DOCSLIB.ORG

2. First-Order Logic (FOL)

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
2. First-Order Logic (FOL) The Calculus of Computation: Decision Procedures with Applications to Verification 2. First-Order Logic (FOL) by Aaron Bradley Zohar Manna Springer 2007 2- 1 2- 2 First-Order Logic (FOL) quantifiers existential quantifier ∃x.F [x] Also called Predicate Logic or Predicate Calculus “there exists an x such that F [x]” FOL Syntax universal quantifier ∀x.F [x] “for all x, F [x]” variables x, y, z, · · · a b c constants , , , · · · FOL formula literal, application of logical connectives f g h functions , , , · · · (¬, ∨ , ∧ , → , ↔ ) to formulae, terms variables, constants or or application of a quantifier to a formula n-ary function applied to n terms as arguments a, x, f (a), g(x, b), f (g(x, g(b))) predicates p, q, r, · · · atom ⊤, ⊥, or an n-ary predicate applied to n terms literal atom or its negation p(f (x), g(x, f (x))), ¬p(f (x), g(x, f (x))) Note: 0-ary functions: constant 0-ary predicates: P, Q, R,... 2- 3 2- 4 Example: FOL formula Translations of English Sentences into FOL ◮ ∀x. p(f (x), x) → (∃y. p(f (g(x, y)), g(x, y))) ∧ q(x, f (x)) The length of one side of a triangle is less than the sum of the | {z } lengths of the other two sides G | {z } F ∀x, y, z. triangle(x, y, z) → length(x) < length(y)+length(z) The scope of ∀x is F . ◮ Fermat’s Last Theorem. The scope of ∃y is G. The formula reads: ∀n. integer(n) ∧ n > 2 “for all x, → ∀x, y, z. if p(f (x), x) integer(x) ∧ integer(y) ∧ integer(z) then there exists a y such that ∧ x > 0 ∧ y > 0 ∧ z > 0 n n n p(f (g(x, y)), g(x, y)) and q(x, f (x))” → x + y 6= z 2- 5 2- 6 FOL Semantics Example: F : p(f (x, y), z) → p(y, g(z, x)) An interpretation I : (DI , αI ) consists of: Interpretation I : (DI , αI ) ◮ Domain DI DI = Z = {· · · , −2, −1, 0, 1, 2, ···} integers non-empty set of values or objects αI : {f 7→ +, g 7→ −, p 7→>} cardinality |DI | finite (eg, 52 cards), Therefore, we can write countably infinite (eg, integers), or uncountably infinite (eg, reals) FI : x + y > z → y > z − x ◮ Assignment αI (This is the way we’ll write it in the future!) ◮ each variable x assigned value xI ∈ DI Also ◮ each n-ary function f assigned αI : {x 7→ 13, y 7→ 42, z 7→ 1} n fI : DI → DI Thus In particular, each constant a (0-ary function) assigned value FI : 13 + 42 > 1 → 42 > 1 − 13 aI ∈ DI Compute the truth value of F under I ◮ each n-ary predicate p assigned 1. I |= x + y > z since 13 + 42 > 1 n pI : DI →{true, false} 2. I |= y > z − x since 42 > 1 − 13 In particular, each propositional variable P (0-ary predicate) 