DOCSLIB.ORG

Overview Resolution Proofs Resolution (SAT)

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Overview Resolution Proofs Resolution (SAT) Overview SAT Solving Tremendous gains have been achieved over the ڗ and its last 5 years in SAT solving technology. Systematic backtracking based systems have ڗ Relationship to CSPs become the best method for solving structured SAT instances. New theoretical insights have been gained into ڗ Fahiem Bacchus the behaviour of backtracking SAT solvers via University of Toronto their close relationship with resolution proofs. These insights have direct relevance to finite ڗ domain CSP solvers. 9/15/2005 Fahiem Bacchus 1 9/15/2005 Fahiem Bacchus 2 Resolution (SAT) A complete proof procedure for ڗ Resolution Proofs propositional logic that works on formulas expressed in conjunctive normal form. (Robinson 1965) All (complete) backtracking algorithms (Conjunctive Normal Form (CNF ڗ for CSPs are implicitly generating resolution proofs. Ӎ Literal: a propositional variable p or its On problems with solution, still have negation ¬p to backtrack out of failed subtrees. Ӎ Clause: a disjunction of literals (a set). Baker (1995), Mitchell (2002) Ӎ CNF theory: a conjunction of clauses. 9/15/2005 Fahiem Bacchus 3 9/15/2005 Fahiem Bacchus 4 Resolution Resolution A resolution refutation of a CNF theory ѝ is ڗ From two clauses (A, x) (B, ¬x) the ڗ resolution rule generates the new clause a sequence of clauses C1, C2, …, Cm such that (A, B), where A and B are sets of literals. Ӎ each Ci is either a member of ѝ or Ӎ (A,B) is the resolvant. Ӎ Ci is a resolvant of two previous clauses in the proof: Ci = R(Cj,Ck)j,k< I Ӎ x is the variable resolved on Clauses arising from resolution are called the derived ڗ Ӎ duplicate literals are removed from the clauses of . resolvant Ӎ Cm = () the empty clause. .is also called a resolution proof ڗ Ӎ denote by C = R(C1,C2) .The SIZE of is the number of resolvants in it ڗ 9/15/2005 Fahiem Bacchus 5 9/15/2005 Fahiem Bacchus 6 Resolution DAG Resolution Dag ,(Any resolution proof can be represented () [(¬Y), (Z,¬X),(¬Z,Y),(Y,¬X ڗ (¬X),(Z),(Q,¬Z,X),(¬Q,X), as a DAG. X (¬Z,X),(X),(¬X),()] Ӎ nodes are clauses in the proof. ¬X X Ӎ Every clause Ci that arises from a resolution Y Z step has two incoming edges. One from each Z ¬Z,X of the clauses that were resolved together to ¬Y Y, ¬X obtain Ci. Z Y Q Ӎ The arcs are labeled by the variable that was Z,¬X ¬Z,Y Q,¬Z,X ¬Q,X resolved away to obtain Ci. Ӎ Clauses in ѝ have no incoming edges. 9/15/2005 Fahiem Bacchus 7 9/15/2005 Fahiem Bacchus 8 Restrictions of Resolution Tree Resolution Tree resolution ڗ A number of restricted forms of ڗ resolution can be defined, where, e.g., Ӎ The DAG is required to be a tree. we require the DAG to be a tree. Î Clauses derived during the proof can only be used once. The reason the restricted forms have ڗ been developed is that the restrictions Ӎ Work must be duplicated to rederive clauses that need to be used more than once. can make it easier to find a proof. 9/15/2005 Fahiem Bacchus 9 9/15/2005 Fahiem Bacchus 10 Tree Resolution Ordered Resolution C4 C5 The variables resolved on along any path ڗ C4 C5 in the DAG to the empty clause must C2 C2 C3 C1 C2 C3 C1 respect some fixed ordering of the variables. 