The Samson-Mills-Quine theorem
|
|
- Ashlie Lambert
- 6 years ago
- Views:
Transcription
1 The Samson-Mills-Quine theorem Intermediate Logic The dnf simplification procedure can greatly simplify the complexity of truth-functional formulas, but it does not always turn up a simplest possible Boolean equivalent of a formula. The formula, (p q) ( p q) (q r) ( q r) is an illustrative example. It cannot be further simplified by applying any of the four transformations of the dnf simplification procedure, yet the dnf formula (p q) (q r) ( p r) is truth-functionally equivalent to it and simpler. This note describes a method for finding, for any truth functional formula, a simplest Boolean equivalent. The method is a modification, due to the American logician W. V. O. Quine, of a procedure for electrical circuit minimization discovered by E. W. Samson and B. E. Mills. The proof that the method works is essentially Quine s. It is evident from the above example why the dnf simplification procedure fails to generate the a simplest possible equivalent. That procedure allows one only ever to erase literals or clauses of a pre-recorded formula, while the second formula above contains a clause (the third) that is not a sub-formula of any of the first formula s clauses. One might reasonably wonder where that clause came from, and how one might discover it. To address that wonder, it is helpful to reflect a bit on some basic properties of dnf formulas. Every disjunct of a disjunction implies the whole disjunction. This fact is evident from the definition of disjunction. In particular, then, every clause of a dnf formula implies the whole formula. Inspired by this observation, let us define successively 1
2 1. a conjunction block to be a formula that is a conjunction of literals, in which no literal occurs twice 2. an implicant of a formula to be a conjunction block that implies that formula 3. two conjunction blocks are opposed in a sentence letter if that sentence letter appears as a conjunct of one and its negation appears as a conjunct of the other 4. one conjunction block subsumes another if they are opposed in no sentence letter, and every conjunct of the second is also a conjunct of the first 5. a prime implicant of a formula φ to be an implicant of φ that subsumes no other implicant of φ. It is immediately clear that any simplest dnf equivalent of a formula φ will be a disjunction of prime implicants of φ, for if one disjunct of a dnf formula failed to be prime, then the fourth transformation of the dnf simplification procedure would be applicable, in effect replacing that disjunct with one that it subsumes (i.e., a shorter one). Thus, the dnf simplification procedure can be seen as a way of reducing any dnf formula to an equivalent dnf formula that is a disjunction of some of its prime implicants. It fails to reliably generate a simplest possible dnf equivalent because there may be other prime implicants which, if appended to the formula as additional disjuncts, would allow a greater number of clauses to be erased. The above example illustrates this general phenomenon. Since (p q) ( p q) (q r) ( q r) cannot be further simplified by the dnf simplification procedure, all of its clauses must be prime implicants. But one can verify that the formulas p r and p r are also prime implicants, so that the formula (p q) ( p q) (q r) ( q r) (p r) ( p r) is equivalent to the original formula. Clearly, the fifth and sixth clauses of this new formula could now be erased by successive applications of rule three (each implies the original four clause formula, and thus the fifth implies 2
3 the result of deleting itself from the new six clause formula, and the sixth similarly implies the result of deleting itself from the resulting five clause formula.) But applying rule three differently allows one instead to preserve the sixth clause and erase the second, fourth and fifth, resulting in (p q) (q r) ( p r). Exercise: The last formula is indeed a simplest possible equivalent of the original one. There is another equally simple equivalent. Find it by choosing a different sequence of applications of rule three. Now, a satisfiable formula φ is equivalent to the disjunction of all its prime implicants (an unsatisfiable formula has no prime implicants, and gets obliterated by the dnf simplification procedure anyway.) For φ is implied by each of its prime implicants, so it is implied by their disjunction. And conversely, every interpretation that satisfies φ satisfies some conjunction block that implies φ and thus also some prime implicant of φ. Therefore every interpretation that satisfies φ satisfies also the disjunction of its prime implicants. Thus, to find a simplest dnf equivalent of a formula, is suffices to find all of its prime implicants. Is this a coherent task? One might wonder whether every truth functional formula has prime implicants and whether, assuming that a formula φ does have prime implicants, whether it might have infinitely many such. But these wonders subside when we observe, first, that the truth table of a satisfiable formula indicates some of its implicants (in each row of the table where φ is true, the list of T s and F s in the reference columns describe how to build an implicant of φ: conjoin all the sentence letters interpreted as T with the negation of all sentence letters interpreted as F if these implicants aren t prime, then some sub-formulas of them are), and second, that prime implicants can contain neither two occurrences of the same sentence letter nor any sentence letter that doesn t appear in φ (if a conjunction block contains both p and p, then it is unsatisfiable and therefore does not qualify as an implicant any dnf formula with such a block as one of its clauses is equivalent to the formula that results when that clause is erased). The method of Samson and Mills is an algorithm for finding all of the prime implicants of a dnf formula. To describe it, we need one more definition: 3