3. I |= F by 1, 2, and → assigned truth value (true, false) F is true under I 2- 7 2- 8 Semantics: Quantifiers Example For Q, the set of rational numbers, consider x variable. x-variant of interpretation I is an interpretation J : (DJ , αJ ) such FI : ∀x. ∃y. 2 × y = x that ◮ DI = DJ Compute the value of FI (F under I ): ◮ αI [y]= αJ [y] for all symbols y, except possibly x Let v J1 : I ⊳ {x 7→ v} J2 : J1 ⊳ {y 7→ } That is, I and J agree on everything except possibly the value of x 2 x-variant of I y-variant of J1 Denote J : I ⊳ {x 7→ v} the x-variant of I in which αJ [x] = v for for v ∈ Q. some v ∈ DI . Then Then ◮ I |= ∀x. F iff for all v ∈ DI , I ⊳ {x 7→ v} |= F v ◮ I |= ∃x. F iff there exists v ∈ DI s.t. I ⊳ {x 7→ v} |= F 1. J2 |= 2 × y = x since 2 × 2 = v 2. J1 |= ∃y. 2 × y = x 3. I |= ∀x. ∃y. 2 × y = x since v ∈ Q is arbitrary 2- 9 2- 10 Satisfiability and Validity Second case F is satisfiable iff there exists I s.t. I |= F 1. I 6|= ∀x. p(x) assumption F is valid iff for all I , I |= F 2. I |= ¬∃x. ¬p(x) assumption 3. I ⊳ {x 7→ v} 6|= p(x) 1and ∀, for some v ∈ DI F is valid iff ¬F is unsatisfiable 4. I 6|= ∃x. ¬p(x) 2and ¬ 5. I ⊳ {x 7→ v} 6|= ¬p(x) 4and ∃ 6. I ⊳ {x 7→ v} |= p(x) 5and ¬ Example: F : (∀x. p(x)) ↔ (¬∃x. ¬p(x)) valid? Suppose not. Then there is I s.t. 3 and 6 are contradictory. 0. I 6|= (∀x. p(x)) ↔ (¬∃x. ¬p(x)) Both cases end in contradictions for arbitrary I ⇒ F is valid. First case 1. I |= ∀x. p(x) assumption 2. I 6|= ¬∃x. ¬p(x) assumption 3. I |= ∃x. ¬p(x) 2and ¬ 4. I ⊳ {x 7→ v} |= ¬p(x) 3and ∃, for some v ∈ DI 5. I ⊳ {x 7→ v} |= p(x) 1and ∀ 4 and 5 are contradictory. 2- 11 2- 12 Example: Prove Example: Show F : p(a) → ∃x. p(x) isvalid. F : (∀x. p(x, x)) → (∃x. ∀y. p(x, y)) is invalid. Assume otherwise. Find interpretation I such that 1. I 6|= F assumption I |= ¬[(∀x. p(x, x)) → (∃x. ∀y. p(x, y))] 2. I |= p(a) 1and → 3. I 6|= ∃x. p(x) 1and → i.e. 4. I ⊳ {x 7→ αI [a]} 6|= p(x) 3and ∃ I |= (∀x. p(x, x)) ∧ ¬(∃x. ∀y. p(x, y)) 2 and 4 are contradictory. Thus, F is valid. Choose DI = {0, 1} pI = {(0, 0), (1, 1)} i.e. pI (0, 0) and pI (1, 1) are true pI (1, 0) and pI (1, 0) are false I falsifying interpretation ⇒ F is invalid. 2- 13 2- 14 Safe Substitution F σ Rename x by x′: ′ ′ Example: replace x in ∀x by x and all free x in the scope of ∀x by x . ′ ′ scope of ∀x ∀x. G[x] ⇔ ∀x . G[x ] z }| { F : (∀x. p(x, y) ) → q(f (y), x) Same for ∃x ր տ ր տ bound by ∀x free free free ∃x. G[x] ⇔ ∃x′. G[x′] free(F )= {x, y} where x′ is a fresh variable substitution Proposition (Substitution of Equivalent Formulae) σ : {x 7→ g(x), y 7→ f (x), q(f (y), x) 7→ ∃x. h(x, y)} σ : {F1 