9/15/2005 Fahiem Bacchus 11 9/15/2005 Fahiem Bacchus 12 Ordered Resolution Regular Resolution Along any path in the DAG to the empty ڗ Not ordered () X clause the sequence of variables resolved ¬X X away cannot contain any duplicates. Y Z ¬Y Y, ¬X Z ¬Z,X Z Y Q Z,¬X ¬Z,Y Q,¬Z,X ¬Q,X 9/15/2005 Fahiem Bacchus 13 9/15/2005 Fahiem Bacchus 14 Regular Resolution Negative Resolution Q,Z,Y,P Q,Z,Y,P One of the clauses in each resolution ڗ X H step must contain only negative literals. ¬X,Z X,Q,Y,P Z,Q,¬H, (a negative clause) H X Ӎ This is complete! Q,¬H,X H,P,Y H,P,Y ¬X,Z Q,¬H,X Ӎ Note ѝ must contain at least one negative X X clause else the “all true” truth assignment is P,H,X ¬X,Y P,H,X ¬X,Y a satisfying model. Not Regular Regular 9/15/2005 Fahiem Bacchus 15 9/15/2005 Fahiem Bacchus 16 Relative Power Relative Power R(F)—the minimal size R-refutation of ѝ among all# ڗ .A general formalism for comparing the possible R-refutations of ѝ ڗ power of different proof systems was (For a family of formulas ѝi we look at how #R(ѝi ڗ developed by Cook and Reckhow 1997. grows with i. For our purposes we simply look at the ڗ Let S and T be two restrictions of resolution. S ڗ minimal size refutation proof (the p-simulates T if there exists a polytime computable number of clauses in the proof). function f such that: Ӎ For any S-refutation of a formula ѝ, f() is a T-refutation of ѝ. Ӎ Note that this means that f() can’t be more than polynomially longer than Î #T(F) no more than polynomially larger than #S(F) for any formula F. 9/15/2005 Fahiem Bacchus 17 9/15/2005 Fahiem Bacchus 18 Relative Power Known Results Relative Power Known Results Regular Negative Ordered Tree Buresh-Oppenhiem, Pitassi (2003) many ڗ Regular No Yes Yes new results and a summary of previously Negative No No Yes proved results. Ordered No No No Tree No No No Regular always yields shorter proofs than ڗ either Ordered or Tree Negative and Regular are incomparable ڗ .Ordered and Tree are incomparable ڗ 9/15/2005 Fahiem Bacchus 19 9/15/2005 Fahiem Bacchus 20 Relative Power Known Results It is also known that none of these Solving Sat ڗ restrictions can p-simulate general resolution. 9/15/2005 Fahiem Bacchus 21 9/15/2005 Fahiem Bacchus 22 DP & DPLL (DLL) DP Pick a variable ordering (one that has a low ڗ Two earliest algorithms for solving SAT ڗ elimination width if possible): X1, X2, …, Xn actually predate resolution. Starting with the original set of clauses ѝ ڗ :At the i-th stage ڗ DP: Davis-Putnam (1960) a variable ڗ elimination technique. Ӎ Add to ѝ all possible resolvants that can be generated by resolving on Xi. (DPLL: Davis-Logemann-Loveland (1962 ڗ Ӎ Remove from ѝ all clauses containing Xi or ¬Xi. a backtracking search algorithm. Ӎ If the empty clause is generated stop The input set of clauses (the formula ѝ) is ڗ UNSAT iff this process generates the empty clause. 9/15/2005 Fahiem Bacchus 23 9/15/2005 Fahiem Bacchus 24 DP DP [a] [b] [c] [a] [b] [c] (a,b,c) (b,c) (c) () (a,b,c) (b,c) (c) () (¬a,b,c) (¬b, c) (¬c) (¬a,b,c) (¬b, c) (¬c) (¬b, c) (¬b,¬c) (¬b, c) (¬b,¬c) (a,¬b,¬c) (b,¬c) (a,¬b,¬c) (b,¬c) (¬a,¬b,¬c) (¬a,¬b,¬c) (b,¬c) (b,¬c) Potentially many redundant clauses are generated, but an ordered resolution is contained in these clauses. 9/15/2005 Fahiem Bacchus 25 9/15/2005 Fahiem Bacchus 26 DP DPÙOrdered Resolution Note also that every ordered resolution ڗ Every DP proof contains an ordered ڗ resolution, and thus it can never be shorter can be found inside of a DP refutation: than an ordered resolution refutation. Ӎ just follow the same order. Ӎ Note lower bounds are wrt any possible ordered Ӎ Since DP generates all possible ordered resolution (i.e., any ordering). refutations along that order, it might In practice, DP’s space requirements are terminate before completing the specified ڗ prohibitive ordered refutation (by finding a shorter Ӎ Although some attempts using ZBDDs to represent ordered refutation). the clauses compactly. Ӎ DP can also waste a lot of time generating Ӎ Still not competitive with current best techniques. clauses that are not needed for the refutation. 