DOCSLIB.ORG

2 Ch 2: LOGIC

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

2 Ch 2: LOGIC 2.1 Statements A statement is a sentence that is either true or false, but not both. Ex 1: Today is Monday. Ex 2: The integer 3 is even. Not examples: The equation 3x = 12. This is not a statement b/c it depends on the value of x. There is one value that makes it true, but the sentence is not always true. Every statement has a truth value, namely true T or false F. A sentence containing a variable(s) is called an open sentence. Ex: The integer r is even. Possible truth values are often given in a table called a truth table. Examples: P Truth table for a sentence P: T F P Q T T Truth table for two sentences P and Q: T F F T F F Thus two statements will give us 22 combinations (rows below the one with P and Q) in the table. For three statements we would get 23 combinations, since there are two choices for each of the three statement(either T or F). 2.2 The Negation of a statement The negation of a statement P is the statement ∼ P : not P . Ex: P : 3 is even. ∼ P : 3 is not even. OR: ∼ P : 3 is odd. Observe that when the statement is false, its negation is true. P ∼ P Truth table for sentence ∼ P : T F F T 2.3 The Disjunction and Conjunction of a Statement For two statements P and Q, the disjunction of P and Q is P ∨ Q (P or Q). P Q P ∨ Q T T T Fact: P ∨ Q is true if at least one of P and Q is true T F T F T T F F F For two statements P and Q, the conjunction of P and Q is P ∧ Q (P and Q). 1 Fact: P ∧ Q is true only if both P and Q are true. P Q P ∧ Q T T T T F F F T F F F F 2.4 The Implication (or Conditional statement) P Q P ⇒ Q T T T An implication is a statement P ⇒ Q : If P , then Q (or P implies Q). T F F F T T F F T Ex: A student is currently receiving a B+ just before final exam. Student’s question: Is there a chance that I will get an A in the class? Answer: If you will get an A in the final, then you will receive an A in the class Question: Is the conditional statement above true or false? P: You will get an A in the final Q: You will receive an A in the class. P Q P ⇒ Q T T T T F F F T T F F T The last two statements are still correct since the conditional didn’t say what will happen if the student does not get an A in the final. There are more ways of expressing P ⇒ Q: If P then Q Q if P P implies Q P is sufficient for Q Q is necessary for P 2.5 More on Implication This section considers open sentences: P (x) and Q(x), which depend on x. P(x) Q(x) P(x) ⇒ Q(x) T T T An implication is a statement P (x) ⇒ Q(x) : If P (x), then Q(x). T F F F T T F F T Note that P (x) and Q(x) are open sentences. P (x) ⇒ Q(x) is an implication, thus it is a statement. 2 Generally, we write P and Q instead of P (x) and Q(x) for convenience. However, if we want to emphasis that the statements depend on x, then we write P (x) and Q(x). 2.6 The Biconditional The biconditional of P and Q is P ⇐⇒ Q :(P ⇒ Q) ∧ (Q ⇒ P ), and it means either that P is equivalent to Q, or P if and only if Q. Truth table for P ⇐⇒ Q: P Q P ⇒ Q Q ⇒ P P ⇐⇒ Q T T T T T T F F T F F T T F F F F T T T The biconditional is true when P and Q are either both true, or both false. 2.7 Tautologies and Contradictions We use logical connectives to make new statements. Logical Connectives: ∼, ∨, ∧, ⇒, ⇐⇒ A compound statement is a statement composed of one or more given statements and at least one logical connective. P ∼ P P ∨ ( ∼ P ) Ex: P ∨ (∼ P ) with the truth table: T F T F T T A compound statement S is called a tautology if it is true for all possible combinations of truth values of its component statements: Ex: The sun rises from the East.