7→ G1, · · · , Fn 7→ Gn} F σ? 1. Rename s.t. for each i, Fi ⇔ Gi F ′ : ∀x′. p(x′, y) → q(f (y), x) ↑ ↑ If F σ a safe substitution, then F ⇔ F σ where x′ is a fresh variable 2. F ′σ : ∀x′. p(x′, f (x)) → ∃x. h(x, y) 2- 15 2- 16 Formula Schema Substitution σ of H Formula σ : {F1 7→ ,..., Fn 7→ } (∀x. p(x)) ↔ (¬∃x. ¬p(x)) mapping place holders Fi of H to FOL formulae, Formula Schema (obeying the side conditions of H) H1 : (∀x. F ) ↔ (¬∃x. ¬F ) Proposition (Formula Schema) ↑ place holder If H is valid formula schema and Formula Schema (with side condition) σ is a substitution obeying H’s side conditions H2 : (∀x. F ) ↔ F provided x ∈/ free(F ) then Hσ is also valid. Valid Formula Schema Example: H is valid iff valid for any FOL formula Fi obeying the side H : (∀x. F ) ↔ F provided x ∈/ free(F ) isvalid conditions σ : {F 7→ p(y)} obeys the side condition Example: H1 and H2 are valid. Therefore Hσ : ∀x. p(y) ↔ p(y) isvalid 2- 17 2- 18 Proving Validity of Formula Schema Normal Forms Example: Prove validity of 1. Negation Normal Forms (NNF) H : (∀x. F ) ↔ F provided x ∈/ free(F ) Augment the equivalence with (left-to-right) Proof by contradiction. Consider the two directions of ↔ . ¬∀x. F [x] ⇔ ∃x. ¬F [x] First case: ¬∃x. F [x] ⇔ ∀x. ¬F [x] 1. I |= ∀x. F assumption 2. I 6|= F assumption Example 3. I |= F 1, ∀, since x 6∈ free(F ) 4. I |= ⊥ 2, 3 G : ∀x. (∃y. p(x, y) ∧ p(x, z)) → ∃w.p(x, w) . Second Case: 1. ∀x. (∃y. p(x, y) ∧ p(x, z)) → ∃w. p(x, w) 1. I 6|= ∀x. F assumption 2. ∀x. ¬(∃y. p(x, y) ∧ p(x, z)) ∨ ∃w. p(x, w) 2. I |= F assumption F1 → F2 ⇔ ¬F1 ∨ F2 3. I |= ∃x. ¬F 1 and ¬ 4. I |= ¬F 3, ∃, since x 6∈ free(F ) 3. ∀x. (∀y. ¬(p(x, y) ∧ p(x, z))) ∨ ∃w. p(x, w) 5. I |= ⊥ 2, 4 ¬∃x. F [x] ⇔ ∀x. ¬F [x] 4. ∀x. (∀y. ¬p(x, y) ∨ ¬p(x, z)) ∨ ∃w. p(x, w) H Hence, is a valid formula schema. 2- 19 2- 20 2. Prenex Normal Form (PNF) 2. Rename quantified variables to fresh names All quantifiers appear at the beginning of the formula F2 : ∀x. (∀y. ¬p(x, y) ∨ ¬p(x, z)) ∨ ∃w. p(x, w) ↑ in the scope of ∀x Q1x1 · · · Qnxn. F [x1, · · · , xn] 3. Remove all quantifiers to produce quantifier-free formula where Qi ∈ {∀, ∃} and F is quantifier-free. F3 : ¬p(x, y) ∨ ¬p(x, z) ∨ p(x, w) Every FOL formula F can be transformed to formula F ′ in PNF 4. Add the quantifiers before F3 s.t. F ′ ⇔ F . F4 : ∀x. ∀y. ∃w. ¬p(x, y) ∨ ¬p(x, z) ∨ p(x, w) Example: Find equivalent PNF of Alternately, ′ F : ∀x. ¬(∃y. p(x, y) ∧ p(x, z)) ∨ ∃y. p(x, y) F4 : ∀x. ∃w. ∀y. ¬p(x, y) ∨ ¬p(x, z) ∨ p(x, w) ↑ to the end of the formula Note: In F2, ∀y is in the scope of ∀x, therefore the order of quantifiers must be ···∀x ···∀y · · · 1.