9/15/2005 Fahiem Bacchus 27 9/15/2005 Fahiem Bacchus 28 DPLL DPLL Simplification Given a clausal theory ѝ, we can simplify ڗ Developed shortly after DP, DPLL is based ڗ on backtracking search. The connection it by a literal р as follows: to resolution was realized later. Ӎѝ|р = Ӎ One picks a literal (a true or false variable) Remove from ѝ all clauses containing рڗ .Remove р from all of the remaining clausesڗ Ӎ simplify the formula based on that literal Ӎ recursively solve the simplified formula. Ӎ if the simplified formula is UNSAT, try using the negation of the literal chosen. 9/15/2005 Fahiem Bacchus 29 9/15/2005 Fahiem Bacchus 30 Unit Propagation DPLL (a,b,c) (In addition DPLL employs unit propagation: (¬a,b,c),(¬b, c ڗ (a,¬b,¬c) Ӎ if contains any unit clauses, e.g. (¬x) then ѝ|р (¬a,¬b,¬c) further simplify ѝ|р by the literal in the unit clause, (b,¬c) i.e., generate (ѝ ) |р |¬x C ¬C Unit propagation is the iterative application of this Ӎ (a,b,c) simplification until the resultant theory has no unit b ¬b (a,b,c) clauses (or contains the empty clause). (¬a,b,c),(¬b, c) a a (¬a,b,c),(¬b, c) (a,¬b,¬c) (More powerful forms of propagation examined (a,¬b,¬c ڗ in Bacchus (2002) (¬a,¬b,¬c) (¬a,¬b,¬c) (b,¬c) (b,¬c) 9/15/2005 Fahiem Bacchus 31 9/15/2005 Fahiem Bacchus 32 (a,b,c) DPLL DPLL (¬a,b,c),(¬b, c) () Every DPLL proof contains an tree ڗ (a,¬b,¬c) (¬a,¬b,¬c) C (¬C) ¬C (C) resolution, and thus it can never be (b,¬c) shorter than an tree resolution b: (b,¬c) (¬b,¬c) ¬b: (¬b, c) (b,c) refutation.
Load more

Recommended publications
  • Automated Theorem Proving Introduction
    Automated Theorem Proving Scott Sanner, Guest Lecture Topics in Automated Reasoning Thursday, Jan. 19, 2006 Introduction • Def. Automated Theorem Proving: Proof of mathematical theorems by a computer program. • Depending on underlying logic, task varies from trivial to impossible: – Simple description logic: Poly-time – Propositional logic: NP-Complete (3-SAT) – First-order logic w/ arithmetic: Impossible 1 Applications • Proofs of Mathematical Conjectures – Graph theory: Four color theorem – Boolean algebra: Robbins conjecture • Hardware and Software Verification – Verification: Arithmetic circuits – Program correctness: Invariants, safety • Query Answering – Build domain-specific knowledge bases, use theorem proving to answer queries Basic Task Structure • Given: – Set of axioms (KB encoded as axioms) – Conjecture (assumptions + consequence) • Inference: – Search through space of valid inferences • Output: – Proof (if found, a sequence of steps deriving conjecture consequence from axioms and assumptions) 2 Many Logics / Many Theorem Proving Techniques Focus on theorem proving for logics with a model-theoretic semantics (TBD) • Logics: – Propositional, and first-order logic – Modal, temporal, and description logic • Theorem Proving Techniques: – Resolution, tableaux, sequent, inverse – Best technique depends on logic and app. Example of Propositional Logic Sequent Proof • Given: • Direct Proof: – Axioms: (I) A |- A None (¬R) – Conjecture: |- ¬A, A ∨ A ∨ ¬A ? ( R2) A A ? |- A∨¬A, A (PR) • Inference: |- A, A∨¬A (∨R1) – Gentzen |- A∨¬A, A∨¬A Sequent (CR) ∨ Calculus |- A ¬A 3 Example of First-order Logic Resolution Proof • Given: • CNF: ¬Man(x) ∨ Mortal(x) – Axioms: Man(Socrates) ∀ ⇒ Man(Socrates) x Man(x) Mortal(x) ¬Mortal(y) [Neg. conj.] Man(Socrates) – Conjecture: • Proof: ∃y Mortal(y) ? 1. ¬Mortal(y) [Neg. conj.] 2. ¬Man(x) ∨ Mortal(x) [Given] • Inference: 3.
