DOCSLIB.ORG

Handout 4 VII. First-Order Logic

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

06-26264 Reasoning The University of Birmingham Spring Semester 2019 School of Computer Science Volker Sorge 5 February, 2019 Handout 4 Summary of this handout: First-Order Logic — Syntax — Terms — Predicates — Quantifiers — Seman- tics — Substitution — Renaming — Ground Terms — Prenex Normal Form — Skolemisation — Clause Normal Form VII. First-Order Logic While propositional logic can express some basic facts we want to reason about, it is not rich enough to express many aspects of intelligent discourse or reasoning. We can talk about facts only very broadly, as we have no means of restricting ourselves to talking about what or who these facts apply to. We can express “it is raining” with a proposition, but we can not restrict it to a particular place like “it is raining in Birmingham” versus “it is not raining in London”. Moreover, we can not talk about general or abstract facts. E.g., a classic example which we can not yet express is: Socrates is a man, all men are mortal, therefore Socrates is mortal. In the following we will build the theoretical foundations for first-order predicate logic (FOL) that gives us a much more powerful instrument to reason. VII.1 Syntax Since the main purpose of first-order logic is to allow us to make statements about distinct entities in a particular domain or universe of discourse, we first need some notation to represent this. 29. Constants represent concrete individuals in our universe of discourse. Example: 0; 1; 2; 3 or “Birming- ham, London” or “John, Paul, Mary” are constants. 30. Functions maps individuals in the universe to other individuals, thereby relating them to each other. Example: father(John) = Paul expresses the Paul is John’s father. Functions have different arities, e.g., the sin function has arity one, + has arity two and so on. 31. Variables are symbolic representations of an entity that is not (or not yet) determined. They are similar to the variables you know from programming languages or from mathematics, e.g., as in 2x2 +x+1 = 0. With these three components we now formally define a notion of terms: Definition 20 (Terms). Let V; C; F be disjoint sets of symbols, representing variables, constants and functions, respectively. Then we define terms inductively as follows: 1. Any variable in V is a term. 2. Any constant symbol in C is a term. 3. If t1; : : : ; tn are terms and f 2 F has arity n, then f(t1; : : : ; tn) is a term. 4. Nothing else is an term. Often constant, function and even variables have names that are in someway connected to their semantic meaning (e.g., we use digits for numbers or meaningful function names like sin). However, this is not strictly necessary, and in general, we will avoid confusing syntax and semantic, by using uniform naming conventions for Constants to be c; d; e or c0; c1;::: 24 Functions to be f; g; h or f0; f1;::: Variables to be x; y; z or x0; x1;::: 32. Predicates In propositional logic the basic components were propositional variables that expressed the statements that could either be true or false. In first-order logic this role is played by predicates that can be applied to terms and are similarly statements that are either true or false. Thus, similar to functions, predicates have also an arity and we can view propositional variables as predicates of arity 0. We can then more formally define atomic formulas. For notation we will us capital roman letters, similar to our propositional variables. Normally, we will use P; Q; R; : : : or P0;P1;:::. Definition 21 (Atomic Formulae). Let P be a set of symbols representing predicates. (i) Let P 2 P a predicate taking n arguments and t1; t2; : : : ; tn are terms then P (t1; t2; : : : ; tn) is an atomic formula or atom. (ii) Nothing else is an atomic formula. Example: We can now formalise that Socrates is mortal by M(s), where s denotes Socrates and M denotes that its argument is mortal. 33. Ground Terms and Atoms In the above definitions, both terms and hence atoms can contain variables. We say that a term is ground if it does not contain any variables; that is, it is generated from C; F, only. Similarly we say an atom (or later formula) is ground, if it contains only ground terms. 