Load more

Recommended publications
  • “The Church-Turing “Thesis” As a Special Corollary of Gödel's
    “The Church-Turing “Thesis” as a Special Corollary of Gödel’s Completeness Theorem,” in Computability: Turing, Gödel, Church, and Beyond, B. J. Copeland, C. Posy, and O. Shagrir (eds.), MIT Press (Cambridge), 2013, pp. 77-104. Saul A. Kripke This is the published version of the book chapter indicated above, which can be obtained from the publisher at https://mitpress.mit.edu/books/computability. It is reproduced here by permission of the publisher who holds the copyright. © The MIT Press The Church-Turing “ Thesis ” as a Special Corollary of G ö del ’ s 4 Completeness Theorem 1 Saul A. Kripke Traditionally, many writers, following Kleene (1952) , thought of the Church-Turing thesis as unprovable by its nature but having various strong arguments in its favor, including Turing ’ s analysis of human computation. More recently, the beauty, power, and obvious fundamental importance of this analysis — what Turing (1936) calls “ argument I ” — has led some writers to give an almost exclusive emphasis on this argument as the unique justification for the Church-Turing thesis. In this chapter I advocate an alternative justification, essentially presupposed by Turing himself in what he calls “ argument II. ” The idea is that computation is a special form of math- ematical deduction. Assuming the steps of the deduction can be stated in a first- order language, the Church-Turing thesis follows as a special case of G ö del ’ s completeness theorem (first-order algorithm theorem). I propose this idea as an alternative foundation for the Church-Turing thesis, both for human and machine computation. Clearly the relevant assumptions are justified for computations pres- ently known.
    [Show full text]
  • Logic and Proof
    Logic and Proof Computer Science Tripos Part IB Lent Term Lawrence C Paulson Computer Laboratory University of Cambridge [email protected] Copyright c 2018 by Lawrence C. Paulson I Logic and Proof 101 Introduction to Logic Logic concerns statements in some language. The language can be natural (English, Latin, . ) or formal. Some statements are true, others false or meaningless. Logic concerns relationships between statements: consistency, entailment, . Logical proofs model human reasoning (supposedly). Lawrence C. Paulson University of Cambridge I Logic and Proof 102 Statements Statements are declarative assertions: Black is the colour of my true love’s hair. They are not greetings, questions or commands: What is the colour of my true love’s hair? I wish my true love had hair. Get a haircut! Lawrence C. Paulson University of Cambridge I Logic and Proof 103 Schematic Statements Now let the variables X, Y, Z, . range over ‘real’ objects Black is the colour of X’s hair. Black is the colour of Y. Z is the colour of Y. Schematic statements can even express questions: What things are black? Lawrence C. Paulson University of Cambridge I Logic and Proof 104 Interpretations and Validity An interpretation maps variables to real objects: The interpretation Y 7 coal satisfies the statement Black is the colour→ of Y. but the interpretation Y 7 strawberries does not! A statement A is valid if→ all interpretations satisfy A. Lawrence C. Paulson University of Cambridge I Logic and Proof 105 Consistency, or Satisfiability A set S of statements is consistent if some interpretation satisfies all elements of S at the same time.
    [Show full text]
  • Satisfiability 6 the Decision Problem 7
    Satisfiability Difficult Problems Dealing with SAT Implementation Klaus Sutner Carnegie Mellon University 2020/02/04 Finding Hard Problems 2 Entscheidungsproblem to the Rescue 3 The Entscheidungsproblem is solved when one knows a pro- cedure by which one can decide in a finite number of operations So where would be look for hard problems, something that is eminently whether a given logical expression is generally valid or is satis- decidable but appears to be outside of P? And, we’d like the problem to fiable. The solution of the Entscheidungsproblem is of funda- be practical, not some monster from CRT. mental importance for the theory of all fields, the theorems of which are at all capable of logical development from finitely many The Circuit Value Problem is a good indicator for the right direction: axioms. evaluating Boolean expressions is polynomial time, but relatively difficult D. Hilbert within P. So can we push CVP a little bit to force it outside of P, just a little bit? In a sense, [the Entscheidungsproblem] is the most general Say, up into EXP1? problem of mathematics. J. Herbrand Exploiting Difficulty 4 Scaling Back 5 Herbrand is right, of course. As a research project this sounds like a Taking a clue from CVP, how about asking questions about Boolean fiasco. formulae, rather than first-order? But: we can turn adversity into an asset, and use (some version of) the Probably the most natural question that comes to mind here is Entscheidungsproblem as the epitome of a hard problem. Is ϕ(x1, . , xn) a tautology? The original Entscheidungsproblem would presumable have included arbitrary first-order questions about number theory.
