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COMPSCI 501: Formal Language Theory Insights on Computability Turing Machines Are a Model of Computation Two (No Longer) Surpris
Insights on Computability Turing machines are a model of computation COMPSCI 501: Formal Language Theory Lecture 11: Turing Machines Two (no longer) surprising facts: Marius Minea Although simple, can describe everything [email protected] a (real) computer can do. University of Massachusetts Amherst Although computers are powerful, not everything is computable! Plus: “play” / program with Turing machines! 13 February 2019 Why should we formally define computation? Must indeed an algorithm exist? Back to 1900: David Hilbert’s 23 open problems Increasingly a realization that sometimes this may not be the case. Tenth problem: “Occasionally it happens that we seek the solution under insufficient Given a Diophantine equation with any number of un- hypotheses or in an incorrect sense, and for this reason do not succeed. known quantities and with rational integral numerical The problem then arises: to show the impossibility of the solution under coefficients: To devise a process according to which the given hypotheses or in the sense contemplated.” it can be determined in a finite number of operations Hilbert, 1900 whether the equation is solvable in rational integers. This asks, in effect, for an algorithm. Hilbert’s Entscheidungsproblem (1928): Is there an algorithm that And “to devise” suggests there should be one. decides whether a statement in first-order logic is valid? Church and Turing A Turing machine, informally Church and Turing both showed in 1936 that a solution to the Entscheidungsproblem is impossible for the theory of arithmetic. control To make and prove such a statement, one needs to define computability. In a recent paper Alonzo Church has introduced an idea of “effective calculability”, read/write head which is equivalent to my “computability”, but is very differently defined. -
Use Formal and Informal Language in Persuasive Text
Author’S Craft Use Formal and Informal Language in Persuasive Text 1. Focus Objectives Explain Using Formal and Informal Language In this mini-lesson, students will: Say: When I write a persuasive letter, I want people to see things my way. I use • Learn to use both formal and different kinds of language to gain support. To connect with my readers, I use informal language in persuasive informal language. Informal language is conversational; it sounds a lot like the text. way we speak to one another. Then, to make sure that people believe me, I also use formal language. When I use formal language, I don’t use slang words, and • Practice using formal and informal I am more objective—I focus on facts over opinions. Today I’m going to show language in persuasive text. you ways to use both types of language in persuasive letters. • Discuss how they can apply this strategy to their independent writing. Model How Writers Use Formal and Informal Language Preparation Display the modeling text on chart paper or using the interactive whiteboard resources. Materials Needed • Chart paper and markers 1. As the parent of a seventh grader, let me assure you this town needs a new • Interactive whiteboard resources middle school more than it needs Old Oak. If selling the land gets us a new school, I’m all for it. Advanced Preparation 2. Last month, my daughter opened her locker and found a mouse. She was traumatized by this experience. She has not used her locker since. If you will not be using the interactive whiteboard resources, copy the Modeling Text modeling text and practice text onto chart paper prior to the mini-lesson. -
A Grammar-Based Approach to Class Diagram Validation Faizan Javed Marjan Mernik Barrett R
A Grammar-Based Approach to Class Diagram Validation Faizan Javed Marjan Mernik Barrett R. Bryant, Jeff Gray Department of Computer and Faculty of Electrical Engineering and Department of Computer and Information Sciences Computer Science Information Sciences University of Alabama at Birmingham University of Maribor University of Alabama at Birmingham 1300 University Boulevard Smetanova 17 1300 University Boulevard Birmingham, AL 35294-1170, USA 2000 Maribor, Slovenia Birmingham, AL 35294-1170, USA [email protected] [email protected] {bryant, gray}@cis.uab.edu ABSTRACT between classes to perform the use cases can be modeled by UML The UML has grown in popularity as the standard modeling dynamic diagrams, such as sequence diagrams or activity language for describing software applications. However, UML diagrams. lacks the formalism of a rigid semantics, which can lead to Static validation can be used to check whether a model conforms ambiguities in