– one component Today is Tu or it is not. –2 components. Ex: Let Q and P be two statements. Is (∼ Q) ∨ (P ⇒ Q) a tautology? To check this we look at the truth table: P Q ∼ Q P ⇒ Q (∼ Q) ∨ (P ⇒ Q) T T F T T T F T F T Thus the compound statement is a tautology. F T F T T F F T T T What about P ∧ (∼ P )? Consider the truth table: P ∼ P P ∧ (∼ P) T F F F T F Thus the compound statement is not a tautology. A compound statement that is false for all possible combinations of truth values of its component statements is called a contradiction. 3 2.8 Logical Equivalence A compound statement can be a tautology, contradiction or neither. If P is a tautology, then ∼ P is a contradiction and vice versa. When negating ∧ we get ∨, and negating ∨ we get ∧. Let R and S be two compound statements. Then R and S are logically equivalent if R and S have the same truth values for all possible combinations of truth values of their components. Ex: Show that R : P ⇒ Q and S :(∼ P ) ∨ Q are logically equivalent. P Q P ⇒ Q ∼ P (∼ P ) ∨ Q T T T F T T F F F F F T T T T F F T T T Thus the compound statements are logically equivalent. This means that R ⇐⇒ S is a tautology, or (P ⇒ Q) ⇐⇒ ((∼ P ) ∨ Q) is a tautology. 2.9 Some Fundamental Properties of Logical Equivalence (page 49): Thm. 2.18: For statements P, Q, and R, 1. Commutative Laws: (a) P ∨ Q is equivalent to Q ∨ P (b) P ∧ Q is equivalent to Q ∧ P 2. Associative Laws: (a) P ∨ (Q ∨ R) is equivalent to (P ∨ Q) ∨ R (b) P ∧ (Q ∧ R) is equivalent to (P ∧ Q) ∧ R 3. Distributive Laws: (a) P ∨ (Q ∧ R) is equivalent to (P ∨ Q) ∧ (P ∨ R) (b) P ∧ (Q ∨ R) is equivalent to (P ∧ Q) ∨ (P ∧ R) 4. DeMorgan’s Laws: (a) ∼ (P ∨ Q) is equivalent to (∼ P ) ∧ (∼ Q) (b) ∼ (P ∧ Q) is equivalent to (∼ P ) ∨ (∼ Q) Ex: The statement ∼ (P ⇒ Q) is logically equivalent to P ∧ (∼ Q) Proof: We have the following tautologies (the first one comes from a truth table we’ve done 4 before): ∼ (P ⇒ Q) ⇐⇒ ∼ ((∼ P ) ∨ Q) ⇐⇒ (∼ (∼ P )) ∧ (∼ Q) ⇐⇒ P ∧ (∼ Q). 2.10 Quantified Statements Ex: The following statements have the same meaning, but they are quantified differently: If x is a real number, then x2 ≥ 0. The square of every real number is nonnegative. For every real number x, we have x2 ≥ 0. Each of the the phrases ”‘every”’, ”‘for every”’, ”‘for each”’, and ”‘for all”’ is called the universal quantifier and it is denoted by ∀. Each phrase ”there exists”, ”there is”, ”for some”, ”for at least one” is called the existential quantifier and is denoted by ∃. NEGATION OF QUANTIFIED STATEMENTS: Negation of universal quantifier: ∼ (∀x ∈ R, x < 0) is equivalent to ∃x ∈ R, x > 0. Generally, if P (x) is an open sentence, then: ∼ (∀x ∈ S, P (x)) is equivalent to ∃x ∈ R, (∼ P (x)). Ex: Let A be a set, with P(A) being powerset of A. P : For every set B ∈ P(A), A \ B 6= ∅. What is its negation? ∼ P : There is a set B ∈ P(A) such that A \ B = ∅. Negation of existential quantifier: Generally, if P (x) is an open sentence, then: ∼ (∃x ∈ S, P (x)) is equivalent to ∀x ∈ R, (∼ P (x)). Ex: P : There is a real number x, such that x2 = 4. What is its negation? ∼ P : For all real numbers x, x2 6= 4. 2.11 Characterization of Statements Let’s return to biconditionals: P ⇐⇒ Q = (P ⇒ Q) ∧ (Q ⇒ P ) : If P , then Q and if Q, then P (or simpler: P if and only if Q). Ex: Let P : x = −3, and let Q : |x| = 3. Consider P ⇐⇒ Q which says x = −3 ⇐⇒ |x| = 3. Statement is F because of (⇐). That is to say that |x| = 3 ⇒ x = −3 is false. If P (x) ⇐⇒ Q(x) we say that P (x) is characterized by Q(x). Note: This means that Q(x) is another way to say P (x), but Q(x) cannot be the definition of P (x). 5.
Recommended publications
  • Dialetheists' Lies About the Liar