    [Show full text]
  • Solving the Boolean Satisfiability Problem Using the Parallel Paradigm Jury Composition
    Philosophæ doctor thesis Hoessen Benoît Solving the Boolean Satisfiability problem using the parallel paradigm Jury composition: PhD director Audemard Gilles Professor at Universit´ed'Artois PhD co-director Jabbour Sa¨ıd Assistant Professor at Universit´ed'Artois PhD co-director Piette C´edric Assistant Professor at Universit´ed'Artois Examiner Simon Laurent Professor at University of Bordeaux Examiner Dequen Gilles Professor at University of Picardie Jules Vernes Katsirelos George Charg´ede recherche at Institut national de la recherche agronomique, Toulouse Abstract This thesis presents different technique to solve the Boolean satisfiability problem using parallel and distributed architec- tures. In order to provide a complete explanation, a careful presentation of the CDCL algorithm is made, followed by the state of the art in this domain. Once presented, two proposi- tions are made. The first one is an improvement on a portfo- lio algorithm, allowing to exchange more data without loosing efficiency. The second is a complete library with its API al- lowing to easily create distributed SAT solver. Keywords: SAT, parallelism, distributed, solver, logic R´esum´e Cette th`ese pr´esente diff´erentes techniques permettant de r´esoudre le probl`eme de satisfaction de formule bool´eenes utilisant le parall´elismeet du calcul distribu´e. Dans le but de fournir une explication la plus compl`ete possible, une pr´esentation d´etaill´ee de l'algorithme CDCL est effectu´ee, suivi d'un ´etatde l'art. De ce point de d´epart,deux pistes sont explor´ees. La premi`ereest une am´eliorationd'un algorithme de type portfolio, permettant d'´echanger plus d'informations sans perte d’efficacit´e.
    [Show full text]
  • Foundations of Mathematics
    Foundations of Mathematics November 27, 2017 ii Contents 1 Introduction 1 2 Foundations of Geometry 3 2.1 Introduction . .3 2.2 Axioms of plane geometry . .3 2.3 Non-Euclidean models . .4 2.4 Finite geometries . .5 2.5 Exercises . .6 3 Propositional Logic 9 3.1 The basic definitions . .9 3.2 Disjunctive Normal Form Theorem . 11 3.3 Proofs . 12 3.4 The Soundness Theorem . 19 3.5 The Completeness Theorem . 20 3.6 Completeness, Consistency and Independence . 22 4 Predicate Logic 25 4.1 The Language of Predicate Logic . 25 4.2 Models and Interpretations . 27 4.3 The Deductive Calculus . 29 4.4 Soundness Theorem for Predicate Logic . 33 5 Models for Predicate Logic 37 5.1 Models . 37 5.2 The Completeness Theorem for Predicate Logic . 37 5.3 Consequences of the completeness theorem . 41 5.4 Isomorphism and elementary equivalence . 43 5.5 Axioms and Theories . 47 5.6 Exercises . 48 6 Computability Theory 51 6.1 Introduction and Examples . 51 6.2 Finite State Automata . 52 6.3 Exercises . 55 6.4 Turing Machines . 56 6.5 Recursive Functions . 61 6.6 Exercises . 67 iii iv CONTENTS 7 Decidable and Undecidable Theories 69 7.1 Introduction . 69 7.1.1 Gödel numbering . 69 7.2 Decidable vs. Undecidable Logical Systems . 70 7.3 Decidable Theories . 71 7.4 Gödel’s Incompleteness Theorems . 74 7.5 Exercises . 81 8 Computable Mathematics 83 8.1 Computable Combinatorics . 83 8.2 Computable Analysis . 85 8.2.1 Computable Real Numbers . 85 8.2.2 Computable Real Functions .
