First-Order Logic
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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. -
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. -
Propositional Logic
Propositional Logic - Review Predicate Logic - Review Propositional Logic: formalisation of reasoning involving propositions Predicate (First-order) Logic: formalisation of reasoning involving predicates. • Proposition: a statement that can be either true or false. • Propositional variable: variable intended to represent the most • Predicate (sometimes called parameterized proposition): primitive propositions relevant to our purposes a Boolean-valued function. • Given a set S of propositional variables, the set F of propositional formulas is defined recursively as: • Domain: the set of possible values for a predicate’s Basis: any propositional variable in S is in F arguments. Induction step: if p and q are in F, then so are ⌐p, (p /\ q), (p \/ q), (p → q) and (p ↔ q) 1 2 Predicate Logic – Review cont’ Predicate Logic - Review cont’ Given a first-order language L, the set F of predicate (first-order) •A first-order language consists of: formulas is constructed inductively as follows: - an infinite set of variables Basis: any atomic formula in L is in F - a set of predicate symbols Inductive step: if e and f are in F and x is a variable in L, - a set of constant symbols then so are the following: ⌐e, (e /\ f), (e \/ f), (e → f), (e ↔ f), ∀ x e, ∃ s e. •A term is a variable or a constant symbol • An occurrence of a variable x is free in a formula f if and only •An atomic formula is an expression of the form p(t1,…,tn), if it does not occur within a subformula e of f of the form ∀ x e where p is a n-ary predicate symbol and each ti is a term. -
First Order Logic and Nonstandard Analysis
First Order Logic and Nonstandard Analysis Julian Hartman September 4, 2010 Abstract This paper is intended as an exploration of nonstandard analysis, and the rigorous use of infinitesimals and infinite elements to explore properties of the real numbers. I first define and explore first order logic, and model theory. Then, I prove the compact- ness theorem, and use this to form a nonstandard structure of the real numbers. Using this nonstandard structure, it it easy to to various proofs without the use of limits that would otherwise require their use. Contents 1 Introduction 2 2 An Introduction to First Order Logic 2 2.1 Propositional Logic . 2 2.2 Logical Symbols . 2 2.3 Predicates, Constants and Functions . 2 2.4 Well-Formed Formulas . 3 3 Models 3 3.1 Structure . 3 3.2 Truth . 4 3.2.1 Satisfaction . 5 4 The Compactness Theorem 6 4.1 Soundness and Completeness . 6 5 Nonstandard Analysis 7 5.1 Making a Nonstandard Structure . 7 5.2 Applications of a Nonstandard Structure . 9 6 Sources 10 1 1 Introduction The founders of modern calculus had a less than perfect understanding of the nuts and bolts of what made it work. Both Newton and Leibniz used the notion of infinitesimal, without a rigorous understanding of what they were. Infinitely small real numbers that were still not zero was a hard thing for mathematicians to accept, and with the rigorous development of limits by the likes of Cauchy and Weierstrass, the discussion of infinitesimals subsided. Now, using first order logic for nonstandard analysis, it is possible to create a model of the real numbers that has the same properties as the traditional conception of the real numbers, but also has rigorously defined infinite and infinitesimal elements. -
Tarski's Threat to the T-Schema
Tarski’s Threat to the T-Schema MSc Thesis (Afstudeerscriptie) written by Cian Chartier (born November 10th, 1986 in Montr´eal, Canada) under the supervision of Prof. dr. Frank Veltman, and submitted to the Board of Examiners in partial fulfillment of the requirements for the degree of MSc in Logic at the Universiteit van Amsterdam. Date of the public defense: Members of the Thesis Committee: August 30, 2011 Prof. Dr. Frank Veltman Prof. Dr. Michiel van Lambalgen Prof. Dr. Benedikt L¨owe Dr. Robert van Rooij Abstract This thesis examines truth theories. First, four relevant programs in philos- ophy are considered. Second, four truth theories are compared according to a range of criteria. The truth theories are categorised according to Leitgeb’s eight criteria for a truth theory. The four truth theories are then compared with each-other based on three new criteria. In this way their relative use- fulness in pursuing some of the aims of the four programs is evaluated. This presents a springboard for future work on truth: proposing ideas for differ- ent truth theories that advance unambiguously different programs on the basis of the four truth theories proposed. 1 Acknowledgements I first thank my supervisor Prof. Dr. Frank Veltman for regularly read- ing and providing constructive feedback, recommending some papers along the way. This thesis has been much improved by his corrections and input, including extensive discussions and suggestions of the ideas proposed here. His patience in working with me on the thesis has been much appreciated. Among the other professors in the ILLC, Dr. Robert van Rooij also provided helpful discussion. -
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 . -
Self-Similarity in the Foundations
