Formal Languages
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Chapter 2 Introduction to Classical Propositional
CHAPTER 2 INTRODUCTION TO CLASSICAL PROPOSITIONAL LOGIC 1 Motivation and History The origins of the classical propositional logic, classical propositional calculus, as it was, and still often is called, go back to antiquity and are due to Stoic school of philosophy (3rd century B.C.), whose most eminent representative was Chryssipus. But the real development of this calculus began only in the mid-19th century and was initiated by the research done by the English math- ematician G. Boole, who is sometimes regarded as the founder of mathematical logic. The classical propositional calculus was ¯rst formulated as a formal ax- iomatic system by the eminent German logician G. Frege in 1879. The assumption underlying the formalization of classical propositional calculus are the following. Logical sentences We deal only with sentences that can always be evaluated as true or false. Such sentences are called logical sentences or proposi- tions. Hence the name propositional logic. A statement: 2 + 2 = 4 is a proposition as we assume that it is a well known and agreed upon truth. A statement: 2 + 2 = 5 is also a proposition (false. A statement:] I am pretty is modeled, if needed as a logical sentence (proposi- tion). We assume that it is false, or true. A statement: 2 + n = 5 is not a proposition; it might be true for some n, for example n=3, false for other n, for example n= 2, and moreover, we don't know what n is. Sentences of this kind are called propositional functions. We model propositional functions within propositional logic by treating propositional functions as propositions. -
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. -
Data Structure
EDUSAT LEARNING RESOURCE MATERIAL ON DATA STRUCTURE (For 3rd Semester CSE & IT) Contributors : 1. Er. Subhanga Kishore Das, Sr. Lect CSE 2. Mrs. Pranati Pattanaik, Lect CSE 3. Mrs. Swetalina Das, Lect CA 4. Mrs Manisha Rath, Lect CA 5. Er. Dillip Kumar Mishra, Lect 6. Ms. Supriti Mohapatra, Lect 7. Ms Soma Paikaray, Lect Copy Right DTE&T,Odisha Page 1 Data Structure (Syllabus) Semester & Branch: 3rd sem CSE/IT Teachers Assessment : 10 Marks Theory: 4 Periods per Week Class Test : 20 Marks Total Periods: 60 Periods per Semester End Semester Exam : 70 Marks Examination: 3 Hours TOTAL MARKS : 100 Marks Objective : The effectiveness of implementation of any application in computer mainly depends on the that how effectively its information can be stored in the computer. For this purpose various -structures are used. This paper will expose the students to various fundamentals structures arrays, stacks, queues, trees etc. It will also expose the students to some fundamental, I/0 manipulation techniques like sorting, searching etc 1.0 INTRODUCTION: 04 1.1 Explain Data, Information, data types 1.2 Define data structure & Explain different operations 1.3 Explain Abstract data types 1.4 Discuss Algorithm & its complexity 1.5 Explain Time, space tradeoff 2.0 STRING PROCESSING 03 2.1 Explain Basic Terminology, Storing Strings 2.2 State Character Data Type, 2.3 Discuss String Operations 3.0 ARRAYS 07 3.1 Give Introduction about array, 3.2 Discuss Linear arrays, representation of linear array In memory 3.3 Explain traversing linear arrays, inserting & deleting elements 3.4 Discuss multidimensional arrays, representation of two dimensional arrays in memory (row major order & column major order), and pointers 3.5 Explain sparse matrices. -
Thinking Recursively Part Four
Thinking Recursively Part Four Announcements ● Assignment 2 due right now. ● Assignment 3 out, due next Monday, April 30th at 10:00AM. ● Solve cool problems recursively! ● Sharpen your recursive skillset! A Little Word Puzzle “What nine-letter word can be reduced to a single-letter word one letter at a time by removing letters, leaving it a legal word at each step?” Shrinkable Words ● Let's call a word with this property a shrinkable word. ● Anything that isn't a word isn't a shrinkable word. ● Any single-letter word is shrinkable ● A, I, O ● Any multi-letter word is shrinkable if you can remove a letter to form a word, and that word itself is shrinkable. ● So how many shrinkable words are there? Recursive Backtracking ● The