understanding the specifications. We propose a to a valid syntax. Techniques supporting static validation can also grammar-based approach to validating class diagrams and check whether a model includes some related snapshots (i.e., illustrate this technique using a simple case-study. Our technique system states consisting of objects possessing attribute values and involves converting UML representations into an equivalent links) desired by the end-user, but perhaps missing from the grammar form, and then using existing language transformation current model. We believe that the latter problem can occur as a and development tools to assist in the validation process. A string system becomes more intricate; in this situation, it can become comparison metric is also used which provides feedback, allowing hard for a developer to detect whether a state envisaged by the the user to modify the original class diagram according to the user is included in the model. -
Chapter 6 Formal Language Theory
Chapter 6 Formal Language Theory In this chapter, we introduce formal language theory, the computational theories of languages and grammars. The models are actually inspired by formal logic, enriched with insights from the theory of computation. We begin with the definition of a language and then proceed to a rough characterization of the basic Chomsky hierarchy. We then turn to a more de- tailed consideration of the types of languages in the hierarchy and automata theory. 6.1 Languages What is a language? Formally, a language L is defined as as set (possibly infinite) of strings over some finite alphabet. Definition 7 (Language) A language L is a possibly infinite set of strings over a finite alphabet Σ. We define Σ∗ as the set of all possible strings over some alphabet Σ. Thus L ⊆ Σ∗. The set of all possible languages over some alphabet Σ is the set of ∗ all possible subsets of Σ∗, i.e. 2Σ or ℘(Σ∗). This may seem rather simple, but is actually perfectly adequate for our purposes. 6.2 Grammars A grammar is a way to characterize a language L, a way to list out which strings of Σ∗ are in L and which are not. If L is finite, we could simply list 94 CHAPTER 6. FORMAL LANGUAGE THEORY 95 the strings, but languages by definition need not be finite. In fact, all of the languages we are interested in are infinite. This is, as we showed in chapter 2, also true of human language. Relating the material of this chapter to that of the preceding two, we can view a grammar as a logical system by which we can prove things. -
Axiomatic Set Teory P.D.Welch
Axiomatic Set Teory P.D.Welch. August 16, 2020 Contents Page 1 Axioms and Formal Systems 1 1.1 Introduction 1 1.2 Preliminaries: axioms and formal systems. 3 1.2.1 The formal language of ZF set theory; terms 4 1.2.2 The Zermelo-Fraenkel Axioms 7 1.3 Transfinite Recursion 9 1.4 Relativisation of terms and formulae 11 2 Initial segments of the Universe 17 2.1 Singular ordinals: cofinality 17 2.1.1 Cofinality 17 2.1.2 Normal Functions and closed and unbounded classes 19 2.1.3 Stationary Sets 22 2.2 Some further cardinal arithmetic 24 2.3 Transitive Models 25 2.4 The H sets 27 2.4.1 H - the hereditarily finite sets 28 2.4.2 H - the hereditarily countable sets 29 2.5 The Montague-Levy Reflection theorem 30 2.5.1 Absoluteness 30 2.5.2 Reflection Theorems 32 2.6 Inaccessible Cardinals 34 2.6.1 Inaccessible cardinals 35 2.6.2 A menagerie of other large cardinals 36 3 Formalising semantics within ZF 39 3.1 Definite terms and formulae 39 3.1.1 The non-finite axiomatisability of ZF 44 3.2 Formalising syntax 45 3.3 Formalising the satisfaction relation 46 3.4 Formalising definability: the function Def. 47 3.5 More on correctness and consistency 48 ii iii 3.5.1 Incompleteness and Consistency Arguments 50 4 The Constructible Hierarchy 53 4.1 The L -hierarchy 53 4.2 The Axiom of Choice in L 56 4.3 The Axiom of Constructibility 57 4.4 The Generalised Continuum Hypothesis in L. -
An Introduction to Formal Language Theory That Integrates Experimentation and Proof
An Introduction to Formal Language Theory that Integrates Experimentation and Proof Allen Stoughton Kansas State University Draft of Fall 2004 Copyright °c 2003{2004 Allen Stoughton Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sec- tions, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled \GNU Free Documentation License". The LATEX source of this book and associated lecture slides, and the distribution of the Forlan toolset are available on the WWW at http: //www.cis.ksu.edu/~allen/forlan/. Contents Preface v 1 Mathematical Background 1 1.1 Basic Set Theory . 1 1.2 Induction Principles for the Natural Numbers . 11 1.3 Trees and Inductive De¯nitions . 16 2 Formal Languages 21 2.1 Symbols, Strings, Alphabets and (Formal) Languages . 21 2.2 String Induction Principles . 26 2.3 Introduction to Forlan . 34 3 Regular Languages 44 3.1 Regular Expressions and Languages . 44 3.2 Equivalence and Simpli¯cation of Regular Expressions . 54 3.3 Finite Automata and Labeled Paths . 78 3.4 Isomorphism of Finite Automata . 86 3.5 Algorithms for Checking Acceptance and Finding Accepting Paths . 94 3.6 Simpli¯cation of Finite Automata . 99 3.7 Proving the Correctness of Finite Automata . 103 3.8 Empty-string Finite Automata . 114 3.9 Nondeterministic Finite Automata . 120 3.10 Deterministic Finite Automata . 129 3.11 Closure Properties of Regular Languages . -