Self-similarity in the Foundations Paul K. Gorbow Thesis submitted for the degree of Ph.D. in Logic, defended on June 14, 2018. Supervisors: Ali Enayat (primary) Peter LeFanu Lumsdaine (secondary) Zachiri McKenzie (secondary) University of Gothenburg Department of Philosophy, Linguistics, and Theory of Science Box 200, 405 30 GOTEBORG,¨ Sweden arXiv:1806.11310v1 [math.LO] 29 Jun 2018 2 Contents 1 Introduction 5 1.1 Introductiontoageneralaudience . ..... 5 1.2 Introduction for logicians . .. 7 2 Tour of the theories considered 11 2.1 PowerKripke-Plateksettheory . .... 11 2.2 Stratifiedsettheory ................................ .. 13 2.3 Categorical semantics and algebraic set theory . ....... 17 3 Motivation 19 3.1 Motivation behind research on embeddings between models of set theory. 19 3.2 Motivation behind stratified algebraic set theory . ...... 20 4 Logic, set theory and non-standard models 23 4.1 Basiclogicandmodeltheory ............................ 23 4.2 Ordertheoryandcategorytheory. ...... 26 4.3 PowerKripke-Plateksettheory . .... 28 4.4 First-order logic and partial satisfaction relations internal to KPP ........ 32 4.5 Zermelo-Fraenkel set theory and G¨odel-Bernays class theory............ 36 4.6 Non-standardmodelsofsettheory . ..... 38 5 Embeddings between models of set theory 47 5.1 Iterated ultrapowers with special self-embeddings . ......... 47 5.2 Embeddingsbetweenmodelsofsettheory . ..... 57 5.3 Characterizations.................................. .. 66 6 Stratified set theory and categorical semantics 73 6.1 Stratifiedsettheoryandclasstheory . ...... 73 6.2 Categoricalsemantics ............................... .. 77 7 Stratified algebraic set theory 85 7.1 Stratifiedcategoriesofclasses . ..... 85 7.2 Interpretation of the Set-theories in the Cat-theories ................ 90 7.3 ThesubtoposofstronglyCantorianobjects . ....... 99 8 Where to go from here? 103 8.1 Category theoretic approach to embeddings between models of settheory . -
Monadic Decomposition
Monadic Decomposition Margus Veanes1, Nikolaj Bjørner1, Lev Nachmanson1, and Sergey Bereg2 1 Microsoft Research {margus,nbjorner,levnach}@microsoft.com 2 The University of Texas at Dallas [email protected] Abstract. Monadic predicates play a prominent role in many decid- able cases, including decision procedures for symbolic automata. We are here interested in discovering whether a formula can be rewritten into a Boolean combination of monadic predicates. Our setting is quantifier- free formulas over a decidable background theory, such as arithmetic and we here develop a semi-decision procedure for extracting a monadic decomposition of a formula when it exists. 1 Introduction Classical decidability results of fragments of logic [7] are based on careful sys- tematic study of restricted cases either by limiting allowed symbols of the lan- guage, limiting the syntax of the formulas, fixing the background theory, or by using combinations of such restrictions. Many decidable classes of problems, such as monadic first-order logic or the L¨owenheim class [29], the L¨ob-Gurevich class [28], monadic second-order logic with one successor (S1S) [8], and monadic second-order logic with two successors (S2S) [35] impose at some level restric- tions to monadic or unary predicates to achieve decidability. Here we propose and study an orthogonal problem of whether and how we can transform a formula that uses multiple free variables into a simpler equivalent formula, but where the formula is not a priori syntactically or semantically restricted to any fixed fragment of logic. Simpler in this context means that we have eliminated all theory specific dependencies between the variables and have transformed the formula into an equivalent Boolean combination of predicates that are “essentially” unary. -
Mathematical Logic
Copyright c 1998–2005 by Stephen G. Simpson Mathematical Logic Stephen G. Simpson December 15, 2005 Department of Mathematics The Pennsylvania State University University Park, State College PA 16802 http://www.math.psu.edu/simpson/ This is a set of lecture notes for introductory courses in mathematical logic offered at the Pennsylvania State University. Contents Contents 1 1 Propositional Calculus 3 1.1 Formulas ............................... 3 1.2 Assignments and Satisfiability . 6 1.3 LogicalEquivalence. 10 1.4 TheTableauMethod......................... 12 1.5 TheCompletenessTheorem . 18 1.6 TreesandK¨onig’sLemma . 20 1.7 TheCompactnessTheorem . 21 1.8 CombinatorialApplications . 22 2 Predicate Calculus 24 2.1 FormulasandSentences . 24 2.2 StructuresandSatisfiability . 26 2.3 TheTableauMethod......................... 31 2.4 LogicalEquivalence. 37 2.5 TheCompletenessTheorem . 40 2.6 TheCompactnessTheorem . 46 2.7 SatisfiabilityinaDomain . 47 3 Proof Systems for Predicate Calculus 50 3.1 IntroductiontoProofSystems. 50 3.2 TheCompanionTheorem . 51 3.3 Hilbert-StyleProofSystems . 56 3.4 Gentzen-StyleProofSystems . 61 3.5 TheInterpolationTheorem . 66 4 Extensions of Predicate Calculus 71 4.1 PredicateCalculuswithIdentity . 71 4.2 TheSpectrumProblem . .. .. .. .. .. .. .. .. .. 75 4.3 PredicateCalculusWithOperations . 78 4.4 Predicate Calculus with Identity and Operations . ... 82 4.5 Many-SortedPredicateCalculus . 84 1 5 Theories, Models, Definability 87 5.1 TheoriesandModels ......................... 87 5.2 MathematicalTheories. 89 5.3 DefinabilityoveraModel . 97 5.4 DefinitionalExtensionsofTheories . 100 5.5 FoundationalTheories . 103 5.6 AxiomaticSetTheory . 106 5.7 Interpretability . 111 5.8 Beth’sDefinabilityTheorem. 112 6 Arithmetization of Predicate Calculus 114 6.1 Primitive Recursive Arithmetic . 114 6.2 Interpretability of PRA in Z1 ....................114 6.3 G¨odelNumbers ............................ 114 6.4 UndefinabilityofTruth. 117 6.5 TheProvabilityPredicate . -
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.