function we wrote last time is an example of recursive backtracking. ● At each step, we try one of many possible options. ● If any option succeeds, that's great! We're done. ● If none of the options succeed, then this particular problem can't be solved. Recursive Backtracking if (problem is sufficiently simple) { return whether or not the problem is solvable } else { for (each choice) { try out that choice. if it succeeds, return success. } return failure } Failure in Backtracking S T A R T L I N G Failure in Backtracking S T A R T L I N G S T A R T L I G Failure in Backtracking S T A R T L I N G S T A R T L I G Failure in Backtracking S T A R T L I N G Failure in Backtracking S T A R T L I N G S T A R T I N G Failure in Backtracking S T A R T L I N G S T A R T I N G S T R T I N G Failure in Backtracking S T A R T L I N G S T A R T I N G S T R T I N G Failure in Backtracking S T A R T L I N G S T A R T I N G Failure in Backtracking S T A R T L I N G S T A R T I N G S T A R I N G Failure in Backtracking ● Returning false in recursive backtracking does not mean that the entire problem is unsolvable! ● Instead, it just means that the current subproblem is unsolvable. -
Precise Analysis of String Expressions
Precise Analysis of String Expressions Aske Simon Christensen, Anders Møller?, and Michael I. Schwartzbach BRICS??, Department of Computer Science University of Aarhus, Denmark aske,amoeller,mis @brics.dk f g Abstract. We perform static analysis of Java programs to answer a simple question: which values may occur as results of string expressions? The answers are summarized for each expression by a regular language that is guaranteed to contain all possible values. We present several ap- plications of this analysis, including statically checking the syntax of dynamically generated expressions, such as SQL queries. Our analysis constructs flow graphs from class files and generates a context-free gram- mar with a nonterminal for each string expression. The language of this grammar is then widened into a regular language through a variant of an algorithm previously used for speech recognition. The collection of resulting regular languages is compactly represented as a special kind of multi-level automaton from which individual answers may be extracted. If a program error is detected, examples of invalid strings are automat- ically produced. We present extensive benchmarks demonstrating that the analysis is efficient and produces results of useful precision. 1 Introduction To detect errors and perform optimizations in Java programs, it is useful to know which values that may occur as results of string expressions. The exact answer is of course undecidable, so we must settle for a conservative approxima- tion. The answers we provide are summarized for each expression by a regular language that is guaranteed to contain all its possible values. Thus we use an upper approximation, which is what most client analyses will find useful. -
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. -
Regular Languages
id+id*id / (id+id)*id - ( id + +id ) +*** * (+)id + Formal languages Strings Languages Grammars 2 Marco Maggini Language Processing Technologies Symbols, Alphabet and strings 3 Marco Maggini Language Processing Technologies String operations 4 • Marco Maggini Language Processing Technologies Languages 5 • Marco Maggini Language Processing Technologies Operations on languages 6 Marco Maggini Language Processing Technologies Concatenation and closure 7 Marco Maggini Language Processing Technologies Closure and linguistic universe 8 Marco Maggini Language Processing Technologies Examples 9 • Marco Maggini Language Processing Technologies Grammars 10 Marco Maggini Language Processing Technologies Production rules L(G) = {anbncn | n ≥ 1} T = {a,b,c} N = {A,B,C} production rules 11 Marco Maggini Language Processing Technologies Derivation 12 Marco Maggini Language Processing Technologies The language of G • Given a grammar G the generated language is the set of strings, made up of terminal symbols, that can be derived in 0 or more steps from the start symbol LG = { x ∈ T* | S ⇒* x} Example: grammar for the arithmetic expressions T = {num,+,*,(,)} start symbol N = {E} S = E E ⇒ E * E ⇒ ( E ) * E ⇒ ( E + E ) * E ⇒ ( num + E ) * E ⇒ (num+num)* E ⇒ (num + num) * num language phrase 13 Marco Maggini Language Processing Technologies Classes of generative grammars 14 Marco Maggini Language Processing Technologies Regular languages • Regular languages can be described by different formal models, that are equivalent ▫ Finite State Automata (FSA) ▫ Regular -