    Dialetheists' Lies About the Liar

    PRINCIPIA 22(1): 59–85 (2018) doi: 10.5007/1808-1711.2018v22n1p59 Published by NEL — Epistemology and Logic Research Group, Federal University of Santa Catarina (UFSC), Brazil. DIALETHEISTS’LIES ABOUT THE LIAR JONAS R. BECKER ARENHART Departamento de Filosofia, Universidade Federal de Santa Catarina, BRAZIL [email protected] EDERSON SAFRA MELO Departamento de Filosofia, Universidade Federal do Maranhão, BRAZIL [email protected] Abstract. Liar-like paradoxes are typically arguments that, by using very intuitive resources of natural language, end up in contradiction. Consistent solutions to those paradoxes usually have difficulties either because they restrict the expressive power of the language, orelse because they fall prey to extended versions of the paradox. Dialetheists, like Graham Priest, propose that we should take the Liar at face value and accept the contradictory conclusion as true. A logical treatment of such contradictions is also put forward, with the Logic of Para- dox (LP), which should account for the manifestations of the Liar. In this paper we shall argue that such a formal approach, as advanced by Priest, is unsatisfactory. In order to make contradictions acceptable, Priest has to distinguish between two kinds of contradictions, in- ternal and external, corresponding, respectively, to the conclusions of the simple and of the extended Liar. Given that, we argue that while the natural interpretation of LP was intended to account for true and false sentences, dealing with internal contradictions, it lacks the re- sources to tame external contradictions. Also, the negation sign of LP is unable to represent internal contradictions adequately, precisely because of its allowance of sentences that may be true and false.
  • 1 Elementary Set Theory

    1 Elementary Set Theory

    1 Elementary Set Theory Notation: fg enclose a set. f1; 2; 3g = f3; 2; 2; 1; 3g because a set is not defined by order or multiplicity. f0; 2; 4;:::g = fxjx is an even natural numberg because two ways of writing a set are equivalent. ; is the empty set. x 2 A denotes x is an element of A. N = f0; 1; 2;:::g are the natural numbers. Z = f:::; −2; −1; 0; 1; 2;:::g are the integers. m Q = f n jm; n 2 Z and n 6= 0g are the rational numbers. R are the real numbers. Axiom 1.1. Axiom of Extensionality Let A; B be sets. If (8x)x 2 A iff x 2 B then A = B. Definition 1.1 (Subset). Let A; B be sets. Then A is a subset of B, written A ⊆ B iff (8x) if x 2 A then x 2 B. Theorem 1.1. If A ⊆ B and B ⊆ A then A = B. Proof. Let x be arbitrary. Because A ⊆ B if x 2 A then x 2 B Because B ⊆ A if x 2 B then x 2 A Hence, x 2 A iff x 2 B, thus A = B. Definition 1.2 (Union). Let A; B be sets. The Union A [ B of A and B is defined by x 2 A [ B if x 2 A or x 2 B. Theorem 1.2. A [ (B [ C) = (A [ B) [ C Proof. Let x be arbitrary. x 2 A [ (B [ C) iff x 2 A or x 2 B [ C iff x 2 A or (x 2 B or x 2 C) iff x 2 A or x 2 B or x 2 C iff (x 2 A or x 2 B) or x 2 C iff x 2 A [ B or x 2 C iff x 2 (A [ B) [ C Definition 1.3 (Intersection).
  • Chapter 5: Methods of Proof for Boolean Logic

    Chapter 5: Methods of Proof for Boolean Logic

    Chapter 5: Methods of Proof for Boolean Logic § 5.1 Valid inference steps Conjunction elimination Sometimes called simplification. From a conjunction, infer any of the conjuncts. • From P ∧ Q, infer P (or infer Q). Conjunction introduction Sometimes called conjunction. From a pair of sentences, infer their conjunction. • From P and Q, infer P ∧ Q. § 5.2 Proof by cases This is another valid inference step (it will form the rule of disjunction elimination in our formal deductive system and in Fitch), but it is also a powerful proof strategy. In a proof by cases, one begins with a disjunction (as a premise, or as an intermediate conclusion already proved). One then shows that a certain consequence may be deduced from each of the disjuncts taken separately. One concludes that that same sentence is a consequence of the entire disjunction. • From P ∨ Q, and from the fact that S follows from P and S also follows from Q, infer S. The general proof strategy looks like this: if you have a disjunction, then you know that at least one of the disjuncts is true—you just don’t know which one. So you consider the individual “cases” (i.e., disjuncts), one at a time. You assume the first disjunct, and then derive your conclusion from it. You repeat this process for each disjunct. So it doesn’t matter which disjunct is true—you get the same conclusion in any case. Hence you may infer that it follows from the entire disjunction. In practice, this method of proof requires the use of “subproofs”—we will take these up in the next chapter when we look at formal proofs.
  • In Defence of Constructive Empiricism: Metaphysics Versus Science

