Computability and Complexity
The Boolean Satisfiability Problem The Boolean satisfiability problem (SAT) is the problem of determining whether there exists an interpretation that satisfies a given Boolean formula. More formally, SAT asks whether there exists an assignment of truth values to the variables of a Boolean formula that makes the formula evaluate to true. SAT is a canonical…
Quantum Merlin-Arthur (QMA) proof systems are a model of quantum computation whereby a computationally unbounded prover Merlin sends a quantum state to the verifier Arthur, who checks the state by performing an efficient quantum computation. The class QMA captures the computational power of quantum proofs. In contrast, Probabilistic Polynomial-time (PP) defines the class of decision…
The Promise of GCT In the early days of geometric complexity theory (GCT), researchers were filled with optimism about the potential for this new approach to resolve fundamental questions in computational complexity. GCT, first proposed by mathematician Ketan Mulmuley, attempts to characterize the inherent difficulty of computational problems using techniques from representation theory and algebraic…
Defining Obstacles in Innovative Endeavors As researchers and innovators seek to push boundaries and make new discoveries, they inevitably encounter impediments along the way. These obstacles arise from both tangible constraints as well as psychological barriers. Tangible constraints include technical limitations of tools and technologies, lack of resources, and gaps in knowledge. The mathematical intractability…
In complexity theory and related fields, we often aim to prove lower bounds on the resources required to solve computational problems. However, directly proving strong lower bounds has remained notoriously difficult for many critical problems. In this article, we will explore an unexpected and indirect strategy for making progress on this challenge – by instead…
The Power of Relativization in Complexity Theory Relativization is a powerful technique in computational complexity theory that allows comparing the relative computational power of complexity classes. It involves considering complexity classes in the context of oracle Turing machines. An oracle is an abstract device that can solve a decision problem in one step. By relativizing…
Definition of the Exponential Time Hypothesis (ETH) The Exponential Time Hypothesis (ETH) conjectures that 3-SAT, the canonical NP-complete problem, cannot be solved in subexponential time in the worst case. More formally, ETH states that there exists no algorithm that can solve 3-SAT in O(2^o(n)) time where n is the number of variables. This implies that…
Harnessing Paradoxes: The Surprising Power of Undecidable Languages This article explores the counterintuitive idea that weaknesses and limitations in computing systems can sometimes be harnessed as strengths. Specifically, we examine the strange properties of undecidable programming languages – languages that can express inherently unsolvable problems. Through examples and philosophical discussion, we uncover the surprising power…
Improving Time Hierarchy Bounds with Non-Constructivity The time hierarchy theorem demonstrates that more powerful computational models, with access to increased resources, can recognize more complex formal languages in less time. However, constructive proofs establishing concrete bounds face obstacles. By incorporating non-constructive arguments, recent work has developed tighter hierarchy bounds, although gaps remain. Formalizing the Time…
The P vs. NP Problem and Its Implications The P vs. NP problem is a central challenge in computational complexity theory. It asks whether all problems whose solutions can be verified in polynomial time can also be solved in polynomial time. The complexity classes P and NP represent problems that can be solved or verified,…