The Core Question: Can Computers Enhance Human Mathematical Intuition? Mathematical intuition, the ability to grasp concepts non-consciously, is central to the practice of mathematics. Computers, with their precision, speed, and reliability, seem almost antithetical to fuzzy, subjective intuition. Yet mathematical software has advanced to the point where it can augment and enhance human intuitive abilities…
Locally decodable codes (LDCs) are a special type of error-correcting code that enable the extraction of individual message symbols by only querying a small subset of the encoded symbols. Unlike traditional decoding which requires reading the entire codeword, LDCs allow for localized decoding whereby each symbol can be recovered by looking at a few codeword…
Decision trees are a popular supervised learning method used for classification and regression tasks. They work by recursively partitioning the input space and fitting simple prediction models in each partition. Understanding the theoretical properties of how decision trees can reliably learn patterns from data, known as their learnability, has been an important area of research….
The Boolean satisfiability problem (SAT) asks whether there exists an assignment to variables that satisfies a given Boolean formula. SAT is a canonical NP-complete problem with widespread applications in areas like hardware verification, software testing, and artificial intelligence. Conjunctive normal form SAT (CNF-SAT) focuses on formulas expressed as ANDs (conjunctions) of ORs (disjunctions), a practical…
Formalizing the Problem This section establishes quantitative complexity measures for Boolean functions and formally defines depth-2 circuits with bounded fan-in gates. It discusses the challenges in proving lower bounds against circuits with small depth due to limited computational power. Establishing complexity measures We introduce notation and terminology to quantify the complexity of Boolean functions in…
Overcoming Barriers to BPP Lower Bounds The complexity class BPP, consisting of problems solvable in probabilistic polynomial time with bounded error, has proven notoriously difficult to establish lower bounds for. While NP-hardness proofs establish computational intractability assuming P ≠ NP, proving problems are hard for BPP requires defeating the power of randomness and is thus…
The P vs NP Problem and Its Implications The P vs NP problem refers to the open question of whether or not the complexity classes P and NP are equal. The class P contains decision problems that can be solved in polynomial time by a deterministic Turing machine. The class NP contains problems where solutions…
Defining the Complexity Classes BPP and P Probability distributions play a key role in defining randomized computational complexity classes. The class BPP consists of decision problems that can be solved by a probabilistic Turing machine in polynomial time, with an error probability of at most 1/3 for all instances. In contrast, P is the class…
Is BPP Closed Under Complement? The complexity class BPP (Bounded-error Probabilistic Polynomial time) contains decision problems solvable by a probabilistic Turing machine in polynomial time with an error probability bounded away from 1/2 for all instances. A key question is whether BPP is closed under complement – that is, if a problem is in BPP,…
The Complexity Class Conundrum Defining the key complexity classes BPP, RP, and co-RP is critical for understanding the relationships between them. BPP, or Bounded-error Probabilistic Polynomial time, contains decision problems solvable in polynomial time by a probabilistic Turing machine with an error probability bounded by 1/3 for all instances. RP, or Randomized Polynomial time, denotes…