs in computational complexity theory. Quantum proofs and algorithms purportedly wield strange, almost magical powers exceeding their classical counterparts. However, recent research has begun demystifying the differences between quantum and classical techniques, clarifying misconceptions and highlighting surprising equivalences. A key concept…

Limiting Expressiveness with Least Herbrand Models Least Herbrand models play a pivotal role in restricting the expressive capacity of logic programs based on Horn clause logic. By grounding predicates and functions to a finite domain, least Herbrand models impose limits on what can be represented. This has implications for knowledge representation and reasoning systems built…

The P vs. NP Problem: A Fundamental Question in Complexity Theory The relationship between the complexity classes P and NP is one of the most central open questions in theoretical computer science. The classes P and NP relate to the fundamental resources of time and space needed by algorithms to solve computational problems. Informally, P…