First-Order Logic
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CS 512, Spring 2017, Handout 10 [1Ex] Propositional Logic: [2Ex
CS 512, Spring 2017, Handout 10 Propositional Logic: Conjunctive Normal Forms, Disjunctive Normal Forms, Horn Formulas, and other special forms Assaf Kfoury 5 February 2017 Assaf Kfoury, CS 512, Spring 2017, Handout 10 page 1 of 28 CNF L ::= p j :p literal D ::= L j L _ D disjuntion of literals C ::= D j D ^ C conjunction of disjunctions DNF L ::= p j :p literal C ::= L j L ^ C conjunction of literals D ::= C j C _ D disjunction of conjunctions conjunctive normal form & disjunctive normal form Assaf Kfoury, CS 512, Spring 2017, Handout 10 page 2 of 28 DNF L ::= p j :p literal C ::= L j L ^ C conjunction of literals D ::= C j C _ D disjunction of conjunctions conjunctive normal form & disjunctive normal form CNF L ::= p j :p literal D ::= L j L _ D disjuntion of literals C ::= D j D ^ C conjunction of disjunctions Assaf Kfoury, CS 512, Spring 2017, Handout 10 page 3 of 28 conjunctive normal form & disjunctive normal form CNF L ::= p j :p literal D ::= L j L _ D disjuntion of literals C ::= D j D ^ C conjunction of disjunctions DNF L ::= p j :p literal C ::= L j L ^ C conjunction of literals D ::= C j C _ D disjunction of conjunctions Assaf Kfoury, CS 512, Spring 2017, Handout 10 page 4 of 28 I A disjunction of literals L1 _···_ Lm is valid (equivalently, is a tautology) iff there are 1 6 i; j 6 m with i 6= j such that Li is :Lj. -
An Algorithm for the Satisfiability Problem of Formulas in Conjunctive
Journal of Algorithms 54 (2005) 40–44 www.elsevier.com/locate/jalgor An algorithm for the satisfiability problem of formulas in conjunctive normal form Rainer Schuler Abt. Theoretische Informatik, Universität Ulm, D-89069 Ulm, Germany Received 26 February 2003 Available online 9 June 2004 Abstract We consider the satisfiability problem on Boolean formulas in conjunctive normal form. We show that a satisfying assignment of a formula can be found in polynomial time with a success probability − − + of 2 n(1 1/(1 logm)),wheren and m are the number of variables and the number of clauses of the formula, respectively. If the number of clauses of the formulas is bounded by nc for some constant c, − + this gives an expected run time of O(p(n) · 2n(1 1/(1 c logn))) for a polynomial p. 2004 Elsevier Inc. All rights reserved. Keywords: Complexity theory; NP-completeness; CNF-SAT; Probabilistic algorithms 1. Introduction The satisfiability problem of Boolean formulas in conjunctive normal form (CNF-SAT) is one of the best known NP-complete problems. The problem remains NP-complete even if the formulas are restricted to a constant number k>2 of literals in each clause (k-SAT). In recent years several algorithms have been proposed to solve the problem exponentially faster than the 2n time bound, given by an exhaustive search of all possible assignments to the n variables. So far research has focused in particular on k-SAT, whereas improvements for SAT in general have been derived with respect to the number m of clauses in a formula [2,7]. -
Propositional Logic - Basics
Outline Propositional Logic - Basics K. Subramani1 1Lane Department of Computer Science and Electrical Engineering West Virginia University 14 January and 16 January, 2013 Subramani Propositonal Logic Outline Outline 1 Statements, Symbolic Representations and Semantics Boolean Connectives and Semantics Subramani Propositonal Logic (i) The Law! (ii) Mathematics. (iii) Computer Science. Definition Statement (or Atomic Proposition) - A sentence that is either true or false. Example (i) The board is black. (ii) Are you John? (iii) The moon is made of green cheese. (iv) This statement is false. (Paradox). Statements, Symbolic Representations and Semantics Boolean Connectives and Semantics Motivation Why Logic? Subramani Propositonal Logic (ii) Mathematics. (iii) Computer Science. Definition Statement (or Atomic Proposition) - A sentence that is either true or false. Example (i) The board is black. (ii) Are you John? (iii) The moon is made of green cheese. (iv) This statement is false. (Paradox). Statements, Symbolic Representations and Semantics Boolean Connectives and Semantics Motivation Why Logic? (i) The Law! Subramani Propositonal Logic (iii) Computer Science. Definition Statement (or Atomic Proposition) - A sentence that is either true or false. Example (i) The board is black. (ii) Are you John? (iii) The moon is made of green cheese. (iv) This statement is false. (Paradox). Statements, Symbolic Representations and Semantics Boolean Connectives and Semantics Motivation Why Logic? (i) The Law! (ii) Mathematics. Subramani Propositonal Logic (iii) Computer Science. Definition Statement (or Atomic Proposition) - A sentence that is either true or false. Example (i) The board is black. (ii) Are you John? (iii) The moon is made of green cheese. (iv) This statement is false. -
