A Brutal Look at Balanced Parentheses, Computing Machines, and Pushdown Automata
17 days ago
- #formal-languages
- #balanced-parentheses
- #automata-theory
- The article discusses the problem of determining whether a string of parentheses is balanced, meaning each opening parenthesis has a corresponding closing parenthesis and they are properly nested.
- It explores the concept of formal languages and recognizers, explaining that a formal language is a set of strings with an unambiguous way to determine membership.
- The article introduces deterministic finite automata (DFA) as the simplest machines to recognize regular languages, providing examples like binary numbers and simple name recognition.
- It proves that balanced parentheses cannot be recognized by a DFA, establishing that balanced parentheses is not a regular language.
- The article then introduces deterministic pushdown automata (DPA) as more powerful machines capable of recognizing deterministic context-free languages, including balanced parentheses.
- It demonstrates how to implement a DPA in JavaScript to recognize balanced parentheses and nested parentheses.
- The article further explores recursive regular expressions in languages like Ruby, showing they can recognize balanced parentheses and nested structures.
- It discusses the limitations of DPAs, proving they cannot recognize palindromes, which require more powerful non-deterministic pushdown automata (PDA).
- The article provides an implementation of a PDA in JavaScript to recognize binary palindromes, illustrating the difference between deterministic and non-deterministic context-free languages.
- It concludes with a summary of language families (regular, deterministic context-free, context-free) and their corresponding automata, along with practical advice for recognizing balanced parentheses in code.