34. Free and bound variables So far in our terms and atoms variables would occur free that is they serve the role of a placeholder for some term by which they can be substituted. We call a variable bound if it either is either substituted by a specific value or restricted to a set of values. For example, the x in sin(x) is free, while in the following expression it is bound to a specific interval 0 ≤ x ≤ π : sin(x) ≥ 0 35. Quantifiers While one can think of different ways to bind variables to sets of values, we are primarily interested in two ways: binding a variable to all possible values or to at least on value. As an example consider the sentences For all x, x2 ≥ 0 and there exists x, x2 = 1. More formally we will use the quantifiers 8 (for all) and 9 (there exists): 8x x2 ≥ 0 9x x2 = 1 Observe that in the quantifier notation we use the as a shorthand for parentheses. That is, we could have also written 8x(x2 ≥ 0) and 9x(x2 = 1). But since in our notation quantifiers will only ever be applied from the left, this notation will save us a lot of writing (and counting) of parentheses. 36. Scope of a quantifier is how far the “influence” of a quantifier reaches in our formula. This is given by the dot and parentheses notation. This point is best illustrated with an example: scope of 8x z }| { scope of 8z z }| { 8x (9y P (x; y) ) _ 8y Q(y; x) _ 8z R(x; z) | {z } | {z } scope of 9y scope of 8y We can now finalise the definition of the full syntax of first-order logic. 25 Definition 22 (Well Formed Fomulae). We define the well formed formulae of first-order logic as follows: (i) > and ? are well-formed formulae. (ii) Every atomic formula is a well-formed formula. (iii) If ' and are well-formed formulas, then so are: :'; ' ^ ; ' _ ; ' ! ; ' $ (iv) If ' is a well-formed formula and x 2 V is a variable, then 9x ' and 8x ' are well-formed formulae. (v) Nothing else is a well-formed formula. We will sometimes denote the set of well formed formulae in first-order logic as Wff(FOL), in particular, if we do not want to specify the underlying term set in detail. Example: This now allows us finally to simply denote our statement on Socrates formally as: (H(s) ^ 8x H(x) ! M(x)) ! M(s) VII.2 Semantics In propositional logic we gave the propositional variables meaning by interpreting them into a set of truth values. We now do something very similar, however we have to start by interpreting terms first. This is done by mapping them into a universe, which is a non-empty set U. Apart from the non-emptiness requirement, we have no restrictions on U; that is, it can be finite of any size, or infinite countable (e.g., the natural numbers) or uncountable (e.g., the real numbers). We formally define an interpretation as follows: Definition 23 (Interpretation). Let U be a non-empty set called universe. An interpretation I over U is a function that (i) maps each c 2 C to an element of U, (ii) maps each f 2 F with n arguments to a concrete function f I : U n ! U, i.e., the set of n-tuples over U to U, (iii) maps echo P 2 P with n arguments to a concrete function P I : U n ! fT; Fg. Observe that we use the superscript notation to distinguish between the syntactic and semantic entities, and in particular that I(f) = f I and I(P ) = P I . The distinction between f and f I and between P and P I is important. The symbols f and P are just that: symbols. Whereas f I and P I denote a concrete function and a concrete relation on an interpretation I, respectively. Example: Let C = fc0; c1; c2g, F = fsg, P = fMg. Assume we have the following universe U = fSocrates; Zeus; Heraclesg. We can now construct the following interpretation: I(c0) = Socrates I(c1) = Zeus I(c2) = Heracles I(s) = Son I(M) = is mortal Similar to propositional variables in an interpretation I an atomic formula is interpreted either as true, T, or as false, F. Also I is called a model of an atomic formula P (t1; : : : ; tn), i.e., I j= P (t1; : : : ; tn), I if and only if P (I(t1);:::;I(tn)) = T for ground terms t1; : : : ; tn. Example: Let C = fcg, F = ffg, and P = fP g; and given the atomic formulae P (c; c) P (c; f(c)) P (f(c); c) One possible interpretation is for example into the universe of natural numbers, where U= IN0 and 26 1. I(c) = 0, 2. I(f) = s, where s is the successor function (for example, ++ in Java or incr in Ocaml), then f I (I(c)) = s(0) = 1. 3. I(P ) =≤, the usual less-or-equal, which we normally write in infix notation. The interpretation of the three formulas is then I(P (c; c)) = P I (I(c); I(c))= 0 ≤ 0 = T I(P (c; f(c))) = 0 ≤ 1 = T I(P (f(c); c)) = 1 ≤ 0 = F Observe that = is not part of the syntax of our language! 37.
Recommended publications
  • Skolemization Modulo Theories