4 6 If two conjunction blocks φ and ψ are opposed in exactly one sentence letter, then their consensus is the result of deleting the two opposed literals and all repeated literals from φ ψ Notice that the consensus of two conjunction blocks is another conjunction block. Our algorithm has only two transformation rules: i Erase any clause of φ that subsumes another. ii Adjoin to φ as an additional clause the consensus of any two clauses whose consensus doesn t subsume a clause that s already present. Observe that these two transformations, applied to a dnf formula φ, preserve the property of being in dnf and also preserve equivalence. The Samson-Mills-Quine theorem says that successive applications of these two rules will inevitably result in a formula that is the disjunction of all the prime implicants of φ. Proof: The theorem is a consequence of three lemmas. Lemma 1: A dnf formula φ remains susceptible to transformation (ii) so long as some prime implicant of it is not among its clauses. Lemma 2: A dnf formula φ remains susceptible either to transformation (i) or to transformation (ii) or to both so long as some clause of it is not one of its prime implicants. Lemma 3: It is not possible to apply transformations (i) and (ii) indefinitely. Thus, no matter how one applies the transformations (i) and (ii) to a formula φ, eventually one will reach a point where it is not possible to do anything more, and at that point, all and only prime implicants of φ will appear as clauses of the dnf equivalent of φ that one has produced. To prove the first lemma, let χ be a prime implicant of φ not appearing as a clause of φ. (Notice that, for there to be such, φ cannot be valid.) Since χ is prime, it subsumes no clause of φ. Therefore there is at least one conjunction block with these three properties: 4
5 a it subsumes X b it subsumes no clause of φ c it contains no letter not in φ (After all, χ itself is such a formula.) Let ψ be a longest such conjunction block. (Should one think that, for any conjunction block with these three properties, there is another that is larger still, observe that there are only finitely many sentence letters in φ, and that no implicant of φ can contain a sentence letter other than one of these and still be prime.) Notice that ψ does not contain every sentence letter in φ for if it did, then by (b), it would oppose every clause of φ in at least one letter and thus not imply φ, contradicting (a). Let p be a sentence letter contained by φ but not by ψ. Since ψ was a longest formula with properties (a), (b), and (c), the formulas p ψ and p ψ cannot satisfy all three properties. But both these formulas clearly satisfy (a) and (c), so they must both fail (b). Thus there are clauses C 1 and C 2 of φ subsumed respectively by p ψ and p ψ, although ψ subsumes neither. So C 1 must contain p; C 2, p. Since all their other literals are in ψ, C 1 and C 2 cannot be opposed in any letter other than p, though they must each contain letters other than p in order that φ not be valid. i.e., C 1 and C 2 have a consensus. Call it ω. Observe that ω subsumes no clause of φ, for ψ subsumes ω and ψ doesn t (by (b)). φ is therefore susceptible to transformation (ii). To prove lemma 2, let C be the clause of φ that is not one of its prime implicants. Becase φ in in dnf, C is one of its implicants. But C is not itself prime, so C subsumes some prime implicant χ of φ. Is χ already a clause of φ? If so, then it is possible to apply transformation (i) to φ by erasing C. If not, then not all of φ s prime implicants appear as clauses of φ, and therefore φ is susceptible to transformation (ii), by lemma 1. Lemma 3 is proved with two observations. First, only a finite number of clauses can ever be appended with applications of transformation (ii), and no one of these can ever be appended twice. The number of append-able clauses is finite, because any such clause will have to be built up from sentence letters appearing in the original φ. No clause can be twice appended because, at any time one considers appending 5
6 a clause for the second time, either it will already appear in the formula or, if it doesn t, it will have once appeared but since been erased. In the latter case, some clause still in the current formula will necessarily be subsumed by the clause. In neither case is transformation (ii) applicable. Second, the subtractive nature of transformation (i) delimits the number of times it can be applied. For after one has applied transformation (ii) for the last time, which time one must reach eventually in light of the last observation, one can apply transformation (i) at most as many times as there are clauses in the current formula. This completes the proof of the Samson-Mills-Quine theorem. Notice how unhelpful the proof is: It sheds no light on how one might best go about applying the transformations in order to attain prime implicants and obliterate other clauses expeditiously. In fact, it is not at all clear from the proof why it is that prime