Computability and Complexity
The P versus NP Problem The P versus NP problem is a major unsolved problem in computer science and mathematics. It asks whether every problem whose solution can be quickly verified by a computer can also be quickly solved by a computer. P refers to the complexity class containing decision problems that can be solved…
Formalizing Computational Complexity To formally analyze the computational complexity of algorithms, computer scientists have developed mathematical abstractions and models. These formalisms allow for quantifying the amount of computational resources like time and memory used by an algorithm. Key concepts in studying computational complexity formally include: Complexity Classes – Sets of problems with related resource usage…
Defining Leaf Language Complexity Classes A leaf language is a set of strings over some alphabet that can be decided by a decider machine. Leaf language complexity classes group together leaf languages based on the resources required by their decider machines. Common resource bounds include time, space, randomness, and more. Some examples of leaf language…
Defining #P-completeness The complexity class #P consists of counting problems associated with decision problems in NP. Specifically, #P contains the counting versions of NP problems where the goal is to count the number of solutions or witnesses that satisfy the decision problem. For example, while the NP-complete problem SAT asks “Is there a satisfying assignment…
The Gap Between Computability and Complexity Computability theory and complexity theory are two fundamental pillars of theoretical computer science. Computability theory deals with what can and cannot be computed by algorithms, regardless of resource constraints. Complexity theory analyzes the time, space, and other resource requirements for algorithms to solve computational problems. There exists a conceptual…
The P vs. NP problem is one of the most profound open questions in computer science and mathematics. It asks whether every problem whose solution can be efficiently verified by a computer can also be efficiently solved by a computer. The Clay Mathematics Institute has called it one of the seven most important open questions…
This article explores the power of logical relations and sheaf models as techniques for proving properties about programs and systems. We show how these techniques enable establishing powerful equivalences between programs, modeling complex proofs, verifying system properties, and more. In particular, logical relations and sheaf models overcome limitations of natural proofs for areas like cryptography,…
The CLIQUE problem is a canonical NP-hard problem with several applications in bioinformatics, social network analysis, and other domains. In this article, we provide an in-depth overview of recent advancements in understanding the intermediate complexity of CLIQUE under different parameterizations. Formalizing the CLIQUE Problem The CLIQUE problem takes as input an undirected graph G =…
The Power of Interaction Interactive proofs are a model of computation where a verifier has a conversation with a prover to check the validity of a statement. This interaction between two parties, rather than just a one-way verification, adds considerable power. The class IP captures the complexity of problems that have interactive proofs. In this…
The P vs. NP Problem and Its Implications The P vs. NP problem asks whether the complexity classes P and NP are equal. P contains decision problems that can be solved in polynomial time by a deterministic Turing machine. NP contains problems with solutions that can be verified in polynomial time. If P=NP, then every…