    Skolemization Modulo Theories

    Skolemization Modulo Theories Konstantin Korovin1 and Margus Veanes2 1 The University of Manchester [email protected] 2 Microsoft Research [email protected] Abstract. Combining classical automated theorem proving techniques with theory based reasoning, such as satisfiability modulo theories, is a new approach to first-order reasoning modulo theories. Skolemization is a classical technique used to transform first-order formulas into eq- uisatisfiable form. We show how Skolemization can benefit from a new satisfiability modulo theories based simplification technique of formulas called monadic decomposition. The technique can be used to transform a theory dependent formula over multiple variables into an equivalent form as a Boolean combination of unary formulas, where a unary for- mula depends on a single variable. In this way, theory specific variable dependencies can be eliminated and consequently, Skolemization can be refined by minimizing variable scopes in the decomposed formula in order to yield simpler Skolem terms. 1 The role of Skolemization In classical automated theorem proving, Skolemization [8, 3] is a technique used to transform formulas into equisatisfiable form by replacing existentially quanti- fied variables by Skolem terms. In resolution based methods using clausal normal form (CNF) this is a necessary preprocessing step of the input formula. A CNF represents a universally quantified conjunction of clauses, where each clause is a disjunction of literals, a literal being an atom or a negated atom. The arguments of the atoms are terms, some of which may contain Skolem terms as subterms where a Skolem term has the form f(¯x) for some Skolem function symbol f and a sequencex ¯ of variables; f may also be a constant (¯x is empty).
  • (Aka “Geometric”) Iff It Is Axiomatised by “Coherent Implications”

    (Aka “Geometric”) Iff It Is Axiomatised by “Coherent Implications”

    GEOMETRISATION OF FIRST-ORDER LOGIC ROY DYCKHOFF AND SARA NEGRI Abstract. That every first-order theory has a coherent conservative extension is regarded by some as obvious, even trivial, and by others as not at all obvious, but instead remarkable and valuable; the result is in any case neither sufficiently well-known nor easily found in the literature. Various approaches to the result are presented and discussed in detail, including one inspired by a problem in the proof theory of intermediate logics that led us to the proof of the present paper. It can be seen as a modification of Skolem's argument from 1920 for his \Normal Form" theorem. \Geometric" being the infinitary version of \coherent", it is further shown that every infinitary first-order theory, suitably restricted, has a geometric conservative extension, hence the title. The results are applied to simplify methods used in reasoning in and about modal and intermediate logics. We include also a new algorithm to generate special coherent implications from an axiom, designed to preserve the structure of formulae with relatively little use of normal forms. x1. Introduction. A theory is \coherent" (aka \geometric") iff it is axiomatised by \coherent implications", i.e. formulae of a certain simple syntactic form (given in Defini- tion 2.4). That every first-order theory has a coherent conservative extension is regarded by some as obvious (and trivial) and by others (including ourselves) as non- obvious, remarkable and valuable; it is neither well-enough known nor easily found in the literature. We came upon the result (and our first proof) while clarifying an argument from our paper [17].
  • Axiomatic Set Teory P.D.Welch

    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

    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

    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.
  • Self-Similarity in the Foundations

    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

    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

    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 .
  • Abduction Algorithm for First-Order Logic Reasoning Frameworks

    Abduction Algorithm for First-Order Logic Reasoning Frameworks

    Abduction Algorithm for First-Order Logic Reasoning Frameworks Andre´ de Almeida Tavares Cruz de Carvalho Thesis to obtain the Master of Science Degree in Mathematics and Applications Advisor: Prof. Dr. Jaime Ramos Examination Committee Chairperson: Prof. Dr. Cristina Sernadas Advisor: Prof. Dr. Jaime Ramos Member of the Committee: Prof. Dr. Francisco Miguel Dion´ısio June 2020 Acknowledgments I want to thank my family and friends, who helped me to overcome many issues that emerged throughout my personal and academic life. Without them, this project would not be possible. I also want to thank professor Cristina Sernadas and professor Jaime Ramos for all the help they provided during this entire process, whether regarding theoretical and practical advise or other tips concerning my academic life. i Abstract Abduction is a process through which we can formulate hypotheses that justify a set of observed facts, using a background theory as a basis. Eventhough the process used to formulate these hypotheses may vary, the innerent advantages are universal. It approaches real life problems within fields like medicine and criminality, while maintaining their usefulness within more theoretical subjects, like fibring of logic proof systems. There are, however, some setbacks, such as the time complexity associated with these algorithms, and the expressive power of the formulas used. In this thesis, we tackle this last issue. We study the notions of consequence system and of proof system, as well as provide relevant ex- amples of these systems. We formalize some theoretical concepts regarding the Resolution principle, including refutation completeness and soundness. We then extend these concepts to First-Order Logic.
  • Easy Proofs of L\" Owenheim-Skolem Theorems by Means of Evaluation