implicants turn up as the algorithm runs. The consensus of two clauses might be prime, but more often it will not be it will generally even be longer than both of the clauses it is the consensus of. The theorem seems almost miraculous, and, for this reason, I allege that the proof is worth some philosophical attention. Indeed, a ballistic application of the transformation rules might be very inefficient. They are not set-up for efficiency, but to make the proof as transparent as possible. However, one can approach the task of finding all of a formula s prime implicants prudently, by conjoining the second transformation rule with the original rules from the dnf simplification procedure. More precisely, if one simplifies a dnf formula as much as possible with the old procedure, then one will be left with a disjunction, each clause of which is a prime implicant. To find the remaining implicants, a relatively small number of consensus taking measures need to be done. Indeed, this is how the fifth and sixth clauses ( p r and p r ) in the above example were generated. In the formula: (p q) ( p q) (q r) ( q r), p r is the consensus of the first and third clauses, and p r is the consensus of the second and fourth. After these two clauses are appended to yield (p q) ( p q) (q r) ( q r) (p r) ( p r), consensuses of new pairs of clauses (the second and fifth, as well as the third and sixth) can be taken, but further application of transformation (ii) is 6
7 nevertheless impossible because these consensuses already appear as clauses (as the third and second). Thus we can be assured that the last formula is a disjunction of all its own prime implicants. This whole enterprise of consensus taking appears to be overly complex and unnecessary when one observes a more easily describable way to generate a list of all a formula s prime implicants. Any prime implicant of φ must be built out of sentence letters appearing in φ, and there are only finitely many conjunction blocks built out of these sentence letters. One could simply write down all the possible conjunction blocks, test each to see if it is an implicant, and then pare down the list of implicants by checking for pairwise subsumption. The problem with this procedure is that the number of conjunction blocks to consider is super-exponential in the number of sentence letters that appear in the formula. More precisely, for a formula with n sentence letters, one must check n i=0 ( n i) (2 i ) conjunction blocks. Thus, for example, with seven sentence letters, one must check 2186 conjunction blocks. An efficient use of the consensus method would turn up those prime implicants in much less computation time than would the brute force checking method. On the other hand, it is conceivable that a method might be discovered for finding simplest dnf equivalents that bypasses any need to find all prime implicants. The sheer number of a formula s prime implicants can itself be exponential in the number of the formula s sentence letters (according to an analysis of a seven switch Boolean circuit by Fridshal, as many as 1698 different formulas could be prime implicants of a seven letter formula). A direct simplification method that bypasses any need to consider exponentially more data than a given formula contains would be a discovery of great practical and theoretical interest. 7
The Samson-Mills-Quine theorem
The Samson-Mills-Quine theorem Intermediate Logic The dnf simplification procedure can greatly simplify the complexity of truth-functional formulas, but it does not always turn up a simplest possible Boolean
More informationECE473 Lecture 15: Propositional Logic
ECE473 Lecture 15: Propositional Logic Jeffrey Mark Siskind School of Electrical and Computer Engineering Spring 2018 Siskind (Purdue ECE) ECE473 Lecture 15: Propositional Logic Spring 2018 1 / 23 What
More informationPropositional Logic Basics Propositional Equivalences Normal forms Boolean functions and digital circuits. Propositional Logic.
Propositional Logic Winter 2012 Propositional Logic: Section 1.1 Proposition A proposition is a declarative sentence that is either true or false. Which ones of the following sentences are propositions?
More informationLecture 9: The Splitting Method for SAT
Lecture 9: The Splitting Method for SAT 1 Importance of SAT Cook-Levin Theorem: SAT is NP-complete. The reason why SAT is an important problem can be summarized as below: 1. A natural NP-Complete problem.
More informationCritical Reading of Optimization Methods for Logical Inference [1]
Critical Reading of Optimization Methods for Logical Inference [1] Undergraduate Research Internship Department of Management Sciences Fall 2007 Supervisor: Dr. Miguel Anjos UNIVERSITY OF WATERLOO Rajesh
More informationEquivalences. Proposition 2.8: The following equivalences are valid for all formulas φ, ψ, χ: (φ φ) φ. Idempotency Idempotency Commutativity
Substitution Theorem Proposition 2.7: Let φ 1 and φ 2 be equivalent formulas, and ψ[φ 1 ] p be a formula in which φ 1 occurs as a subformula at position p. Then ψ[φ 1 ] p is equivalent to ψ[φ 2 ] p. 84
More informationUnary negation: T F F T
Unary negation: ϕ 1 ϕ 1 T F F T Binary (inclusive) or: ϕ 1 ϕ 2 (ϕ 1 ϕ 2 ) T T T T F T F T T F F F Binary (exclusive) or: ϕ 1 ϕ 2 (ϕ 1 ϕ 2 ) T T F T F T F T T F F F Classical (material) conditional: ϕ 1
More informationSection 1.2 Propositional Equivalences. A tautology is a proposition which is always true. A contradiction is a proposition which is always false.