    In Defence of Constructive Empiricism: Metaphysics Versus Science

    To appear in Journal for General Philosophy of Science (2004) In Defence of Constructive Empiricism: Metaphysics versus Science F.A. Muller Institute for the History and Philosophy of Science and Mathematics Utrecht University, P.O. Box 80.000 3508 TA Utrecht, The Netherlands E-mail: [email protected] August 2003 Summary Over the past years, in books and journals (this journal included), N. Maxwell launched a ferocious attack on B.C. van Fraassen's view of science called Con- structive Empiricism (CE). This attack has been totally ignored. Must we con- clude from this silence that no defence is possible against the attack and that a fortiori Maxwell has buried CE once and for all, or is the attack too obviously flawed as not to merit exposure? We believe that neither is the case and hope that a careful dissection of Maxwell's reasoning will make this clear. This dis- section includes an analysis of Maxwell's `aberrance-argument' (omnipresent in his many writings) for the contentious claim that science implicitly and per- manently accepts a substantial, metaphysical thesis about the universe. This claim generally has been ignored too, for more than a quarter of a century. Our con- clusions will be that, first, Maxwell's attacks on CE can be beaten off; secondly, his `aberrance-arguments' do not establish what Maxwell believes they estab- lish; but, thirdly, we can draw a number of valuable lessons from these attacks about the nature of science and of the libertarian nature of CE. Table of Contents on other side −! Contents 1 Exordium: What is Maxwell's Argument? 1 2 Does Science Implicitly Accept Metaphysics? 3 2.1 Aberrant Theories .
  • Boolean Satisfiability Solvers: Techniques and Extensions

    Boolean Satisfiability Solvers: Techniques and Extensions

    Boolean Satisfiability Solvers: Techniques and Extensions Georg WEISSENBACHER a and Sharad MALIK a a Princeton University Abstract. Contemporary satisfiability solvers are the corner-stone of many suc- cessful applications in domains such as automated verification and artificial intelli- gence. The impressive advances of SAT solvers, achieved by clever engineering and sophisticated algorithms, enable us to tackle Boolean Satisfiability (SAT) problem instances with millions of variables – which was previously conceived as a hope- less problem. We provide an introduction to contemporary SAT-solving algorithms, covering the fundamental techniques that made this revolution possible. Further, we present a number of extensions of the SAT problem, such as the enumeration of all satisfying assignments (ALL-SAT) and determining the maximum number of clauses that can be satisfied by an assignment (MAX-SAT). We demonstrate how SAT solvers can be leveraged to solve these problems. We conclude the chapter with an overview of applications of SAT solvers and their extensions in automated verification. Keywords. Satisfiability solving, Propositional logic, Automated decision procedures 1. Introduction Boolean Satisfibility (SAT) is the problem of checking if a propositional logic formula can ever evaluate to true. This problem has long enjoyed a special status in computer science. On the theoretical side, it was the first problem to be classified as being NP- complete. NP-complete problems are notorious for being hard to solve; in particular, in the worst case, the computation time of any known solution for a problem in this class increases exponentially with the size of the problem instance. On the practical side, SAT manifests itself in several important application domains such as the design and verification of hardware and software systems, as well as applications in artificial intelligence.
  • Deduction (I) Tautologies, Contradictions And

