Formal Languages
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…
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…
Formal languages as mathematical models Formal languages provide a precise mathematical framework for analyzing computational models and their capabilities. A formal language consists of strings of symbols drawn from a defined alphabet and governed by specific grammatical rules. The language contains all possible symbol strings generable under those rules. Studying formal languages enables systematic classification…
The Need for Software Reliability Software pervades every aspect of modern technological and information systems, yet software bugs and vulnerabilities remain prevalent, posing rising economic and safety risks. High-profile software failures like the Mars Climate Orbiter loss, Therac-25 radiation overdoses, and Boeing 737 MAX crashes demonstrate the potential for catastrophic consequences when software systems fail….
Topological data analysis (TDA) leverages concepts and tools from algebraic topology to analyze complex datasets. The key insight is that topology captures meaningful properties of shapes and spaces that are invariant under continuous deformation, allowing the mathematical formalism to be applied for revealing intrinsic structure within data. Leveraging Algebraic Topology for Analyzing Complex Datasets Algebraic…
Formalizing Computational Complexity Defining complexity classes allows us to categorize computational problems based on the resources needed to solve them. The most fundamental distinctions are between problems soluble in polynomial time (class P), polynomial space (class PSPACE), logarithmic space (class L), and exponential time (class EXP). Further refinement leads to central classes like NP and…
Formal Verification of Software Systems Ensuring the reliability and security of software systems is a pivotal concern in software engineering. Formal verification techniques provide mathematically rigorous methods to prove the correctness of software implementations with respect to formal logical specifications. Interactive proof assistants are computer programs that enable the construction of machine-checkable proofs of theorems…
What is Homotopy Type Theory? Homotopy type theory (HoTT) is a new branch of mathematics that combines ideas from homotopy theory, higher category theory, and type theory. It aims to provide a new foundation for mathematics based on a homotopical interpretation of types. Some key aspects of HoTT include: Origin in homotopy theory and higher…
Formal Verification: The Promise of Mathematical Certainty Formal verification refers to the use of mathematical reasoning to ensure that a system satisfies desired properties. Unlike testing, which samples expected behavior, formal verification aims to provide an exhaustive proof that a system works as intended under all circumstances. By constructing a mathematical model and proving theorems…
Intersection of Regular Languages and SAT Solving The intersection operation on regular languages has important connections to the Boolean satisfiability problem. We explore these connections by showing how intersection can be encoded as a conjunctive normal form satisfiability (CNF-SAT) formula. We also examine fixed-parameter tractable reductions between various types of automata that recognize regular languages….