    [Show full text]
  • Turing-Computable Functions  • Let F :(Σ−{ })∗ → Σ∗
    Turing-Computable Functions • Let f :(Σ−{ })∗ → Σ∗. – Optimization problems, root finding problems, etc. • Let M be a TM with alphabet Σ. • M computes f if for any string x ∈ (Σ −{ })∗, M(x)=f(x). • We call f a recursive functiona if such an M exists. aKurt G¨odel (1931, 1934). c 2017 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 60 Kurt G¨odela (1906–1978) Quine (1978), “this the- orem [···] sealed his im- mortality.” aThis photo was taken by Alfred Eisenstaedt (1898–1995). c 2017 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 61 Church’s Thesis or the Church-Turing Thesis • What is computable is Turing-computable; TMs are algorithms.a • No “intuitively computable” problems have been shown not to be Turing-computable, yet.b aChurch (1935); Kleene (1953). bQuantum computer of Manin (1980) and Feynman (1982) and DNA computer of Adleman (1994). c 2017 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 62 Church’s Thesis or the Church-Turing Thesis (concluded) • Many other computation models have been proposed. – Recursive function (G¨odel), λ calculus (Church), formal language (Post), assembly language-like RAM (Shepherdson & Sturgis), boolean circuits (Shannon), extensions of the Turing machine (more strings, two-dimensional strings, and so on), etc. • All have been proved to be equivalent. c 2017 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 63 Alonso Church (1903–1995) c 2017 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 64 Extended Church’s Thesisa • All “reasonably succinct encodings” of problems are polynomially related (e.g., n2 vs. n6). – Representations of a graph as an adjacency matrix and as a linked list are both succinct.
    [Show full text]
  • Classical Metalogic an Introduction to Classical Model Theory & Computability
    Classical Metalogic An Introduction to Classical Model Theory & Computability Notes by R.J. Buehler January 5, 2015 2 Preface What follows are my personal notes created in preparation for the UC-Berkeley Group in Logic preliminary exam. I am not a computability theorist, nor a model theorist; I am a graduate student with some knowledge who is–alas–quite fallible. Accordingly, this text is made available as a convenient reference, set of notes, and summary, but without even the slight hint of a guarantee that everything contained within is factual and correct (indeed, some areas are entirely unchanged from the moment I copied them off the blackboard). This said, if you find a serious error, I would greatly appreciate it if you would let me know so that it can be corrected. The material for these notes derives from a wide variety of sources: Lectures by Wes Holliday Lectures by Antonio Montalban Lectures by John Steel Kevin Kelly’s computability theory notes David Marker’s “Model Theory: An Introduction” Wilfrid Hodge’s “A Shorter Model Theory” Robert Soare’s “Recursively Enumerable Sets and Degrees" Richard Kaye’s “Models of Peano Arithmetic” Chang and Keisler’s “Model Theory" While I certainly hope my notes are beneficial, if you’re attempting to learn the contained material for the first time, I would highly suggest picking up (at least) a copy of Marker and Soare’s texts in addition. To those Group in Logic students who may be using these notes to help themselves prepare for their preliminary exam, chapters 1-5, 7, and 9-18 contain relevant material, as well as chapter 8, section 3.