    [Show full text]
  • Logic and Proof Computer Science Tripos Part IB
    Logic and Proof Computer Science Tripos Part IB Lawrence C Paulson Computer Laboratory University of Cambridge [email protected] Copyright c 2015 by Lawrence C. Paulson Contents 1 Introduction and Learning Guide 1 2 Propositional Logic 2 3 Proof Systems for Propositional Logic 5 4 First-order Logic 8 5 Formal Reasoning in First-Order Logic 11 6 Clause Methods for Propositional Logic 13 7 Skolem Functions, Herbrand’s Theorem and Unification 17 8 First-Order Resolution and Prolog 21 9 Decision Procedures and SMT Solvers 24 10 Binary Decision Diagrams 27 11 Modal Logics 28 12 Tableaux-Based Methods 30 i 1 INTRODUCTION AND LEARNING GUIDE 1 1 Introduction and Learning Guide 2008 Paper 3 Q6: BDDs, DPLL, sequent calculus • 2008 Paper 4 Q5: proving or disproving first-order formulas, This course gives a brief introduction to logic, including • resolution the resolution method of theorem-proving and its relation 2009 Paper 6 Q7: modal logic (Lect. 11) to the programming language Prolog. Formal logic is used • for specifying and verifying computer systems and (some- 2009 Paper 6 Q8: resolution, tableau calculi • times) for representing knowledge in Artificial Intelligence 2007 Paper 5 Q9: propositional methods, resolution, modal programs. • logic The course should help you to understand the Prolog 2007 Paper 6 Q9: proving or disproving first-order formulas language, and its treatment of logic should be helpful for • 2006 Paper 5 Q9: proof and disproof in FOL and modal logic understanding other theoretical courses. It also describes • 2006 Paper 6 Q9: BDDs, Herbrand models, resolution a variety of techniques and data structures used in auto- • mated theorem provers.
    [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]
  • Exploring Semantic Hierarchies to Improve Resolution Theorem Proving on Ontologies
    The University of Maine DigitalCommons@UMaine Honors College Spring 2019 Exploring Semantic Hierarchies to Improve Resolution Theorem Proving on Ontologies Stanley Small University of Maine Follow this and additional works at: https://digitalcommons.library.umaine.edu/honors Part of the Computer Sciences Commons Recommended Citation Small, Stanley, "Exploring Semantic Hierarchies to Improve Resolution Theorem Proving on Ontologies" (2019). Honors College. 538. https://digitalcommons.library.umaine.edu/honors/538 This Honors Thesis is brought to you for free and open access by DigitalCommons@UMaine. It has been accepted for inclusion in Honors College by an authorized administrator of DigitalCommons@UMaine. For more information, please contact [email protected]. EXPLORING SEMANTIC HIERARCHIES TO IMPROVE RESOLUTION THEOREM PROVING ON ONTOLOGIES by Stanley C. Small A Thesis Submitted in Partial Fulfillment of the Requirements for a Degree with Honors (Computer Science) The Honors College University of Maine May 2019 Advisory Committee: Dr. Torsten Hahmann, Assistant Professor1, Advisor Dr. Mark Brewer, Professor of Political Science Dr. Max Egenhofer, Professor1 Dr. Sepideh Ghanavati, Assistant Professor1 Dr. Roy Turner, Associate Professor1 1School of Computing and Information Science ABSTRACT A resolution-theorem-prover (RTP) evaluates the validity (truthfulness) of conjectures against a set of axioms in a knowledge base. When given a conjecture, an RTP attempts to resolve the negated conjecture with axioms from the knowledge base until the prover finds a contradiction. If the RTP finds a contradiction between the axioms and a negated conjecture, the conjecture is proven. The order in which the axioms within the knowledge-base are evaluated significantly impacts the runtime of the program, as the search-space increases exponentially with the number of axioms.