Section 1.2 Propositional Equivalences A tautology is a proposition which is always true. Classic Example: P P A contradiction is a proposition which is always false. Classic Example: P P A contingency
More informationCS 512, Spring 2017, Handout 10 Propositional Logic: Conjunctive Normal Forms, Disjunctive Normal Forms, Horn Formulas, and other special forms
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
More informationTopic 1: Propositional logic
Topic 1: Propositional logic Guy McCusker 1 1 University of Bath Logic! This lecture is about the simplest kind of mathematical logic: propositional calculus. We discuss propositions, which are statements
More informationComp487/587 - Boolean Formulas
Comp487/587 - Boolean Formulas 1 Logic and SAT 1.1 What is a Boolean Formula Logic is a way through which we can analyze and reason about simple or complicated events. In particular, we are interested
More informationAutomated Program Verification and Testing 15414/15614 Fall 2016 Lecture 2: Propositional Logic
Automated Program Verification and Testing 15414/15614 Fall 2016 Lecture 2: Propositional Logic Matt Fredrikson mfredrik@cs.cmu.edu October 17, 2016 Matt Fredrikson Propositional Logic 1 / 33 Propositional
More information1 Propositional Logic
CS 2800, Logic and Computation Propositional Logic Lectures Pete Manolios Version: 384 Spring 2011 1 Propositional Logic The study of logic was initiated by the ancient Greeks, who were concerned with
More informationTruth-Functional Logic
Truth-Functional Logic Syntax Every atomic sentence (A, B, C, ) is a sentence and are sentences With ϕ a sentence, the negation ϕ is a sentence With ϕ and ψ sentences, the conjunction ϕ ψ is a sentence
More information2.5.2 Basic CNF/DNF Transformation
2.5. NORMAL FORMS 39 On the other hand, checking the unsatisfiability of CNF formulas or the validity of DNF formulas is conp-complete. For any propositional formula φ there is an equivalent formula in
More informationLecture 2. Logic Compound Statements Conditional Statements Valid & Invalid Arguments Digital Logic Circuits. Reading (Epp s textbook)
Lecture 2 Logic Compound Statements Conditional Statements Valid & Invalid Arguments Digital Logic Circuits Reading (Epp s textbook) 2.1-2.4 1 Logic Logic is a system based on statements. A statement (or
More informationDescription Logics. Foundations of Propositional Logic. franconi. Enrico Franconi
(1/27) Description Logics Foundations of Propositional Logic Enrico Franconi franconi@cs.man.ac.uk http://www.cs.man.ac.uk/ franconi Department of Computer Science, University of Manchester (2/27) Knowledge
More informationCS 486: Lecture 2, Thursday, Jan 22, 2009
CS 486: Lecture 2, Thursday, Jan 22, 2009 Mark Bickford January 22, 2009 1 Outline Propositional formulas Interpretations and Valuations Validity and Satisfiability Truth tables and Disjunctive Normal
More informationLogic: Propositional Logic (Part I)
Logic: Propositional Logic (Part I) Alessandro Artale Free University of Bozen-Bolzano Faculty of Computer Science http://www.inf.unibz.it/ artale Descrete Mathematics and Logic BSc course Thanks to Prof.
More informationCSE20: Discrete Mathematics for Computer Science. Lecture Unit 2: Boolan Functions, Logic Circuits, and Implication
CSE20: Discrete Mathematics for Computer Science Lecture Unit 2: Boolan Functions, Logic Circuits, and Implication Disjunctive normal form Example: Let f (x, y, z) =xy z. Write this function in DNF. Minterm
More informationCHAPTER 12 Boolean Algebra
318 Chapter 12 Boolean Algebra CHAPTER 12 Boolean Algebra SECTION 12.1 Boolean Functions 2. a) Since x 1 = x, the only solution is x = 0. b) Since 0 + 0 = 0 and 1 + 1 = 1, the only solution is x = 0. c)
More informationPart 1: Propositional Logic
Part 1: Propositional Logic Literature (also for first-order logic) Schöning: Logik für Informatiker, Spektrum Fitting: First-Order Logic and Automated Theorem Proving, Springer 1 Last time 1.1 Syntax
More informationLogic and Inferences
Artificial Intelligence Logic and Inferences Readings: Chapter 7 of Russell & Norvig. Artificial Intelligence p.1/34 Components of Propositional Logic Logic constants: True (1), and False (0) Propositional
More informationSubstitution Theorem. Equivalences. Proposition 2.8 The following equivalences are valid for all formulas φ, ψ, χ: (φ φ) φ Idempotency
Substitution Theorem Proposition 2.7 Let φ 1 and φ 2 be equivalent formulas, and ψ[φ 1 ] p be a formula in which φ 1 occurs as a subformula at position p. Then ψ[φ 1 ] p is equivalent to ψ[φ 2 ] p. Proof.
More informationPropositional and Predicate Logic - II
Propositional and Predicate Logic - II Petr Gregor KTIML MFF UK WS 2016/2017 Petr Gregor (KTIML MFF UK) Propositional and Predicate Logic - II WS 2016/2017 1 / 16 Basic syntax Language Propositional logic
More informationCSC Discrete Math I, Spring Propositional Logic
CSC 125 - Discrete Math I, Spring 2017 Propositional Logic Propositions A proposition is a declarative sentence that is either true or false Propositional Variables A propositional variable (p, q, r, s,...)