    [Show full text]
  • Λ-Calculus: the Other Turing Machine
    λ-Calculus: The Other Turing Machine Guy Blelloch and Robert Harper July 25, 2015 The early 1930s were bad years for the worldwide economy, but great years for what would even- tually be called Computer Science. In 1932, Alonzo Church at Princeton described his λ-calculus as a formal system for mathematical logic,and in 1935 argued that any function on the natural numbers that can be effectively computed, can be computed with his calculus [4]. Independently in 1935, as a master's student at Cambridge, Alan Turing was developing his machine model of computation. In 1936 he too argued that his model could compute all computable functions on the natural numbers, and showed that his machine and the λ-calculus are equivalent [6]. The fact that two such different models of computation calculate the same functions was solid evidence that they both represented an inherent class of computable functions. From this arose the so-called Church-Turing thesis, which states, roughly, that any function on the natural numbers can be effectively computed if and only if it can be computed with the λ-calculus, or equivalently, the Turing machine. Although the Church-Turing thesis by itself is one of the most important ideas in Computer Science, the influence of Church and Turing's models go far beyond the thesis itself. The Turing machine has become the core of all complexity theory [5]. In the early 1960's Juris Hartmanis and Richard Stearns initiated the study of the complexity of computation. In 1967 Manuel Blum developed his axiomatic theory of complexity, which although independent of any particular machine, was considered in the context of a spectrum of Turing machines (e.g., one tape, two tape, or two stack).
    [Show full text]
  • Lecture 13: Propositional Logic Metatheoretic Definitions
    Valid, Invalid, . Lecture 13: Logical Consequence Propositional Logic Metatheoretic Definitions Logical Equivalence Laws of the Algebra of . Module Home Page Aims: Title Page • To discuss what we mean by valid, invalid, satisfiable and unsatisfiable; JJ II • To discuss what we mean by logical consequence; • To discuss what we mean by logical equivalence; and J I • To look at laws of propositional logic. Page 1 of 12 Back Full Screen Close Quit 13.1. Valid, Invalid, Satisfiable and Unsatisfiable • The truth-tables we saw in the previous lecture show that some wffs evaluate to true no matter how their atomic wffs are interpreted (every interpretation satisfies them), Valid, Invalid, . some evaluate to false for every interpretation of their atomic wffs (no interpretation Logical Consequence satisfies them), and others are true for some interpretations and false for others (some Logical Equivalence interpretations satisfy them, others don’t.) There is some terminology for all of this. Laws of the Algebra of . (There would be, wouldn’t there?) always true sometimes true/sometimes false always false Module Home Page tautology contingency contradiction logically true contingency logically false Title Page valid invalid unfalsifiable falsifiable JJ II satisfiable unsatisfiable Some books also use ‘consistent’ for ‘satisfiable’ and ‘inconsistent’ for ‘unsatisfiable’ J I but this is to be avoided. We will use just the following: valid/invalid and satisfiable/unsatisfiable. I ignore all Page 2 of 12 the others to avoid terminology overload: – A wff is valid if all interpretations satisfy it. Otherwise (i.e. if at least one does Back not satisfy it), it is invalid.
    [Show full text]
  • My Slides for a Course on Satisfiability
    Satisfiability Victor W. Marek Computer Science University of Kentucky Spring Semester 2005 Satisfiability – p.1/97 Satisfiability story ie× Á Satisfiability - invented in the ¾¼ of XX century by philosophers and mathematicians (Wittgenstein, Tarski) ie× Á Shannon (late ¾¼ ) applications to what was then known as electrical engineering ie× ie× Á Fundamental developments: ¼ and ¼ – both mathematics of it, and fundamental algorithms ie× Á ¼ - progress in computing speed and solving moderately large problems Á Emergence of “killer applications” in Computer Engineering, Bounded Model Checking Satisfiability – p.2/97 Current situation Á SAT solvers as a class of software Á Solving large cases generated by industrial applications Á Vibrant area of research both in Computer Science and in Computer Engineering ¯ Various CS meetings (SAT, AAAI, CP) ¯ Various CE meetings (CAV, FMCAD, DATE) ¯ CE meetings Stressing applications Satisfiability – p.3/97 This Course Á Providing mathematical and computer science foundations of SAT ¯ General mathematical foundations ¯ Two- and Three- valued logics ¯ Complete sets of functors ¯ Normal forms (including ite) ¯ Compactness of propositional logic ¯ Resolution rule, completeness of resolution, semantical resolution ¯ Fundamental algorithms for SAT ¯ Craig Lemma Satisfiability – p.4/97 And if there is enough of time and will... Á Easy cases of SAT ¯ Horn ¯ 2SAT ¯ Linear formulas Á And if there is time (but notes will be provided anyway) ¯ Expressing runs of polynomial-time NDTM as SAT, NP completeness ¯ “Mix and match” ¯ Learning in SAT, partial closure under resolution ¯ Bounded Model Checking Satisfiability – p.5/97 Various remarks Á Expected involvement of Dr. Truszczynski Á An extensive set of notes (> 200 pages), covering most of topics will be provided f.o.c.