    [Show full text]
  • Deduction and Resolution
    Lecture 12: Deduction and Resolution 1 Hilbert’s System Hilbert discovered a very elegant deductive system that is both sound and com- plete. His system uses only the connectives {¬, →}, which we know are an adequate basis. It has also been shown that each of the three axioms given below are necessary for the system to be sound and complete. Hilbert’s System consists of the following: • Three axioms (all are strongly sound): – A1: (p → (q → p)) Axiom of simplification. – A2: ((p → (q → r)) → ((p → q) → (p → r))) Frege’s axiom. – A3: ((¬p → q) → ((¬p → ¬q) → p)) Reductio ad absurdum. • Inference Rules: p,p→q – Modus Ponens: q (Strongly Sound) ϕ – Substitution: ϕ[θ] , where θ is a substitution (p → ψ). (Sound) The Substitution Inference Rule is shorthand for all sequences of the form hϕ, ϕ[θ]i. It is sometimes called a schema for generating inference rules. Note that in general, it is not the case that ϕ |= ϕ[θ], for example p 6|= q. However, in assignment 2, we showed that if |= ϕ, then |= ϕ[θ]. This is why this rule is sound but not strongly sound. Theorem 1. Hilbert’s system is sound, complete, and tractable. Proof. We prove each property separately. • Soundness: Since all axioms are strongly sound, and the two inference rules are sound, the whole system is sound. • Completeness: The proof of this property is fairly involved and thus it is outside the scope of the class. 1 + • Tractability: Given Γ ∈ Form and ϕ1, . , ϕn, is ϕ1, . , ϕn ∈ Deductions(Γ)? Clearly, < ϕ1 > is in Γ only if it is an axiom.
    [Show full text]
  • Inference (Proof) Procedures for FOL (Chapter 9)
    Inference (Proof) Procedures for FOL (Chapter 9) Torsten Hahmann, CSC384, University of Toronto, Fall 2011 85 Proof Procedures • Interestingly, proof procedures work by simply manipulating formulas. They do not know or care anything about interpretations. • Nevertheless they respect the semantics of interpretations! • We will develop a proof procedure for first-order logic called resolution . • Resolution is the mechanism used by PROLOG • Also used by automated theorem provers such as Prover9 (together with other techniques) Torsten Hahmann, CSC384, University of Toronto, Fall 2011 86 Properties of Proof Procedures • Before presenting the details of resolution, we want to look at properties we would like to have in a (any) proof procedure. • We write KB ⊢ f to indicate that f can be proved from KB (the proof procedure used is implicit). Torsten Hahmann, CSC384, University of Toronto, Fall 2011 87 Properties of Proof Procedures • Soundness • KB ⊢ f → KB ⊨ f i.e all conclusions arrived at via the proof procedure are correct: they are logical consequences. • Completeness • KB ⊨ f → KB ⊢ f i.e. every logical consequence can be generated by the proof procedure. • Note proof procedures are computable, but they might have very high complexity in the worst case. So completeness is not necessarily achievable in practice. Torsten Hahmann, CSC384, University of Toronto, Fall 2011 88 Forward Chaining • First-order inference mechanism (or proof procedure) that generalizes Modus Ponens: • From C1 ∧ … ∧ Cn → Cn+1 with C1, …, Cn in KB, we can infer Cn+1 • We can chain this until we can infer some Ck such that each Ci is either in KB or is the result of Modus Ponens involving only prior Cm (m<i) • We then have that KB ⊢ Ck.
