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…
Counting and Enumerating Algorithmic Possibilities Combinatorics provides powerful mathematical techniques for systematically enumerating and counting the possible states and executions of algorithms. These counting arguments can yield tight asymptotic bounds on the resource consumption of algorithms in terms of time complexity and space complexity. A core technique is to map out the state space of…
Implications for derandomization and circuit lower bounds Understanding Average-Case Complexity Average-case complexity analyzes the performance of algorithms on randomly selected inputs, in contrast to worst-case complexity which focuses on the most difficult inputs. Defining the “average case” formally requires specifying a probability distribution over inputs. Intuitively, an algorithm has low average-case complexity if it performs…
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…
Promises and Derandomization A promise problem is a generalization of a language or decision problem where the input is promised to have certain properties. More formally, a promise problem P contains two disjoint sets Pyes and Pno, and an input x is promised to be in either Pyes or Pno. A correct algorithm for P…
Defining Randomness in SAT Problems Boolean satisfiability (SAT) problems are core challenges in computer science and mathematics. However, what makes some SAT instances considerably more difficult to solve than others remains unclear. Specifically, the role of randomness in inducing SAT hardness is not fully formalized. Here, we introduce rigorous quantitative definitions and measures to capture…
The SAT Problem and Its Variants The boolean satisfiability problem (SAT) aims to determine if there exists an interpretation that satisfies a given boolean formula. In formal terms, we are given a boolean formula φ with n variables x1, x2, …, xn that can take on true or false values. The goal is to find…
The Fundamental Limits of Computing Properties of Algorithms Rice’s theorem constitutes a fundamental limit on the ability to automatically verify or infer semantic properties of computer programs. At its core, the theorem states that no general algorithm can decide whether an arbitrary computer program possesses non-trivial semantic properties. As such, it establishes inherent undecidability in…
Derandomizing Valiant-Vazirani: The Core Problem The Valiant-Vazirani theorem shows that any Boolean formula can be made satisfiable by a random assignment of truth values to its variables with high probability. However, generating truly random bits requires specialized hardware and can be inefficient. Derandomization aims to reduce or eliminate the need for randomness while preserving the…
The Fundamental Challenge Hierarchy theorems are fundamental results in computational complexity theory that formalize the intuitive notion that more complex computational problems require more resources to solve. These theorems characterize broad complexity classes such as P and NP in terms of the number of steps required for a Turing machine or circuit family to solve…