Randomness refers to the lack of pattern or predictability in events or data. True randomness involves events that cannot be predicted even with complete knowledge of the system. Pseudorandomness refers to data that appears random but is actually generated by a deterministic algorithm. Understanding the difference between true and pseudorandomness is important for many areas…
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…
What is Rice’s Theorem and why it matters Rice’s Theorem establishes fundamental limits on what can be algorithmically computed. Specifically, it states that for any non-trivial property of partial functions, there exists no general algorithm that can decide whether an arbitrary program computing a partial function has that property. This powerful result delineates the boundary…
Defining the Counting Hierarchy The counting hierarchy refers to a collection of complexity classes that categorize counting problems based on their computational difficulty. At the lowest level is the class PP, which contains decision problems where the number of accepting inputs can be counted in polynomial time by a nondeterministic Turing machine. The next level…
The quest for more constructive proofs in mathematics while retaining high levels of generality has been aided by the introduction of powerful new tools like polynomial interpolation and pinning lemmas. These approaches aim to bridge the gap between the concrete guarantees provided by constructive proofs and the wide applicability offered by probabilistic methods. Seeking Pinning…
Reducing Complexity through Problem Transformations The concept of polynomial-time reductions allows translating instances of one computational problem to instances of another problem. This translation enables leveraging algorithms that efficiently solve the second problem in order to solve the first problem. Polynomial-time reductions have become a cornerstone technique for establishing relationships between hardness classes such as…
Deciphering Non-Constructivity in Graph Minor Theory What are Graph Minors? A graph minor of a graph G is a graph that can be obtained from G by contracting edges, deleting edges and deleting isolated vertices. More formally, a graph H is called a minor of a graph G if H can be formed from G…
Defining Uniform Algorithms A uniform algorithm is a computational process that operates on all possible inputs of a given size in a consistent way. Unlike a standard computer program that contains specific instructions for each input, a uniform algorithm follows the same procedure regardless of the particular input values. This property gives uniform algorithms wide…
Defining Nondeterministic Time Complexity Classes A nondeterministic Turing machine is a theoretical model of computation that, unlike a standard deterministic Turing machine, can follow multiple computational paths simultaneously. Formally, a nondeterministic Turing machine has a separate transition function that, for each combination of current state and symbol being read, specifies any number of possible next…
Formalizing the Basic Concepts A relativized world is a possible world that is accessible from another possible world based on a specified accessibility relation. The accessibility relation defines which possible worlds an agent in a given world can access or conceive of. To formalize reasoning about knowledge and belief using relativized worlds, we need to…