Boolean Algebra
Boolean Algebra Definition: A Boolean Algebra is a math construct (B,+, . , ‘, 0,1) where B is a non-empty set, + and . are binary operations in B, ‘ is a unary operation in B, 0 and 1 are special elements of B, such that: a) + and . are communative: for all x and y in B, x+y=y+x, and x.y=y.x b) + and . are associative: for all x, y and z in B, x+(y+z)=(x+y)+z, and x.(y.z)=(x.y).z c) + and . are distributive over one another: x.(y+z)=xy+xz, and x+(y.z)=(x+y).(x+z) d) Identity laws: 1.x=x.1=x and 0+x=x+0=x for all x in B e) Complementation laws: x+x’=1 and x.x’=0 for all x in B Examples: • (B=set of all propositions, or, and, not, T, F) • (B=2A, U, ∩, c, Φ,A) Theorem 1: Let (B,+, . , ‘, 0,1) be a Boolean Algebra. Then the following hold: a) x+x=x and x.x=x for all x in B b) x+1=1 and 0.x=0 for all x in B c) x+(xy)=x and x.(x+y)=x for all x and y in B Proof: a) x = x+0 Identity laws = x+xx’ Complementation laws = (x+x).(x+x’) because + is distributive over . = (x+x).1 Complementation laws = x+x Identity laws x = x.1 Identity laws = x.(x+x’) Complementation laws = x.x +x.x’ because + is distributive over . -
Normal Forms
Propositional Logic Normal Forms 1 Literals Definition A literal is an atom or the negation of an atom. In the former case the literal is positive, in the latter case it is negative. 2 Negation Normal Form (NNF) Definition A formula is in negation formal form (NNF) if negation (:) occurs only directly in front of atoms. Example In NNF: :A ^ :B Not in NNF: :(A _ B) 3 Transformation into NNF Any formula can be transformed into an equivalent formula in NNF by pushing : inwards. Apply the following equivalences from left to right as long as possible: ::F ≡ F :(F ^ G) ≡ (:F _:G) :(F _ G) ≡ (:F ^ :G) Example (:(A ^ :B) ^ C) ≡ ((:A _ ::B) ^ C) ≡ ((:A _ B) ^ C) Warning:\ F ≡ G ≡ H" is merely an abbreviation for \F ≡ G and G ≡ H" Does this process always terminate? Is the result unique? 4 CNF and DNF Definition A formula F is in conjunctive normal form (CNF) if it is a conjunction of disjunctions of literals: n m V Wi F = ( ( Li;j )), i=1 j=1 where Li;j 2 fA1; A2; · · · g [ f:A1; :A2; · · · g Definition A formula F is in disjunctive normal form (DNF) if it is a disjunction of conjunctions of literals: n m W Vi F = ( ( Li;j )), i=1 j=1 where Li;j 2 fA1; A2; · · · g [ f:A1; :A2; · · · g 5 Transformation into CNF and DNF Any formula can be transformed into an equivalent formula in CNF or DNF in two steps: 1. Transform the initial formula into its NNF 2. -
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). -
Extended Conjunctive Normal Form and an Efficient Algorithm For
Proceedings of the Twenty-Ninth International Joint Conference on Artificial Intelligence (IJCAI-20) Extended Conjunctive Normal Form and An Efficient Algorithm for Cardinality Constraints Zhendong Lei1;2 , Shaowei Cai1;2∗ and Chuan Luo3 1State Key Laboratory of Computer Science, Institute of Software, Chinese Academy of Sciences, China 2School of Computer Science and Technology, University of Chinese Academy of Sciences, China 3Microsoft Research, China fleizd, [email protected] Abstract efficient PMS solvers, including SAT-based solvers [Naro- dytska and Bacchus, 2014; Martins et al., 2014; Berg et Satisfiability (SAT) and Maximum Satisfiability al., 2019; Nadel, 2019; Joshi et al., 2019; Demirovic and (MaxSAT) are two basic and important constraint Stuckey, 2019] and local search solvers [Cai et al., 2016; problems with many important applications. SAT Luo et al., 2017; Cai et al., 2017; Lei and Cai, 2018; and MaxSAT are expressed in CNF, which is dif- Guerreiro et al., 2019]. ficult to deal with cardinality constraints. In this One of the most important drawbacks of these logical lan- paper, we introduce Extended Conjunctive Normal guages is the difficulty to deal with cardinality constraints. Form (ECNF), which expresses cardinality con- Indeed, cardinality constraints arise frequently in the encod- straints straightforward and does not need auxiliary ing of many real world situations such as scheduling, logic variables or clauses. Then, we develop a simple and synthesis or verification, product configuration and data min- efficient local search solver LS-ECNF with a well ing. For the above reasons, many works have been done on designed scoring function under ECNF. We also de- finding an efficient encoding of cardinality constraints in CNF velop a generalized Unit Propagation (UP) based formulas [Sinz, 2005; As´ın et al., 2009; Hattad et al., 2017; algorithm to generate the initial solution for local Boudane et al., 2018; Karpinski and Piotrow,´ 2019]. -