    Easy Proofs of L\" Owenheim-Skolem Theorems by Means of Evaluation

    Easy Proofs of Löwenheim-Skolem Theorems by Means of Evaluation Games Jacques Duparc1,2 1 Department of Information, Systems Faculty of Business and Economics University of Lausanne, CH-1015 Lausanne Switzerland [email protected] 2 Mathematics Section, School of Basic Sciences Ecole polytechnique fédérale de Lausanne, CH-1015 Lausanne Switzerland [email protected] Abstract We propose a proof of the downward Löwenheim-Skolem that relies on strategies deriving from evaluation games instead of the Skolem normal forms. This proof is simpler, and easily under- stood by the students, although it requires, when defining the semantics of first-order logic to introduce first a few notions inherited from game theory such as the one of an evaluation game. 1998 ACM Subject Classification F.4.1 Mathematical Logic Keywords and phrases Model theory, Löwenheim-Skolem, Game Theory, Evaluation Game, First-Order Logic 1 Introduction Each mathematical logic course focuses on first-order logic. Once the basic definitions about syntax and semantics have been introduced and the notion of the cardinality of a model has been exposed, sooner or later at least a couple of hours are dedicated to the Löwenheim-Skolem theorem. This statement holds actually two different results: the downward Löwenheim-Skolem theorem (LS↓) and the upward Löwenheim-Skolem theorem (LS↑). I Theorem 1 (Downward Löwenheim-Skolem). Let L be a first-order language, T some L-theory, and κ = max{card (L) , ℵ0}. If T has a model of cardinality λ > κ, then T has a model of cardinality κ. I Theorem 2 (Upward Löwenheim-Skolem). Let L be some first-order language with equality, T some L-theory, and κ = max{card (L) , ℵ0}.
  • Models and Games

    Models and Games

    Incomplete version for students of easllc2012 only. Models and Games JOUKO VÄÄNÄNEN University of Helsinki University of Amsterdam Incomplete version for students of easllc2012 only. cambridge university press Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi, Tokyo, Mexico City Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521518123 © J. Väänänen 2011 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2011 Printed in the United Kingdom at the University Press, Cambridge A catalog record for this publication is available from the British Library ISBN 978-0-521-51812-3 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. Incomplete version for students of easllc2012 only. Contents Preface page xi 1 Introduction 1 2 Preliminaries and Notation 3 2.1 Finite Sequences 3 2.2 Equipollence 5 2.3 Countable sets 6 2.4 Ordinals 7 2.5 Cardinals 9 2.6 Axiom of Choice 10 2.7 Historical Remarks and References 11 Exercises 11 3 Games 14 3.1 Introduction 14 3.2 Two-Person Games of Perfect Information 14 3.3 The Mathematical Concept of Game 20 3.4 Game Positions 21 3.5 Infinite Games 24 3.6 Historical Remarks and References 28 Exercises 28 4 Graphs 35 4.1 Introduction 35 4.2 First-Order Language of Graphs 35 4.3 The Ehrenfeucht–Fra¨ısse´ Game on Graphs 38 4.4 Ehrenfeucht–Fra¨ısse´ Games and Elementary Equivalence 43 4.5 Historical Remarks and References 48 Exercises 49 Incomplete version for students of easllc2012 only.
  • 3. Temporal Logics and Model Checking

    3. Temporal Logics and Model Checking

    3. Temporal Logics and Model Checking Page Temporal Logics 3.2 Linear Temporal Logic (PLTL) 3.4 Branching Time Temporal Logic (BTTL) 3.8 Computation Tree Logic (CTL) 3.9 Linear vs. Branching Time TL 3.16 Structure of Model Checker 3.19 Notion of Fixpoint 3.20 Fixpoint Characterization of CTL 3.25 CTL Model Checking Algorithm 3.30 Symbolic Model Checking 3.34 Model Checking Tools 3.42 References 3.46 3.1 (of 47) Temporal Logics Temporal Logics • Temporal logic is a type of modal logic that was originally developed by philosophers to study different modes of “truth” • Temporal logic provides a formal system for qualitatively describing and reasoning about how the truth values of assertions change over time • It is appropriate for describing the time-varying behavior of systems (or programs) Classification of Temporal Logics • The underlying nature of time: Linear: at any time there is only one possible future moment, linear behavioral trace Branching: at any time, there are different possible futures, tree-like trace structure • Other considerations: Propositional vs. first-order Point vs. intervals Discrete vs. continuous time Past vs. future 3.2 (of 47) Linear Temporal Logic • Time lines Underlying structure of time is a totally ordered set (S,<), isomorphic to (N,<): Discrete, an initial moment without predecessors, infinite into the future. • Let AP be set of atomic propositions, a linear time structure M=(S, x, L) S: a set of states x: NS an infinite sequence of states, (x=s0,s1,...) L: S2AP labeling each state with the set of atomic propositions in AP true at the state.