More informationIntroduction to Metalogic
Philosophy 135 Spring 2008 Tony Martin Introduction to Metalogic 1 The semantics of sentential logic. The language L of sentential logic. Symbols of L: Remarks: (i) sentence letters p 0, p 1, p 2,... (ii)
More information1.1 Statements and Compound Statements
Chapter 1 Propositional Logic 1.1 Statements and Compound Statements A statement or proposition is an assertion which is either true or false, though you may not know which. That is, a statement is something
More informationC241 Homework Assignment 4
C241 Homework Assignment 4 1. Which of the following formulas are tautologies and which are contradictions? Which of the formulas are logically equivalent to each other? (a) p (q r) (b) ( p r) (q r) (c)
More informationConjunctive Normal Form and SAT
Notes on Satisfiability-Based Problem Solving Conjunctive Normal Form and SAT David Mitchell mitchell@cs.sfu.ca September 19, 2013 This is a preliminary draft of these notes. Please do not distribute without
More information2.2: Logical Equivalence: The Laws of Logic
Example (2.7) For primitive statement p and q, construct a truth table for each of the following compound statements. a) p q b) p q Here we see that the corresponding truth tables for two statement p q
More informationTautologies, Contradictions, and Contingencies
Section 1.3 Tautologies, Contradictions, and Contingencies A tautology is a proposition which is always true. Example: p p A contradiction is a proposition which is always false. Example: p p A contingency
More informationTecniche di Verifica. Introduction to Propositional Logic
Tecniche di Verifica Introduction to Propositional Logic 1 Logic A formal logic is defined by its syntax and semantics. Syntax An alphabet is a set of symbols. A finite sequence of these symbols is called
More informationPhilosophy 220. Truth-Functional Equivalence and Consistency
Philosophy 220 Truth-Functional Equivalence and Consistency Review Logical equivalency: The members of a pair of sentences [of natural language] are logically equivalent if and only if it is not [logically]
More information3 The Semantics of the Propositional Calculus
3 The Semantics of the Propositional Calculus 1. Interpretations Formulas of the propositional calculus express statement forms. In chapter two, we gave informal descriptions of the meanings of the logical
More informationCOMP9414: Artificial Intelligence Propositional Logic: Automated Reasoning
COMP9414, Monday 26 March, 2012 Propositional Logic 2 COMP9414: Artificial Intelligence Propositional Logic: Automated Reasoning Overview Proof systems (including soundness and completeness) Normal Forms
More informationLogic Design I (17.341) Fall Lecture Outline
Logic Design I (17.341) Fall 2011 Lecture Outline Class # 06 October 24, 2011 Dohn Bowden 1 Today s Lecture Administrative Main Logic Topic Homework 2 Course Admin 3 Administrative Admin for tonight Syllabus
More informationA Little Deductive Logic
A Little Deductive Logic In propositional or sentential deductive logic, we begin by specifying that we will use capital letters (like A, B, C, D, and so on) to stand in for sentences, and we assume that
More informationConjunctive Normal Form and SAT
Notes on Satisfiability-Based Problem Solving Conjunctive Normal Form and SAT David Mitchell mitchell@cs.sfu.ca October 4, 2015 These notes are a preliminary draft. Please use freely, but do not re-distribute
More informationSec$on Summary. Tautologies, Contradictions, and Contingencies. Logical Equivalence. Normal Forms (optional, covered in exercises in text)
Section 1.3 1 Sec$on Summary Tautologies, Contradictions, and Contingencies. Logical Equivalence Important Logical Equivalences Showing Logical Equivalence Normal Forms (optional, covered in exercises
More informationPart 1: Propositional Logic
Part 1: Propositional Logic Literature (also for first-order logic) Schöning: Logik für Informatiker, Spektrum Fitting: First-Order Logic and Automated Theorem Proving, Springer 1 Last time 1.1 Syntax
More informationCHAPTER 6 - THINKING ABOUT AND PRACTICING PROPOSITIONAL LOGIC
1 CHAPTER 6 - THINKING ABOUT AND PRACTICING PROPOSITIONAL LOGIC Here, you ll learn: what it means for a logic system to be finished some strategies for constructing proofs Congratulations! Our system of
More informationFormal Verification Methods 1: Propositional Logic
Formal Verification Methods 1: Propositional Logic John Harrison Intel Corporation Course overview Propositional logic A resurgence of interest Logic and circuits Normal forms The Davis-Putnam procedure
More informationBoolean Algebra CHAPTER 15
CHAPTER 15 Boolean Algebra 15.1 INTRODUCTION Both sets and propositions satisfy similar laws, which are listed in Tables 1-1 and 4-1 (in Chapters 1 and 4, respectively). These laws are used to define an
More informationConjunctive Normal Form and SAT
Notes on Satisfiability-Based Problem Solving Conjunctive Normal Form and SAT David Mitchell mitchell@cs.sfu.ca September 10, 2014 These notes are a preliminary draft. Please use freely, but do not re-distribute