    [Show full text]
  • An Introduction to Prolog
    Appendix A An Introduction to Prolog A.1 A Short Background Prolog was designed in the 1970s by Alain Colmerauer and a team of researchers with the idea – new at that time – that it was possible to use logic to represent knowledge and to write programs. More precisely, Prolog uses a subset of predicate logic and draws its structure from theoretical works of earlier logicians such as Herbrand (1930) and Robinson (1965) on the automation of theorem proving. Prolog was originally intended for the writing of natural language processing applications. Because of its conciseness and simplicity, it became popular well beyond this domain and now has adepts in areas such as: • Formal logic and associated forms of programming • Reasoning modeling • Database programming • Planning, and so on. This chapter is a short review of Prolog. In-depth tutorials include: in English, Bratko (2012), Clocksin and Mellish (2003), Covington et al. (1997), and Sterling and Shapiro (1994); in French, Giannesini et al. (1985); and in German, Baumann (1991). Boizumault (1988, 1993) contain a didactical implementation of Prolog in Lisp. Prolog foundations rest on first-order logic. Apt (1997), Burke and Foxley (1996), Delahaye (1986), and Lloyd (1987) examine theoretical links between this part of logic and Prolog. Colmerauer started his work at the University of Montréal, and a first version of the language was implemented at the University of Marseilles in 1972. Colmerauer and Roussel (1996) tell the story of the birth of Prolog, including their try-and-fail experimentation to select tractable algorithms from the mass of results provided by research in logic.
    [Show full text]
  • Propositional Logic. Inferences
    CS 1571 Introduction to AI Lecture 12 Propositional logic Milos Hauskrecht [email protected] 5329 Sennott Square CS 1571 Intro to AI M. Hauskrecht Logical inference problem Logical inference problem: • Given: – a knowledge base KB (a set of sentences) and – a sentence (called a theorem), • Does a KB semantically entail ? KB | ? In other words: In all interpretations in which sentences in the KB are true, is also true? CS 1571 Intro to AI M. Hauskrecht 1 Sound and complete inference Inference is a process by which conclusions are reached. • We want to implement the inference process on a computer !! Assume an inference procedure i that KB • derives a sentence from the KB : i Properties of the inference procedure in terms of entailment • Soundness: An inference procedure is sound • Completeness: An inference procedure is complete CS 1571 Intro to AI M. Hauskrecht Sound and complete inference. Inference is a process by which conclusions are reached. • We want to implement the inference process on a computer !! Assume an inference procedure i that KB • derives a sentence from the KB : i Properties of the inference procedure in terms of entailment • Soundness: An inference procedure is sound If KB then it is true that KB | i • Completeness: An inference procedure is complete If then it is true that KB KB | i CS 1571 Intro to AI M. Hauskrecht 2 Solving logical inference problem In the following: How to design the procedure that answers: KB | ? Three approaches: • Truth-table approach • Inference rules • Conversion to the inverse SAT problem – Resolution-refutation CS 1571 Intro to AI M.