Adding the Everywhere Operator to Propositional Logic
Adding the Everywhere Operator to Propositional Logic David Gries∗ and Fred B. Schneider† Computer Science Department, Cornell University Ithaca, New York 14853 USA September 25, 2002 Abstract Sound and complete modal propositional logic C is presented, in which 2P has the interpretation “ P is true in all states”. The interpretation is already known as the Carnapian extension of S5. A new axiomatization for C provides two insights. First, introducing an inference rule textual substitution allows seamless integration of the propositional and modal parts of the logic, giving a more practical system for writing formal proofs. Second, the two following approaches to axiomatizing a logic are shown to be not equivalent: (i) give axiom schemes that denote an infinite number of axioms and (ii) write a finite number of axioms in terms of propositional variables and introduce a substitution inference rule. 1Introduction Logic gives a syntactic way to derive or certify truths that can be expressed in the language of the logic. The expressiveness of the language impacts the logic’s utility —the more expressive the language, the more useful the logic (at least if the intended use is to prove theorems, as opposed to, say, studying logics). We wish to calculate with logic, much as one calculates in algebra to derive equations, and we find it useful for 2P to be a formula of the logic and have the meaning “ P is true in all states”. When a propositional logic extended with 2 is further extended to predicate logic and then to other theories, the logic can be used for proving theorems that could otherwise be ∗Supported by NSF grant CDA-9214957 and ARPA/ONR grant N00014-91-J-4123. -
Logic and Proof
Logic and Proof Computer Science Tripos Part IB Michaelmas Term Lawrence C Paulson Computer Laboratory University of Cambridge [email protected] Copyright c 2007 by Lawrence C. Paulson Contents 1 Introduction and Learning Guide 1 2 Propositional Logic 3 3 Proof Systems for Propositional Logic 13 4 First-order Logic 20 5 Formal Reasoning in First-Order Logic 27 6 Clause Methods for Propositional Logic 33 7 Skolem Functions and Herbrand’s Theorem 41 8 Unification 50 9 Applications of Unification 58 10 BDDs, or Binary Decision Diagrams 65 11 Modal Logics 68 12 Tableaux-Based Methods 74 i ii 1 1 Introduction and Learning Guide This course gives a brief introduction to logic, with including the resolution method of theorem-proving and its relation to the programming language Prolog. Formal logic is used for specifying and verifying computer systems and (some- times) for representing knowledge in Artificial Intelligence programs. The course should help you with Prolog for AI and its treatment of logic should be helpful for understanding other theoretical courses. Try to avoid getting bogged down in the details of how the various proof methods work, since you must also acquire an intuitive feel for logical reasoning. The most suitable course text is this book: Michael Huth and Mark Ryan, Logic in Computer Science: Modelling and Reasoning about Systems, 2nd edition (CUP, 2004) It costs £35. It covers most aspects of this course with the exception of resolution theorem proving. It includes material (symbolic model checking) that should be useful for Specification and Verification II next year. -
Solving QBF by Combining Conjunctive and Disjunctive Normal Forms