More informationCHAPTER 2 INTRODUCTION TO CLASSICAL PROPOSITIONAL LOGIC
CHAPTER 2 INTRODUCTION TO CLASSICAL PROPOSITIONAL LOGIC 1 Motivation and History The origins of the classical propositional logic, classical propositional calculus, as it was, and still often is called,
More informationExercises 1 - Solutions
Exercises 1 - Solutions SAV 2013 1 PL validity For each of the following propositional logic formulae determine whether it is valid or not. If it is valid prove it, otherwise give a counterexample. Note
More informationPropositional Logic. Testing, Quality Assurance, and Maintenance Winter Prof. Arie Gurfinkel
Propositional Logic Testing, Quality Assurance, and Maintenance Winter 2018 Prof. Arie Gurfinkel References Chpater 1 of Logic for Computer Scientists http://www.springerlink.com/content/978-0-8176-4762-9/
More information1 FUNDAMENTALS OF LOGIC NO.10 HERBRAND THEOREM Tatsuya Hagino hagino@sfc.keio.ac.jp lecture URL https://vu5.sfc.keio.ac.jp/slide/ 2 So Far Propositional Logic Logical connectives (,,, ) Truth table Tautology
More informationNatural Deduction for Propositional Logic
Natural Deduction for Propositional Logic Bow-Yaw Wang Institute of Information Science Academia Sinica, Taiwan September 10, 2018 Bow-Yaw Wang (Academia Sinica) Natural Deduction for Propositional Logic
More informationcse541 LOGIC FOR COMPUTER SCIENCE
cse541 LOGIC FOR COMPUTER SCIENCE Professor Anita Wasilewska Spring 2015 LECTURE 2 Chapter 2 Introduction to Classical Propositional Logic PART 1: Classical Propositional Model Assumptions PART 2: Syntax
More informationNormal Forms of Propositional Logic
Normal Forms of Propositional Logic Bow-Yaw Wang Institute of Information Science Academia Sinica, Taiwan September 12, 2017 Bow-Yaw Wang (Academia Sinica) Normal Forms of Propositional Logic September
More informationA Little Deductive Logic
A Little Deductive Logic In propositional or sentential deductive logic, we begin by specifying that we will use capital letters (like A, B, C, D, and so on) to stand in for sentences, and we assume that
More informationBoolean Algebra. Philipp Koehn. 9 September 2016
Boolean Algebra Philipp Koehn 9 September 2016 Core Boolean Operators 1 AND OR NOT A B A and B 0 0 0 0 1 0 1 0 0 1 1 1 A B A or B 0 0 0 0 1 1 1 0 1 1 1 1 A not A 0 1 1 0 AND OR NOT 2 Boolean algebra Boolean
More informationWarm-Up Problem. Is the following true or false? 1/35
Warm-Up Problem Is the following true or false? 1/35 Propositional Logic: Resolution Carmen Bruni Lecture 6 Based on work by J Buss, A Gao, L Kari, A Lubiw, B Bonakdarpour, D Maftuleac, C Roberts, R Trefler,
More informationLogical Design of Digital Systems
Lecture 4 Table of Content 1. Combinational circuit design 2. Elementary combinatorial circuits for data transmission 3. Memory structures 4. Programmable logic devices 5. Algorithmic minimization approaches
More informationSymbolic Logic 3. For an inference to be deductively valid it is impossible for the conclusion to be false if the premises are true.
Symbolic Logic 3 Testing deductive validity with truth tables For an inference to be deductively valid it is impossible for the conclusion to be false if the premises are true. So, given that truth tables
More informationDigital Logic Design ENEE x. Lecture 12
Digital Logic Design ENEE 244-010x Lecture 12 Announcements HW5 due today HW6 up on course webpage, due at the beginning of class on Wednesday, 10/28. Agenda Last time: Quine-McClusky (4.8) Petrick s Method
More informationPropositional Calculus: Formula Simplification, Essential Laws, Normal Forms
P Formula Simplification, Essential Laws, Normal Forms Lila Kari University of Waterloo P Formula Simplification, Essential Laws, Normal CS245, Forms Logic and Computation 1 / 26 Propositional calculus
More informationPropositional and Predicate Logic - V
Propositional and Predicate Logic - V Petr Gregor KTIML MFF UK WS 2016/2017 Petr Gregor (KTIML MFF UK) Propositional and Predicate Logic - V WS 2016/2017 1 / 21 Formal proof systems Hilbert s calculus
More informationPropositional logic. Programming and Modal Logic
Propositional logic Programming and Modal Logic 2006-2007 4 Contents Syntax of propositional logic Semantics of propositional logic Semantic entailment Natural deduction proof system Soundness and completeness
More informationAgenda. Artificial Intelligence. Reasoning in the Wumpus World. The Wumpus World
Agenda Artificial Intelligence 10. Propositional Reasoning, Part I: Principles How to Think About What is True or False 1 Introduction Álvaro Torralba Wolfgang Wahlster 2 Propositional Logic 3 Resolution
More informationLogical Agent & Propositional Logic
Logical Agent & Propositional Logic Berlin Chen 2005 References: 1. S. Russell and P. Norvig. Artificial Intelligence: A Modern Approach. Chapter 7 2. S. Russell s teaching materials Introduction The representation
More informationOn the Structure and the Number of Prime Implicants of 2-CNFs
On the Structure and the Number of Prime Implicants of 2-CNFs Navid Talebanfard Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Meguro-ku Ookayama 2-12-1, Japan 152-8552
