The Problem of NFA Size Nondeterministic finite automata (NFAs) provide a compact way to specify patterns and regular languages. However, the size of an NFA can grow exponentially compared to the size of an equivalent regular expression. As more states and transitions are added to an NFA to capture complex patterns, the computational and memory…
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…
Understanding Algorithmic Bias and Fairness What is algorithmic bias and why does it matter? Algorithmic bias occurs when automatic decision-making systems produce unfair outcomes due to errors, assumptions, or discrimination in the way they were designed or the data they were trained on. This matters because algorithms now play a huge role in determining access…
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,…
The Inadequacy of N-gram Models N-gram models make the Markov assumption that the probability of a word depends only on the previous n-1 words. This limits expressivity as longer range semantic, syntactic, and discourse dependencies cannot be modeled. For example, pronoun resolution relies on entities introduced much earlier in text. Another limitation is the inability…
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…
Many critical optimization problems in domains like logistics, scheduling, and finance are computationally intractable. Known as NP-hard problems, they cannot be solved exactly in polynomial time. As problem sizes scale up, finding optimal solutions becomes infeasible. Approximation algorithms offer a practical way forward by efficiently finding near-optimal solutions. This article explains what approximation algorithms are,…
The Need for New Computational Paradigms The exponential growth in data generation and computational needs is forcing a migration to scalable cloud platforms. Traditional models of distributed algorithms face significant challenges when deployed across global networks of data centers. New programming models are emerging that abstract away physical infrastructure to focus on logical flows of…