Section 12.3 Context-Free Parsing We Know (Via a Theorem) That the Context-Free Languages Are Exactly Those Languages That Are Accepted by Pdas
Section 12.3 Context-Free Parsing We know (via a theorem) that the context-free languages are exactly those languages that are accepted by PDAs. When a context-free language can be recognized by a deterministic final-state PDA, it is called a deterministic context-free language. An LL(k) grammar has the property that a parser can be constructed to scan an input string from left to right and build a leftmost derivation by examining next k input symbols to determine the unique production for each derivation step. If a language has an LL(k) grammar, it is called an LL(k) language. LL(k) languages are deterministic context-free languages, but there are deterministic context-free languages that are not LL(k). (See text for an example on page 789.) Example. Consider the language {anb | n ∈ N}. (1) It has the LL(1) grammar S → aS | b. A parser can examine one input letter to decide whether to use S → aS or S → b for the next derivation step. (2) It has the LL(2) grammar S → aaS | ab | b. A parser can examine two input letters to determine whether to use S → aaS or S → ab for the next derivation step. Notice that the grammar is not LL(1). (3) Quiz. Find an LL(3) grammar that is not LL(2). Solution. S → aaaS | aab | ab | b. 1 Example/Quiz. Why is the following grammar S → AB n n + k for {a b | n, k ∈ N} an-LL(1) grammar? A → aAb | Λ B → bB | Λ. Answer: Any derivation starts with S ⇒ AB. -
Formal Grammar Specifications of User Interface Processes
FORMAL GRAMMAR SPECIFICATIONS OF USER INTERFACE PROCESSES by MICHAEL WAYNE BATES ~ Bachelor of Science in Arts and Sciences Oklahoma State University Stillwater, Oklahoma 1982 Submitted to the Faculty of the Graduate College of the Oklahoma State University iri partial fulfillment of the requirements for the Degree of MASTER OF SCIENCE July, 1984 I TheSIS \<-)~~I R 32c-lf CO'f· FORMAL GRAMMAR SPECIFICATIONS USER INTER,FACE PROCESSES Thesis Approved: 'Dean of the Gra uate College ii tta9zJ1 1' PREFACE The benefits and drawbacks of using a formal grammar model to specify a user interface has been the primary focus of this study. In particular, the regular grammar and context-free grammar models have been examined for their relative strengths and weaknesses. The earliest motivation for this study was provided by Dr. James R. VanDoren at TMS Inc. This thesis grew out of a discussion about the difficulties of designing an interface that TMS was working on. I would like to express my gratitude to my major ad visor, Dr. Mike Folk for his guidance and invaluable help during this study. I would also like to thank Dr. G. E. Hedrick and Dr. J. P. Chandler for serving on my graduate committee. A special thanks goes to my wife, Susan, for her pa tience and understanding throughout my graduate studies. iii TABLE OF CONTENTS Chapter Page I. INTRODUCTION . II. AN OVERVIEW OF FORMAL LANGUAGE THEORY 6 Introduction 6 Grammars . • . • • r • • 7 Recognizers . 1 1 Summary . • • . 1 6 III. USING FOR~AL GRAMMARS TO SPECIFY USER INTER- FACES . • . • • . 18 Introduction . 18 Definition of a User Interface 1 9 Benefits of a Formal Model 21 Drawbacks of a Formal Model . -
Topics in Context-Free Grammar CFG's
Topics in Context-Free Grammar CFG’s HUSSEIN S. AL-SHEAKH 1 Outline Context-Free Grammar Ambiguous Grammars LL(1) Grammars Eliminating Useless Variables Removing Epsilon Nullable Symbols 2 Context-Free Grammar (CFG) Context-free grammars are powerful enough to describe the syntax of most programming languages; in fact, the syntax of most programming languages is specified using context-free grammars. In linguistics and computer science, a context-free grammar (CFG) is a formal grammar in which every production rule is of the form V → w Where V is a “non-terminal symbol” and w is a “string” consisting of terminals and/or non-terminals. The term "context-free" expresses the fact that the non-terminal V can always be replaced by w, regardless of the context in which it occurs. 3 Definition: Context-Free Grammars Definition 3.1.1 (A. Sudkamp book – Language and Machine 2ed Ed.) A context-free grammar is a quadruple (V, Z, P, S) where: V is a finite set of variables. E (the alphabet) is a finite set of terminal symbols. P is a finite set of rules (Ax). Where x is string of variables and terminals S is a distinguished element of V called the start symbol. The sets V and E are assumed to be disjoint. 4 Definition: Context-Free Languages A language L is context-free IF AND ONLY IF there is a grammar G with L=L(G) . 5 Example A context-free grammar G : S aSb S A derivation: S aSb aaSbb aabb L(G) {anbn : n 0} (((( )))) 6 Derivation Order 1. -