Notes on Mathematical Logic David W. Kueker
Notes on Mathematical Logic David W. Kueker University of Maryland, College Park E-mail address: [email protected] URL: http://www-users.math.umd.edu/~dwk/ Contents Chapter 0. Introduction: What Is Logic? 1 Part 1. Elementary Logic 5 Chapter 1. Sentential Logic 7 0. Introduction 7 1. Sentences of Sentential Logic 8 2. Truth Assignments 11 3. Logical Consequence 13 4. Compactness 17 5. Formal Deductions 19 6. Exercises 20 20 Chapter 2. First-Order Logic 23 0. Introduction 23 1. Formulas of First Order Logic 24 2. Structures for First Order Logic 28 3. Logical Consequence and Validity 33 4. Formal Deductions 37 5. Theories and Their Models 42 6. Exercises 46 46 Chapter 3. The Completeness Theorem 49 0. Introduction 49 1. Henkin Sets and Their Models 49 2. Constructing Henkin Sets 52 3. Consequences of the Completeness Theorem 54 4. Completeness Categoricity, Quantifier Elimination 57 5. Exercises 58 58 Part 2. Model Theory 59 Chapter 4. Some Methods in Model Theory 61 0. Introduction 61 1. Realizing and Omitting Types 61 2. Elementary Extensions and Chains 66 3. The Back-and-Forth Method 69 i ii CONTENTS 4. Exercises 71 71 Chapter 5. Countable Models of Complete Theories 73 0. Introduction 73 1. Prime Models 73 2. Universal and Saturated Models 75 3. Theories with Just Finitely Many Countable Models 77 4. Exercises 79 79 Chapter 6. Further Topics in Model Theory 81 0. Introduction 81 1. Interpolation and Definability 81 2. Saturated Models 84 3. Skolem Functions and Indescernables 87 4. Some Applications 91 5. -
Formal Languages
Formal Languages Wednesday, September 29, 2010 Reading: Sipser pp 13-14, 44-45 Stoughton 2.1, 2.2, end of 2.3, beginning of 3.1 CS235 Languages and Automata Department of Computer Science Wellesley College Alphabets, Strings, and Languages An alphabet is a set of symbols. E.g.: 1 = {0,1}; 2 = {-,0,+} 3 = {a,b, …, y, z}; 4 = {, , a , aa } A string over is a seqqfyfuence of symbols from . The empty string is ofte written (Sipser) or ; Stoughton uses %. * denotes all strings over . E.g.: * • 1 contains , 0, 1, 00, 01, 10, 11, 000, … * • 2 contains , -, 0, +, --, -0, -+, 0-, 00, 0+, +-, +0, ++, ---, … * • 3 contains , a, b, …, aa, ab, …, bar, baz, foo, wellesley, … * • 4 contains , , , a , aa, …, a , …, a aa , aa a ,… A lnlanguage over (Stoug hto nns’s -lnlanguage ) is any sbstsubset of *. I.e., it’s a (possibly countably infinite) set of strings over . E.g.: •L1 over 1 is all sequences of 1s and all sequences of 10s. •L2 over 2 is all strings with equal numbers of -, 0, and +. •L3 over 3 is all lowercase words in the OED. •L4 over 4 is {, , a aa }. Formal Languages 11-2 1 Languages over Finite Alphabets are Countable A language = a set of strings over an alphabet = a subset of *. Suppose is finite. Then * (and any subset thereof) is countable. Why? Can enumerate all strings in lexicographic (dictionary) order! • 1 = {0,1} 1 * = {, 0, 1 00, 01, 10, 11 000, 001, 010, 011, 100, 101, 110, 111, …} • for 3 = {a,b, …, y, z}, can enumerate all elements of 3* in lexicographic order -- we’ll eventually get to any given element. -
String Solving with Word Equations and Transducers: Towards a Logic for Analysing Mutation XSS (Full Version)
String Solving with Word Equations and Transducers: Towards a Logic for Analysing Mutation XSS (Full Version) Anthony W. Lin Pablo Barceló Yale-NUS College, Singapore Center for Semantic Web Research [email protected] & Dept. of Comp. Science, Univ. of Chile, Chile [email protected] Abstract to name a few (e.g. see [3] for others). Certain applications, how- We study the fundamental issue of decidability of satisfiability ever, require more expressive constraint languages. The framework over string logics with concatenations and finite-state transducers of satisfiability modulo theories (SMT) builds on top of the basic as atomic operations. Although restricting to one type of opera- constraint satisfaction problem by interpreting atomic propositions