More informationCHAPTER 4 CLASSICAL PROPOSITIONAL SEMANTICS
CHAPTER 4 CLASSICAL PROPOSITIONAL SEMANTICS 1 Language There are several propositional languages that are routinely called classical propositional logic languages. It is due to the functional dependency
More information7. Propositional Logic. Wolfram Burgard and Bernhard Nebel
Foundations of AI 7. Propositional Logic Rational Thinking, Logic, Resolution Wolfram Burgard and Bernhard Nebel Contents Agents that think rationally The wumpus world Propositional logic: syntax and semantics
More informationChapter 2. Reductions and NP. 2.1 Reductions Continued The Satisfiability Problem (SAT) SAT 3SAT. CS 573: Algorithms, Fall 2013 August 29, 2013
Chapter 2 Reductions and NP CS 573: Algorithms, Fall 2013 August 29, 2013 2.1 Reductions Continued 2.1.1 The Satisfiability Problem SAT 2.1.1.1 Propositional Formulas Definition 2.1.1. Consider a set of
More informationInduction on Failing Derivations
Induction on Failing Derivations Technical Report PL-Sep13 September 2013, with addenda from Spring 2016 ay Ligatti Department of Computer Science and Engineering University of South Florida Abstract A
More informationResolution for Predicate Logic
Resolution for Predicate Logic The connection between general satisfiability and Herbrand satisfiability provides the basis for a refutational approach to first-order theorem proving. Validity of a first-order
More informationMathematics 114L Spring 2018 D.A. Martin. Mathematical Logic
Mathematics 114L Spring 2018 D.A. Martin Mathematical Logic 1 First-Order Languages. Symbols. All first-order languages we consider will have the following symbols: (i) variables v 1, v 2, v 3,... ; (ii)
More informationFoundations of Artificial Intelligence
Foundations of Artificial Intelligence 7. Propositional Logic Rational Thinking, Logic, Resolution Wolfram Burgard, Maren Bennewitz, and Marco Ragni Albert-Ludwigs-Universität Freiburg Contents 1 Agents
More informationFoundations of Artificial Intelligence
Foundations of Artificial Intelligence 7. Propositional Logic Rational Thinking, Logic, Resolution Joschka Boedecker and Wolfram Burgard and Bernhard Nebel Albert-Ludwigs-Universität Freiburg May 17, 2016
More informationPropositional Equivalence
Propositional Equivalence Tautologies and contradictions A compound proposition that is always true, regardless of the truth values of the individual propositions involved, is called a tautology. Example:
More informationA non-classical refinement of the interpolation property for classical propositional logic
Accepted for publication in Logique & Analyse A non-classical refinement of the interpolation property for classical propositional logic Peter Milne Abstract We refine the interpolation property of the
More informationSection 1.1: Logical Form and Logical Equivalence
Section 1.1: Logical Form and Logical Equivalence An argument is a sequence of statements aimed at demonstrating the truth of an assertion. The assertion at the end of an argument is called the conclusion,
More informationDecision Procedures for Satisfiability and Validity in Propositional Logic
Decision Procedures for Satisfiability and Validity in Propositional Logic Meghdad Ghari Institute for Research in Fundamental Sciences (IPM) School of Mathematics-Isfahan Branch Logic Group http://math.ipm.ac.ir/isfahan/logic-group.htm
More informationCSE 20 DISCRETE MATH WINTER
CSE 20 DISCRETE MATH WINTER 2016 http://cseweb.ucsd.edu/classes/wi16/cse20-ab/ Reminders Exam 1 in one week One note sheet ok Review sessions Saturday / Sunday Assigned seats: seat map on Piazza shortly
More informationPropositional and First Order Reasoning
Propositional and First Order Reasoning Terminology Propositional variable: boolean variable (p) Literal: propositional variable or its negation p p Clause: disjunction of literals q \/ p \/ r given by
More informationA brief introduction to Logic. (slides from
A brief introduction to Logic (slides from http://www.decision-procedures.org/) 1 A Brief Introduction to Logic - Outline Propositional Logic :Syntax Propositional Logic :Semantics Satisfiability and validity
More informationA Lower Bound of 2 n Conditional Jumps for Boolean Satisfiability on A Random Access Machine
A Lower Bound of 2 n Conditional Jumps for Boolean Satisfiability on A Random Access Machine Samuel C. Hsieh Computer Science Department, Ball State University July 3, 2014 Abstract We establish a lower
More informationPacket #1: Logic & Proofs. Applied Discrete Mathematics
Packet #1: Logic & Proofs Applied Discrete Mathematics Table of Contents Course Objectives Page 2 Propositional Calculus Information Pages 3-13 Course Objectives At the conclusion of this course, you should
More informationThe statement calculus and logic
Chapter 2 Contrariwise, continued Tweedledee, if it was so, it might be; and if it were so, it would be; but as it isn t, it ain t. That s logic. Lewis Carroll You will have encountered several languages
