Computability and Complexity

Factoring Integers in Polynomial Time The problem of factoring large integers, or finding the prime divisors of a given composite number, has long captivated mathematicians and computer scientists. The difficulty of integer factorization is closely tied to the security of popular public-key cryptography schemes. The breakthrough discovery of efficient quantum algorithms for factorization could render…

Defining the Core Problem We formalize the computational problem of systematically generating hard or computationally intractable instances for NP-complete decision problems under the assumption that integer factorization is an NP-complete problem. NP-complete problems have two key properties: (i) they can be verified in polynomial time, and (ii) if any NP-complete problem has a polynomial-time solution,…

The Fundamental Limits of Computation Landauer’s principle establishes the lower theoretical energy limit for irreversible computational operations. All logical operations conducted on physical systems generate thermodynamic entropy, dissipating heat. Each bit operation must expend at least kTln2 joules, where k is the Boltzmann constant and T is temperature. This creates an absolute physical lower bound…

Definition of Factoring and Its Computational Complexity Factoring is the process of decomposing a composite integer into its prime factors. For example, factoring 15 would yield 3 x 5. Factoring the product of two large primes is considered one of the fundamental hard problems in computer science and mathematics, with major implications for cryptography and…

Computability theory examines the inherent capabilities and limitations of computation in the abstract. It asks foundational questions like: What functions can be computed at all? How efficiently can they be computed? What problems cannot be solved by any computer program? While computability theory focuses on computation in the Platonic realm of ideals, the practice of…

Definition of nondeterminism and nondeterministic Turing machines A nondeterministic Turing machine (NDTM) is a theoretical model of computation that extends the standard deterministic Turing machine (DTM) with an additional capability: at each step in its operation, an NDTM can transition into any one of multiple possible next states, rather than just a single determined next…

Theoretical computer science (TCS) is a broad and dynamic field encompassing foundational mathematics, logic, algorithms, complexity theory, programming languages, quantum computing, and more. As computer technology continues its relentless march, TCS provides the conceptual bedrock guiding innovations in computing. This article charts the growth trajectories, intellectual currents, and research frontiers shaping TCS. Quantifiable rises in…

The Problem of Logarithmic Slowdown When simulating a computational model M1 using another model M2, we often encounter logarithmic slowdowns in the simulation. For example, simulating a logarithmic depth circuit on a sequential machine incurs an unavoidable Ω(log n) factor overhead on the runtime. This logarithmic gap manifests due to mismatches between the parallelism afforded…

Formal Languages as Probes The mathematical formalisms known as formal languages provide a powerful set of tools for investigating the nature and boundaries of computational complexity classes. Complexity classes such as P, NP, and beyond categorize broad sets of computational problems by resource usage factors like time and memory. Formal language theory gives computer scientists…

What is Computational Complexity? Computational complexity theory is the study of the inherent difficulty of computational problems. At its core are complexity classes – formal definitions that categorize problems based on the computational resources required to solve them. The most well-known classes are: P – The set of problems that can be solved in polynomial…