    Deduction (I) Tautologies, Contradictions And

    D (I) T, & L L October , Tautologies, contradictions and contingencies Consider the truth table of the following formula: p (p ∨ p) () If you look at the final column, you will notice that the truth value of the whole formula depends on the way a truth value is assigned to p: the whole formula is true if p is true and false if p is false. Contrast the truth table of (p ∨ p) in () with the truth table of (p ∨ ¬p) below: p ¬p (p ∨ ¬p) () If you look at the final column, you will notice that the truth value of the whole formula does not depend on the way a truth value is assigned to p. The formula is always true because of the meaning of the connectives. Finally, consider the truth table table of (p ∧ ¬p): p ¬p (p ∧ ¬p) () This time the formula is always false no matter what truth value p has. Tautology A statement is called a tautology if the final column in its truth table contains only ’s. Contradiction A statement is called a contradiction if the final column in its truth table contains only ’s. Contingency A statement is called a contingency or contingent if the final column in its truth table contains both ’s and ’s. Let’s consider some examples from the book. Can you figure out which of the following sentences are tautologies, which are contradictions and which contingencies? Hint: the answer is the same for all the formulas with a single row. () a. (p ∨ ¬p), (p → p), (p → (q → p)), ¬(p ∧ ¬p) b.
  • Leibniz's Ontological Proof of the Existence of God and the Problem Of

    Leibniz's Ontological Proof of the Existence of God and the Problem Of

    Leibniz’s Ontological Proof of the Existence of God and the Problem of »Impossible Objects« Wolfgang Lenzen (Osnabrück) Abstract The core idea of the ontological proof is to show that the concept of existence is somehow contained in the concept of God, and that therefore God’s existence can be logically derived – without any further assumptions about the external world – from the very idea, or definition, of God. Now, G.W. Leibniz has argued repeatedly that the traditional versions of the ontological proof are not fully conclusive, because they rest on the tacit assumption that the concept of God is possible, i.e. free from contradiction. A complete proof will rather have to consist of two parts. First, a proof of premise (1) God is possible. Second, a demonstration of the “remarkable proposition” (2) If God is possible, then God exists. The present contribution investigates an interesting paper in which Leibniz tries to prove proposition (2). It will be argued that the underlying idea of God as a necessary being has to be interpreted with the help of a distinguished predicate letter ‘E’ (denoting the concept of existence) as follows: (3) g =df ιxE(x). Principle (2) which Leibniz considered as “the best fruit of the entire logic” can then be formalized as follows: (4) ◊E(ιxE(x)) → E(ιxE(x)). At first sight, Leibniz’s proof appears to be formally correct; but a closer examination reveals an ambiguity in his use of the modal notions. According to (4), the possibility of the necessary being has to be understood in the sense of something which possibly exists.
  • Sets, Propositional Logic, Predicates, and Quantifiers

    Sets, Propositional Logic, Predicates, and Quantifiers

    COMP 182 Algorithmic Thinking Sets, Propositional Logic, Luay Nakhleh Computer Science Predicates, and Quantifiers Rice University !1 Reading Material ❖ Chapter 1, Sections 1, 4, 5 ❖ Chapter 2, Sections 1, 2 !2 ❖ Mathematics is about statements that are either true or false. ❖ Such statements are called propositions. ❖ We use logic to describe them, and proof techniques to prove whether they are true or false. !3 Propositions ❖ 5>7 ❖ The square root of 2 is irrational. ❖ A graph is bipartite if and only if it doesn’t have a cycle of odd length. ❖ For n>1, the sum of the numbers 1,2,3,…,n is n2. !4 Propositions? ❖ E=mc2 ❖ The sun rises from the East every day. ❖ All species on Earth evolved from a common ancestor. ❖ God does not exist. ❖ Everyone eventually dies. !5 ❖ And some of you might already be wondering: “If I wanted to study mathematics, I would have majored in Math. I came here to study computer science.” !6 ❖ Computer Science is mathematics, but we almost exclusively focus on aspects of mathematics that relate to computation (that can be implemented in software and/or hardware). !7 ❖Logic is the language of computer science and, mathematics is the computer scientist’s most essential toolbox. !8 Examples of “CS-relevant” Math ❖ Algorithm A correctly solves problem P. ❖ Algorithm A has a worst-case running time of O(n3). ❖ Problem P has no solution. ❖ Using comparison between two elements as the basic operation, we cannot sort a list of n elements in less than O(n log n) time. ❖ Problem A is NP-Complete.
  • On Axiomatizations of General Many-Valued Propositional Calculi