    [Show full text]
  • Some Topics Related to Propositional Logic
    Some Topics Related to Propositional Logic Xishun Zhao [email protected] Institute of Logic and Cognition, Sun Yat-sen University Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 1/42 Outline Propositional Logic The Satisfiability Problem Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 2/42 Outline Propositional Logic The Satisfiability Problem Why SAT is so important? PversusNP Applications Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 2/42 Outline Propositional Logic The Satisfiability Problem Why SAT is so important? PversusNP Applications Challenges for SAT Solving and Its Applications Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 2/42 Outline Propositional Logic The Satisfiability Problem Why SAT is so important? PversusNP Applications Challenges for SAT Solving and Its Applications Minimal Unsatisfiability Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 2/42 Outline Propositional Logic The Satisfiability Problem Why SAT is so important? PversusNP Applications Challenges for SAT Solving and Its Applications Minimal Unsatisfiability Quantified Boolean Formulas (QBF) Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 2/42 Outline Propositional Logic The Satisfiability Problem Why SAT is so important? PversusNP Applications Challenges for SAT Solving and Its Applications Minimal Unsatisfiability Quantified Boolean Formulas (QBF) Expressive Power and Complexity Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p. 2/42 Outline Propositional Logic The Satisfiability Problem Why SAT is so important? PversusNP Applications Challenges for SAT Solving and Its Applications Minimal Unsatisfiability Quantified Boolean Formulas (QBF) Expressive Power and Complexity Future Work Sun Yat-sen University Topics Related to Propositional Logic December 5, 2010 – p.
    [Show full text]
  • The Explanatory Power of a New Proof: Henkin's Completeness Proof
    The explanatory power of a new proof: Henkin’s completeness proof John Baldwin February 25, 2017 Mancosu writes But explanations in mathematics do not only come in the form of proofs. In some cases expla- nations are sought in a major recasting of an entire discipline. ([Mancosu, 2008], 142) This paper takes up both halves of that statement. On the one hand we provide a case study of the explanatory value of a particular milestone proof. In the process we examine how it began the recasting of a discipline. Hafner and Mancosu take a broad view towards the nature of mathematical explanation. They ar- gue that before attempting to establish a model of explanation, one should develop a ‘taxonomy of re- current types of mathematical explanation’ ([Hafner and Mancosu, 2005], 221) and preparatory to such a taxonomy propose to examine in depth various examples of proofs. In [Hafner and Mancosu, 2005] and [Hafner and Mancosu, 2008], they study deep arguments in real algebraic geometry and analysis to test the models of explanation of Kitcher and Steiner. In their discussion of Steiner’s model they challenge1 the as- sertion [Resnik and Kushner, 1987] that Henkin’s proof of the completeness theorem is explanatory, asking ‘what the explanatory features of this proof are supposed to consist of?’ As a model theorist the challenge to ‘explain’ the explanatory value of this fundamental argument is irresistible. In contrasting the proofs of Henkin and Godel,¨ we seek for the elements of Henkin’s proofs that permit its numerous generalizations. In Section 2 we try to make this analysis more precise through Steiner’s notion of characterizing property.
    [Show full text]