Using Parentheses in Quantifiers: Enhancing SEO and Human Appeal. Analyzing Formulaic Matrices: Embracing or Avoiding Brackets?

Introduction:

Introduction:

In the book “Computational Intractability: A Guide to Algorithmic Lower Bounds” by Demaine-Gasarch-Hajiaghayi (see here for a link to a first draft), various choices were made regarding the notation used. This article explores the use of quantifiers in math and theoretical CS books, highlighting their rarity and the absence of parentheses and brackets in more recent works.

Full Article: Using Parentheses in Quantifiers: Enhancing SEO and Human Appeal. Analyzing Formulaic Matrices: Embracing or Avoiding Brackets?

The Choices and Standard Practices in Notation for Computational Intractability

Introduction

When writing the book “Computational Intractability: A Guide to Algorithmic Lower Bounds” by Demaine-Gasarch-Hajiaghayi, the authors were faced with several choices regarding notation. One of these choices pertained to the representation of quantifiers. In this article, we explore the different options and examine the standard practices in mathematical and theoretical computer science literature.

The Notational Dilemma

When defining the complexity class NP and in a few other instances, the authors had to decide between two notations:

Option 1: (\exists y)(\forall y)[B(x,y)]
Option 2: \exists x : \forall y : B(x,y)

Ultimately, the authors chose the second option. However, they were curious to know if either of these notations was considered standard in the field. To find the answer, they conducted an extensive search through numerous mathematical and theoretical computer science books.

The Findings

During their search, the authors made several interesting observations:

a) Most papers and books do not use quantifiers extensively, which came as a surprise.

b) When quantifiers are used, they are primarily found in definitions rather than theorems.

c) An exception can be seen in logic, where quantifiers are utilized when dealing with formulas as objects.

Examples from Literature

The authors compiled a list of instances where quantifiers were used, noting whether parentheses and brackets were present:

1) Cook’s classic paper (1971) – Parentheses, no brackets

2) Stockmeyer’s paper on defining PH (1976) – Parentheses and brackets

3) Computers and Intractability by Garey & Johnson (1979) – Parentheses and brackets

4) Morass-like construction of aleph_2 trees in L by Devlin (1979) – Parentheses and matrix in parentheses

5) Descriptive Complexity by Immerman (1999) – Parentheses, no brackets

6) Bounded Queries in Recursion Theory by Gasarch and Martin (1999) – Parentheses and brackets

7) Complexity Theory from Godel to Feynman by Rudich (2003) – No parentheses, no brackets in definition of PH

8) Parameterized Complexity Theory by Flum & Grohe (2007) – No parentheses and no brackets

9) Computational Complexity: A Conceptual Perspective by Goldreich (2008) – No parentheses and no brackets

10) On Quantifier Rank Equivalence between linear orders by Siders – Parentheses, no brackets (2018)

11) Presburger arithmetic, Rational Generating Functions, and quasi polynomials by Woods (2012) – Parentheses, no brackets

12) Who Witnesses the Witness by Abel et al. (2018) – No parentheses, no brackets, colons between quantifiers

Analysis and Conclusion

Based on the findings, two key observations can be made:

Firstly, the rarity of the use of quantifiers in literature was surprising. Only the book co-authored by Georgie Martin, Bounded Queries in Recursion Theory, extensively utilized quantifiers.

Secondly, it was evident that later works favored notations without parentheses and brackets. This trend can be witnessed in complexity theory books such as those by Garey & Johnson, Flum & Grohe, Arora & Barak, and Goldreich.

To further analyze this observation, the authors encourage readers to examine other complexity theory books and provide feedback on whether parentheses are still commonly used for quantifiers. This investigation aims to test the hypothesis that the use of parentheses may be outdated and less prevalent in contemporary literature.

Conclusion

In conclusion, while writing their book on computational intractability, the authors faced a notational dilemma when deciding how to represent quantifiers. Through an examination of various math and theoretical CS books, they discovered that quantifiers are not heavily used in literature. Moreover, recent works tend to avoid parentheses and brackets when representing quantifiers. This unique aspect of notation warrants further investigation and discussion within the field.

Summary: Using Parentheses in Quantifiers: Enhancing SEO and Human Appeal. Analyzing Formulaic Matrices: Embracing or Avoiding Brackets?

The article discusses the choices made regarding the notation used in the book “Computational Intractability: A Guide to Algorithmic Lower Bounds” by Demaine-Gasarch-Hajiaghayi. It compares different usages of quantifiers in math and theoretical computer science books. The author found that quantifiers are rarely used, and when they are, they are typically used in definitions rather than theorems. The use of parentheses and brackets also varies among different publications. The article concludes by inviting readers to share their observations on the use of quantifiers in complexity theory books.

Frequently Asked Questions

Quantifiers: To Parenthesize or not to Parenthesize?

Q: When should I use parentheses with quantifiers?

A: Parentheses should be used with quantifiers when you want to specify the order of operations within a logical expression. This helps to avoid ambiguity and clearly define the scope of the quantifiers.

Q: Can I omit parentheses with quantifiers?

A: Yes, you can omit parentheses with quantifiers if there is no ambiguity in the expression. However, it is recommended to use parentheses to enhance readability and reduce confusion, especially in complex logical formulas.

Matrix of Formula: To Bracket or not to Bracket?

Q: Should I use brackets in matrix formulas?

A: Whether to use brackets in matrix formulas depends on the specific context and requirements. Brackets are typically used to enclose matrices or indicate the grouping of elements. Consider the mathematical conventions and stylistic guidelines to make an informed decision.

Q: Can I omit brackets in matrix formulas?

A: Yes, brackets can be omitted in certain cases, especially when working with simple matrices or when the absence of brackets does not affect the clarity or correctness of the formula. However, it is generally advised to include brackets to clearly represent the structure and organization of the matrices.

General FAQs

Q: Can I mix quantifiers and matrix formulas in one expression?

A: Yes, you can mix quantifiers and matrix formulas in one expression. Ensure that the desired logical operations and matrix manipulations are properly represented and clearly understood by using appropriate notations and symbols.

Q: Are there any additional guidelines for using quantifiers and matrix formulas together?

A: While there are no strict rules, it is crucial to maintain consistency and clarity throughout the expression. Clearly define the scope and order of operations using parentheses and brackets, if necessary. Use proper mathematical notation and formatting to make the expression easily understandable.