Solving QBF with Combined Conjunctive and Disjunctive Normal Form Lintao Zhang Microsoft Research Silicon Valley Lab 1065 La Avenida, Mountain View, CA 94043, USA [email protected] Abstract of QBF solvers have been proposed based on a number of Similar to most state-of-the-art Boolean Satisfiability (SAT) different underlying principles such as Davis-Logemann- solvers, all contemporary Quantified Boolean Formula Loveland (DLL) search (Cadoli et al., 1998, Giunchiglia et (QBF) solvers require inputs to be in the Conjunctive al. 2002a, Zhang & Malik, 2002, Letz, 2002), resolution Normal Form (CNF). Most of them also store the QBF in and expansion (Biere 2004), Binary Decision Diagrams CNF internally for reasoning. In order to use these solvers, (Pan & Vardi, 2005) and symbolic Skolemization arbitrary Boolean formulas have to be transformed into (Benedetti, 2004). Unfortunately, unlike the SAT solvers, equi-satisfiable formulas in Conjunctive Normal Form by which enjoy huge success in solving problems generated introducing additional variables. In this paper, we point out from real world applications, QBF solvers are still only an inherent limitation of this approach, namely the able to tackle trivial problems and remain limited in their asymmetric treatment of satisfactions and conflicts. This deficiency leads to artificial increase of search space for usefulness in practice. QBF solving. To overcome the limitation, we propose to Even though the current QBF solvers are based on many transform a Boolean formula into a combination of an equi- different reasoning principles, they all require the input satisfiable CNF formula and an equi-tautological DNF QBF to be in Conjunctive Normal Form (CNF). -
On the Logic of Left-Continuous T-Norms and Right-Continuous T-Conorms
On the Logic of Left-Continuous t-Norms and Right-Continuous t-Conorms B Llu´ıs Godo1( ),Mart´ın S´ocola-Ramos2, and Francesc Esteva1 1 Artificial Intelligence Research Institute (IIIA-CSIC), Campus de la UAB, Bellaterra, 08193 Barcelona, Spain {godo,esteva}@iiia.csic.es 2 Juniv¨agen 13, 27172 K¨opingebro, Sweden [email protected] Abstract. Double residuated lattices are expansions of residuated lat- tices with an extra monoidal operator, playing the role of a strong dis- junction operation, together with its dual residuum. They were intro- duced by Orlowska and Radzikowska. In this paper we consider the subclass of double residuated structures that are expansions of MTL- algebras, that is, prelinear, bounded, commutative and integral residu- ated lattices. MTL-algebras constitute the algebraic semantics for the MTL logic, the system of mathematical fuzzy logic that is complete w.r.t. the class of residuated lattices on the real unit interval [0, 1] induced by left-continuous t-norms. Our aim is to axiomatise the logic whose intended semantics are commutative and integral double residu- ated structures on [0, 1], that are induced by an arbitrary left-continuous t-norm, an arbitrary right-continuous t-conorm, and their corresponding residual operations. Keywords: Mathematical fuzzy logic · Double residuated lattices · MTL · DMCTL · Semilinear logics · Standard completeness 1 Introduction Residuated lattices [7] are a usual choice for general algebras of truth-values in systems of mathematical fuzzy logic [1–3,9], as well as for valuation structures in lattice-valued fuzzy sets (L-fuzzy sets) [8] and in multiple-valued generalisations of information relations [11,13,14]. -
Lecture 4 1 Overview 2 Propositional Logic
COMPSCI 230: Discrete Mathematics for Computer Science January 23, 2019 Lecture 4 Lecturer: Debmalya Panigrahi Scribe: Kevin Sun 1 Overview In this lecture, we give an introduction to propositional logic, which is the mathematical formaliza- tion of logical relationships. We also discuss the disjunctive and conjunctive normal forms, how to convert formulas to each form, and conclude with a fundamental problem in computer science known as the satisfiability problem. 2 Propositional Logic Until now, we have seen examples of different proofs by primarily drawing from simple statements in number theory. Now we will establish the fundamentals of writing formal proofs by reasoning about logical statements. Throughout this section, we will maintain a running analogy of propositional logic with the system of arithmetic with which we are already familiar. In general, capital letters (e.g., A, B, C) represent propositional variables, while lowercase letters (e.g., x, y, z) represent arithmetic variables. But what is a propositional variable? A propositional variable is the basic unit of propositional logic. Recall that in arithmetic, the variable x is a placeholder for any real number, such as 7, 0.5, or −3. In propositional logic, the variable A is a placeholder for any truth value. While x has (infinitely) many possible values, there are only two possible values of A: True (denoted T) and False (denoted F). Since propositional variables can only take one of two possible values, they are also known as Boolean variables, named after their inventor George Boole. By letting x denote any real number, we can make claims like “x2 − 1 = (x + 1)(x − 1)” regardless of the numerical value of x.