More informationPROPOSITIONAL CALCULUS
PROPOSITIONAL CALCULUS A proposition is a complete declarative sentence that is either TRUE (truth value T or 1) or FALSE (truth value F or 0), but not both. These are not propositions! Connectives and
More informationPropositional Logic: Models and Proofs
Propositional Logic: Models and Proofs C. R. Ramakrishnan CSE 505 1 Syntax 2 Model Theory 3 Proof Theory and Resolution Compiled at 11:51 on 2016/11/02 Computing with Logic Propositional Logic CSE 505
More informationInference in Propositional Logic
Inference in Propositional Logic Deepak Kumar November 2017 Propositional Logic A language for symbolic reasoning Proposition a statement that is either True or False. E.g. Bryn Mawr College is located
More informationSupplementary exercises in propositional logic
Supplementary exercises in propositional logic The purpose of these exercises is to train your ability to manipulate and analyze logical formulas. Familiarize yourself with chapter 7.3-7.5 in the course
More informationMore Propositional Logic Algebra: Expressive Completeness and Completeness of Equivalences. Computability and Logic
More Propositional Logic Algebra: Expressive Completeness and Completeness of Equivalences Computability and Logic Equivalences Involving Conditionals Some Important Equivalences Involving Conditionals
More informationThe Calculus of Computation: Decision Procedures with Applications to Verification. Part I: FOUNDATIONS. by Aaron Bradley Zohar Manna
The Calculus of Computation: Decision Procedures with Applications to Verification Part I: FOUNDATIONS by Aaron Bradley Zohar Manna 1. Propositional Logic(PL) Springer 2007 1-1 1-2 Propositional Logic(PL)
More informationDeductive Systems. Lecture - 3
Deductive Systems Lecture - 3 Axiomatic System Axiomatic System (AS) for PL AS is based on the set of only three axioms and one rule of deduction. It is minimal in structure but as powerful as the truth
More informationLecture 2: Proof of Switching Lemma
Lecture 2: oof of Switching Lemma Yuan Li June 25, 2014 1 Decision Tree Instead of bounding the probability that f can be written as some s-dnf, we estimate the probability that f can be computed by a
More informationDeMorgan s Laws and the Biconditional. Philosophy and Logic Sections 2.3, 2.4 ( Some difficult combinations )
DeMorgan s aws and the Biconditional Philosophy and ogic Sections 2.3, 2.4 ( Some difficult combinations ) Some difficult combinations Not both p and q = ~(p & q) We won t both sing and dance. A negation
More informationLecture 2 Propositional Logic & SAT
CS 5110/6110 Rigorous System Design Spring 2017 Jan-17 Lecture 2 Propositional Logic & SAT Zvonimir Rakamarić University of Utah Announcements Homework 1 will be posted soon Propositional logic: Chapter
More informationReview CHAPTER. 2.1 Definitions in Chapter Sample Exam Questions. 2.1 Set; Element; Member; Universal Set Partition. 2.
CHAPTER 2 Review 2.1 Definitions in Chapter 2 2.1 Set; Element; Member; Universal Set 2.2 Subset 2.3 Proper Subset 2.4 The Empty Set, 2.5 Set Equality 2.6 Cardinality; Infinite Set 2.7 Complement 2.8 Intersection
More information6. Logical Inference
Artificial Intelligence 6. Logical Inference Prof. Bojana Dalbelo Bašić Assoc. Prof. Jan Šnajder University of Zagreb Faculty of Electrical Engineering and Computing Academic Year 2016/2017 Creative Commons
More informationChapter 2: Introduction to Propositional Logic
Chapter 2: Introduction to Propositional Logic PART ONE: History and Motivation Origins: Stoic school of philosophy (3rd century B.C.), with the most eminent representative was Chryssipus. Modern Origins:
More informationPropositional Logic: Evaluating the Formulas
Institute for Formal Models and Verification Johannes Kepler University Linz VL Logik (LVA-Nr. 342208) Winter Semester 2015/2016 Propositional Logic: Evaluating the Formulas Version 2015.2 Armin Biere
More informationMathematical Logic Propositional Logic - Tableaux*
Mathematical Logic Propositional Logic - Tableaux* Fausto Giunchiglia and Mattia Fumagalli University of Trento *Originally by Luciano Serafini and Chiara Ghidini Modified by Fausto Giunchiglia and Mattia
More information~ p is always false. Based on the basic truth table for disjunction, if q is true then p ~
MAT 101 Solutions Exam 2 (Logic, Part I) Multiple-Choice Questions 1. D Because this sentence contains exactly ten words, it is stating that it is false. But if it is taken to be false, then it has to
More informationAI Principles, Semester 2, Week 2, Lecture 5 Propositional Logic and Predicate Logic
AI Principles, Semester 2, Week 2, Lecture 5 Propositional Logic and Predicate Logic Propositional logic Logical connectives Rules for wffs Truth tables for the connectives Using Truth Tables to evaluate
More informationCMSC 858F: Algorithmic Lower Bounds Fall SAT and NP-Hardness
CMSC 858F: Algorithmic Lower Bounds Fall 2014 3-SAT and NP-Hardness Instructor: Mohammad T. Hajiaghayi Scribe: Philip Dasler September 23, 2014 The most important NP-Complete (logic) problem family! 1
More information