Formal Languages

Faster Regular Language Intersection This section overviews existing algorithms for intersecting regular languages represented as deterministic finite automata (DFAs), analyzing their time complexities. A new sub-quadratic time algorithm for DFA intersection is presented along with a Python implementation and runtime analysis. The key entities explored are the DFA data structure, regular language intersection, and algorithmic…

Defining Nondeterminism in Computation Models Nondeterminism refers to computational models or algorithms that can exhibit different behaviors for the same input. A nondeterministic Turing machine can have multiple possible transition functions at each step, allowing it to explore multiple computational paths in parallel. This contrasts with deterministic models which have only one possible behavior for…

The SETH Conjecture and its Implications The Strong Exponential Time Hypothesis (SETH) is a far-reaching conjecture in complexity theory, postulating a lower bound on the satisfiability problem for CNF formulas. Specifically, SETH asserts that for every ε > 0, there exists an integer k ≥ 3 such that k-SAT cannot be solved in O(2^(n^ε)) time…

Defining the Problem The formal definition of deterministic finite automata (DFA) learnability relates to whether an unknown target DFA can be learned from examples within reasonable time and data constraints. This is connected to core questions in machine learning about model selection and generalization. Despite much research, the learnability of randomly generated DFAs remains an…

The problem of string separation The formal problem of string separation involves designing an automaton that can accept or reject input strings to divide them into two disjoint sets. More precisely, given two finite sets of strings X and Y over some alphabet, the objective is to construct an automaton that accepts all strings in…

Defining Deterministic Finite Automata (DFAs) A deterministic finite automaton (DFA) is a mathematical model used to recognize patterns and languages. Formally, a DFA is defined by a 5-tuple (Q, Σ, δ, q0, F) where: Q is a finite set of states Σ is a finite set of input symbols called the alphabet δ is the…

The Challenge of Active Learning Deterministic finite automata (DFAs) are fundamental computational models used across computer science. However, learning an unknown DFA in the passive setting faces complexity issues. As the number of states in the target DFA grows, passively labeled examples alone are insufficient for efficient learning. This motivates the need for active learning…

Regular expressions are a powerful tool for matching patterns in textual data. However, designing an optimal regular expression to match a specific pattern can be challenging. This article explores the problem of automatically inducing minimal regular expressions from examples, known as regular expression induction. The Challenge of Learning Regular Expressions Manual coding of regular expressions…

The Interplay of Logic and Computation Logic and computation are intrinsically linked disciplines. Boolean logic provides the mathematical foundation for digital circuit design, enabling complex computational machines. Predicate logic formalizes concepts and reasoning used by artificial intelligence systems. Modal logics specify computational behaviors and properties. Logic programming automates reasoning using logical inference. Logical frameworks unify…