    [Show full text]
  • Lecture 8: Rule-Based Reasoning
    Set 6: Knowledge Representation: The Propositional Calculus ICS 271 Fall 2013 Kalev Kask Outline • Representing knowledge using logic – Agent that reason logically – A knowledge based agent • Representing and reasoning with logic – Propositional logic • Syntax • Semantic • validity and models • Rules of inference for propositional logic • Resolution • Complexity of propositional inference. • Reading: Russel and Norvig, Chapter 7 Knowledge bases • Knowledge base = set of sentences in a formal language • • Declarative approach to building an agent (or other system): – Tell it what it needs to know – • Then it can Ask itself what to do - answers should follow from the KB • • Agents can be viewed at the knowledge level i.e., what they know, regardless of how implemented • Or at the implementation level – i.e., data structures in KB and algorithms that manipulate them – Knowledge Representation Defined by: syntax, semantics Computer Inference Assertions Conclusions (knowledge base) Semantics Facts Imply Facts Real-World Reasoning: in the syntactic level Example:x y, y z | x z The party example • If Alex goes, then Beki goes: A B • If Chris goes, then Alex goes: C A • Beki does not go: not B • Chris goes: C • Query: Is it possible to satisfy all these conditions? • Should I go to the party? Example of languages • Programming languages: – Formal languages, not ambiguous, but cannot express partial information. Not expressive enough. • Natural languages: – Very expressive but ambiguous: ex: small dogs and cats. • Good representation language:
    [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]
  • Solvers for the Problem of Boolean Satisfiability (SAT)
    Solvers for the Problem of Boolean Satisfiability (SAT) Will Klieber 15-414 Aug 31, 2011 Why study SAT solvers? Many problems reduce to SAT. Formal verification CAD, VLSI Optimization AI, planning, automated deduction Modern SAT solvers are often fast. Other solvers (QBF, SMT, etc.) borrow techniques from SAT solvers. SAT solvers and related solvers are still active areas of research. 2 Negation-Normal Form (NNF) A formula is in negation-normal form iff: all negations are directly in front of variables, and the only logical connectives are: “ ∧”, “∨”, “ ¬”. A literal is a variable or its negation. Convert to NNF by pushing negations inward: ¬(P ∧ Q) ⇔ (¬P ∨ ¬Q) (De Morgan’s Laws) ¬(P ∨ Q) ⇔ (¬P ∧ ¬Q) 3 Disjunctive Normal Form (DNF) Recall: A literal is a variable or its negation. A formula is in DNF iff: it is a disjunction of conjunctions of literals. (ℓ ∧ ℓ ∧ ℓ ) ∨ (ℓ ∧ ℓ ∧ ℓ ) ∨ (ℓ ∧ ℓ ∧ ℓ ) 11 12 13 21 22 23 31 32 33 conjunction 1 conjunction 2 conjunction 3 Every formula in DNF is also in NNF. A simple (but inefficient) way convert to DNF: Make a truth table for the formula φ. Each row where φ is true corresponds to a conjunct. 4 Conjunctive Normal Form (CNF) A formula is in CNF iff: it is a conjunction of disjunctions of literals. (ℓ ∨ ℓ ∨ ℓ ) ∧ (ℓ ∨ ℓ ∨ ℓ ) ∧ (ℓ ∨ ℓ ∨ ℓ ) 11 12 13 21 22 23 31 32 33 clause 1 clause 2 clause 3 Modern SAT solvers use CNF. Any formula can be converted to CNF. Equivalent CNF can be exponentially larger.
    [Show full text]
  • Traditional Logic and Computational Thinking
    philosophies Article Traditional Logic and Computational Thinking J.-Martín Castro-Manzano Faculty of Philosophy, UPAEP University, Puebla 72410, Mexico; [email protected] Abstract: In this contribution, we try to show that traditional Aristotelian logic can be useful (in a non-trivial way) for computational thinking. To achieve this objective, we argue in favor of two statements: (i) that traditional logic is not classical and (ii) that logic programming emanating from traditional logic is not classical logic programming. Keywords: syllogistic; aristotelian logic; logic programming 1. Introduction Computational thinking, like critical thinking [1], is a sort of general-purpose thinking that includes a set of logical skills. Hence logic, as a scientific enterprise, is an integral part of it. However, unlike critical thinking, when one checks what “logic” means in the computational context, one cannot help but be surprised to find out that it refers primarily, and sometimes even exclusively, to classical logic (cf. [2–7]). Classical logic is the logic of Boolean operators, the logic of digital circuits, the logic of Karnaugh maps, the logic of Prolog. It is the logic that results from discarding the traditional, Aristotelian logic in order to favor the Fregean–Tarskian paradigm. Classical logic is the logic that defines the standards we typically assume when we teach, research, or apply logic: it is the received view of logic. This state of affairs, of course, has an explanation: classical logic works wonders! However, this is not, by any means, enough warrant to justify the absence or the abandon of Citation: Castro-Manzano, J.-M. traditional logic within the computational thinking literature.
    [Show full text]