    On Axiomatizations of General Many-Valued Propositional Calculi

    On axiomatizations of general many-valued propositional calculi Arto Salomaa Turku Centre for Computer Science Joukahaisenkatu 3{5 B, 20520 Turku, Finland asalomaa@utu.fi Abstract We present a general setup for many-valued propositional logics, and compare truth-table and axiomatic stipulations within this setup. Results are obtained concerning cases, where a finitary axiomatization is (resp. is not) possible. Related problems and examples are discussed. 1 Introduction Many-valued systems of logic are constructed by introducing one or more truth- values between truth and falsity. Truth-functions associated with the logical connectives then operate with more than two truth-values. For instance, the truth-function D associated with the 3-valued disjunction might be defined for the truth-values T;I;F (true, intermediate, false) by D(x; y) = max(x; y); where T > I > F: If the truth-function N associated with negation is defined by N(T ) = F; N(F ) = T;N(I) = I; the law of the excluded middle is not valid, provided \validity" means that the truth-value t results for all assignments of values for the variables. On the other hand, many-valuedness is not so obvious if the axiomatic method is used. As such, an ordinary axiomatization is not many-valued. Truth-tables are essential for the latter. However, in many cases it is possi- ble to axiomatize many-valued logics and, conversely, find many-valued models for axiom systems. In this paper, we will investigate (for propositional log- ics) interconnections between such \truth-value stipulations" and \axiomatic stipulations". A brief outline of the contents of this paper follows.
  • Mathematical Logic Part One

    Mathematical Logic Part One

    Mathematical Logic Part One An Important Question How do we formalize the logic we've been using in our proofs? Where We're Going ● Propositional Logic (Today) ● Basic logical connectives. ● Truth tables. ● Logical equivalences. ● First-Order Logic (Today/Friday) ● Reasoning about properties of multiple objects. Propositional Logic A proposition is a statement that is, by itself, either true or false. Some Sample Propositions ● Puppies are cuter than kittens. ● Kittens are cuter than puppies. ● Usain Bolt can outrun everyone in this room. ● CS103 is useful for cocktail parties. ● This is the last entry on this list. More Propositions ● I came in like a wrecking ball. ● I am a champion. ● You're going to hear me roar. ● We all just entertainers. Things That Aren't Propositions CommandsCommands cannotcannot bebe truetrue oror false.false. Things That Aren't Propositions QuestionsQuestions cannotcannot bebe truetrue oror false.false. Things That Aren't Propositions TheThe firstfirst halfhalf isis aa validvalid proposition.proposition. I am the walrus, goo goo g'joob JibberishJibberish cannotcannot bebe truetrue oror false. false. Propositional Logic ● Propositional logic is a mathematical system for reasoning about propositions and how they relate to one another. ● Every statement in propositional logic consists of propositional variables combined via logical connectives. ● Each variable represents some proposition, such as “You liked it” or “You should have put a ring on it.” ● Connectives encode how propositions are related, such as “If you liked it, then you should have put a ring on it.” Propositional Variables ● Each proposition will be represented by a propositional variable. ● Propositional variables are usually represented as lower-case letters, such as p, q, r, s, etc.
  • Solving the Boolean Satisfiability Problem Using the Parallel Paradigm Jury Composition

    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.
  • 12 Propositional Logic

    12 Propositional Logic

    CHAPTER 12 ✦ ✦ ✦ ✦ Propositional Logic In this chapter, we introduce propositional logic, an algebra whose original purpose, dating back to Aristotle, was to model reasoning. In more recent times, this algebra, like many algebras, has proved useful as a design tool. For example, Chapter 13 shows how propositional logic can be used in computer circuit design. A third use of logic is as a data model for programming languages and systems, such as the language Prolog. Many systems for reasoning by computer, including theorem provers, program verifiers, and applications in the field of artificial intelligence, have been implemented in logic-based programming languages. These languages generally use “predicate logic,” a more powerful form of logic that extends the capabilities of propositional logic. We shall meet predicate logic in Chapter 14. ✦ ✦ ✦ ✦ 12.1 What This Chapter Is About Section 12.2 gives an intuitive explanation of what propositional logic is, and why it is useful. The next section, 12,3, introduces an algebra for logical expressions with Boolean-valued operands and with logical operators such as AND, OR, and NOT that Boolean algebra operate on Boolean (true/false) values. This algebra is often called Boolean algebra after George Boole, the logician who first framed logic as an algebra. We then learn the following ideas. ✦ Truth tables are a useful way to represent the meaning of an expression in logic (Section 12.4). ✦ We can convert a truth table to a logical expression for the same logical function (Section 12.5). ✦ The Karnaugh map is a useful tabular technique for simplifying logical expres- sions (Section 12.6).