Languages and Regular Expressions Lecture 2
Languages and Regular expressions Lecture 2 1 Strings, Sets of Strings, Sets of Sets of Strings… • We defined strings in the last lecture, and showed some properties. • What about sets of strings? CS 374 2 Σn, Σ*, and Σ+ • Σn is the set of all strings over Σ of length exactly n. Defined inductively as: – Σ0 = {ε} – Σn = ΣΣn-1 if n > 0 • Σ* is the set of all finite length strings: Σ* = ∪n≥0 Σn • Σ+ is the set of all nonempty finite length strings: Σ+ = ∪n≥1 Σn CS 374 3 Σn, Σ*, and Σ+ • |Σn| = ?|Σ |n • |Øn| = ? – Ø0 = {ε} – Øn = ØØn-1 = Ø if n > 0 • |Øn| = 1 if n = 0 |Øn| = 0 if n > 0 CS 374 4 Σn, Σ*, and Σ+ • |Σ*| = ? – Infinity. More precisely, ℵ0 – |Σ*| = |Σ+| = |N| = ℵ0 no longest • How long is the longest string in Σ*? string! • How many infinitely long strings in Σ*? none CS 374 5 Languages 6 Language • Definition: A formal language L is a set of strings 1 ε 0 over some finite alphabet Σ or, equivalently, an 2 0 0 arbitrary subset of Σ*. Convention: Italic Upper case 3 1 1 letters denote languages. 4 00 0 5 01 1 • Examples of languages : 6 10 1 – the empty set Ø 7 11 0 8 000 0 – the set {ε}, 9 001 1 10 010 1 – the set {0,1}* of all boolean finite length strings. 11 011 0 – the set of all strings in {0,1}* with an odd number 12 100 1 of 1’s. -
15–212: Principles of Programming Some Notes on Grammars and Parsing
15–212: Principles of Programming Some Notes on Grammars and Parsing Michael Erdmann∗ Spring 2011 1 Introduction These notes are intended as a “rough and ready” guide to grammars and parsing. The theoretical foundations required for a thorough treatment of the subject are developed in the Formal Languages, Automata, and Computability course. The construction of parsers for programming languages using more advanced techniques than are discussed here is considered in detail in the Compiler Construction course. Parsing is the determination of the structure of a sentence according to the rules of grammar. In elementary school we learn the parts of speech and learn to analyze sentences into constituent parts such as subject, predicate, direct object, and so forth. Of course it is difficult to say precisely what are the rules of grammar for English (or other human languages), but we nevertheless find this kind of grammatical analysis useful. In an effort to give substance to the informal idea of grammar and, more importantly, to give a plausible explanation of how people learn languages, Noam Chomsky introduced the notion of a formal grammar. Chomsky considered several different forms of grammars with different expressive power. Roughly speaking, a grammar consists of a series of rules for forming sentence fragments that, when used in combination, determine the set of well-formed (grammatical) sentences. We will be concerned here only with one, particularly useful form of grammar, called a context-free grammar. The idea of a context-free grammar is that the rules are specified to hold independently of the context in which they are applied. -
The Formal Language of Recursion Author(S): Yiannis N
The Formal Language of Recursion Author(s): Yiannis N. Moschovakis Source: The Journal of Symbolic Logic, Vol. 54, No. 4 (Dec., 1989), pp. 1216-1252 Published by: Association for Symbolic Logic Stable URL: http://www.jstor.org/stable/2274814 Accessed: 12/10/2008 23:57 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/action/showPublisher?publisherCode=asl. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit organization founded in 1995 to build trusted digital archives for scholarship. We work with the scholarly community to preserve their work and the materials they rely upon, and to build a common research platform that promotes the discovery and use of these resources. For more information about JSTOR, please contact [email protected]. Association for Symbolic Logic is collaborating with JSTOR to digitize, preserve and extend access to The Journal of Symbolic Logic. http://www.jstor.org THE JOURNAL OF SYMBOLIC LoGic Volume 54, Number 4, Dec.