tions yields decidability, little is known about the decidability of as a quantifier-free formula in a certain “background” logical the- their combined theory, which is especially relevant when analysing ory, e.g., linear arithmetic. Today fast SMT-solvers supporting a security vulnerabilities of dynamic web pages in a more realis- multitude of background theories are available including Boolector, tic browser model. On the one hand, word equations (string logic CVC4, Yices, and Z3, to name a few (e.g. see [4] for others). The with concatenations) cannot precisely capture sanitisation func- backbones of fast SMT-solvers are usually a highly efficient SAT- tions (e.g. htmlescape) and implicit browser transductions (e.g. in- solver (to handle disjunctions) and an effective method of dealing nerHTML mutations). On the other hand, transducers suffer from with satisfiability of formulas in the background theory. the reverse problem of being able to model sanitisation functions In the past seven years or so there have been a lot of works and browser transductions, but not string concatenations. -
Propositional Logic: Semantics and an Example
Recap: Syntax PDC: Semantics Using Logic to Model the World Proofs Propositional Logic: Semantics and an Example CPSC 322 { Logic 2 Textbook x5.2 Propositional Logic: Semantics and an Example CPSC 322 { Logic 2, Slide 1 Recap: Syntax PDC: Semantics Using Logic to Model the World Proofs Lecture Overview 1 Recap: Syntax 2 Propositional Definite Clause Logic: Semantics 3 Using Logic to Model the World 4 Proofs Propositional Logic: Semantics and an Example CPSC 322 { Logic 2, Slide 2 Definition (body) A body is an atom or is of the form b1 ^ b2 where b1 and b2 are bodies. Definition (definite clause) A definite clause is an atom or is a rule of the form h b where h is an atom and b is a body. (Read this as \h if b.") Definition (knowledge base) A knowledge base is a set of definite clauses Recap: Syntax PDC: Semantics Using Logic to Model the World Proofs Propositional Definite Clauses: Syntax Definition (atom) An atom is a symbol starting with a lower case letter Propositional Logic: Semantics and an Example CPSC 322 { Logic 2, Slide 3 Definition (definite clause) A definite clause is an atom or is a rule of the form h b where h is an atom and b is a body. (Read this as \h if b.") Definition (knowledge base) A knowledge base is a set of definite clauses Recap: Syntax PDC: Semantics Using Logic to Model the World Proofs Propositional Definite Clauses: Syntax Definition (atom) An atom is a symbol starting with a lower case letter Definition (body) A body is an atom or is of the form b1 ^ b2 where b1 and b2 are bodies. -
Symbol Sense: Informal Sense-Making in Formal Mathematics
Symbol Sense: Informal Sense-making in Formal Mathematics ABRAHAM ARCA VI Prologue Irineo Funes, the "memorious ", was created from Borges ~s fan tions and formulating mathematical arguments (proofs) tastic palette He had an extraordinarity retentive memory, he Instruction does not always provide opportunities not only was able to remember everything, be it the shapes of the southern not to memorize, but also to "forget" rules and details and clouds on the dawn of April 30th, 1882, or all the words in to be able to see through them in order to think, abstract, Engli'>h, French, Portuguese and Latin Each one of his experi generalize, and plan solution strategies Therefore, it ences was fully and accurately registered in his infinite memory would seem reasonable to attempt a description of a paral On one occasion he thought of "reducing" each past day of his lel notion to that of "number sense" in arithmetic: the idea life to s·eventy thousand remembrances to be referred to by num of "symbol sense".[!] bers. Two considerations dissuaded him: the task was inter minable and futile. Towards the end of the story, the real tragedy What is symbol sense? A first round of Funes is revealed He was incapable of general, "platonic " Compared to the attention which has been given to "num ideas For example, it was hard for him to understand that the ber sense", there is very little in the literature on "'symbol generic name dog included so many individuals of diverse sizes sense". One welcome exception is Fey [1990] He does not and shapes Moreover,