1 / 38

Artificial Intelligence Chapter 13 The Propositional Calculus

Artificial Intelligence Chapter 13 The Propositional Calculus. Biointelligence Lab School of Computer Sci. & Eng. Seoul National University. Outline. Using Constraints on Feature Values The Language Rules of Inference Definition of Proof Semantics Soundness and Completeness

gaetan
Download Presentation

Artificial Intelligence Chapter 13 The Propositional Calculus

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Artificial IntelligenceChapter 13The Propositional Calculus Biointelligence Lab School of Computer Sci. & Eng. Seoul National University

  2. Outline • Using Constraints on Feature Values • The Language • Rules of Inference • Definition of Proof • Semantics • Soundness and Completeness • The PSAT Problem • Other Important Topics (c) 1999-2010 SNU CSE Biointelligence Lab

  3. 13.1 Using Constraints on Feature Values • Two different methods for modeling an agent’s world • Feature-based representation : feature vectors • Iconic representation • E.g. maps • Simulations of important aspects of the environment • Sometimes called analogical representations (c) 1999-2010 SNU CSE Biointelligence Lab

  4. Feature based representation– descriptions of the world • Binary-valued features: what is true about the world and what is not true • easy to communicate • In cases where the values of some features cannot be sensed directly, their values can be inferred from the values of other features using constraints (c) 1999-2010 SNU CSE Biointelligence Lab

  5. Iconic representation – simulations of certain aspects of the world • data structures and computations that simulate • aspects of an agent’s environment • the effects of agent actions upon that environment • more direct and more efficient • Requires elaborate processing for construction or modification (c) 1999-2010 SNU CSE Biointelligence Lab

  6. 13.1 Using Constraints on Feature Values (Cont’d) • Difficult or impossible environment to represent iconically • General laws, such as “all blue boxes are pushable” • Negative information, such as “block A is not on the floor” (without saying where block A is) • Uncertain information, such as “either block A is on block B or block A is on block C” • Some of this difficult-to-represent information can be formulated as constraints on the values of features • These constraints can be used to infer the values of features that cannot be sensed directly. • Reasoning • inferring information about an agent’s personal state (c) 1999-2010 SNU CSE Biointelligence Lab

  7. 13.1 Using Constraints on Feature Values (Cont’d) • Applications involving reasoning • Reasoning can enhance the effectiveness of agents • To diagnose malfunction in various physical systems • Represent the functioning of the systems by appropriate set of features • Constraints among features encode physical laws relevant to the organism or device. • Features associated with “causes” can be inferred from features associated with “symptoms,” • Expert Systems (c) 1999-2010 SNU CSE Biointelligence Lab

  8. 13.1 Using Constraints on Feature Values (Cont’d) • Motivating example for reasoning techniques • Consider a robot that is able to lift a block Moves Liftable Causal relation diagram Not Liftable Battery Gauge for battery (c) 1999-2010 SNU CSE Biointelligence Lab

  9. 13.1 Using Constraints on Feature Values (Cont’d) • Condition 1: The block is liftable • Condition 2: The robot’s battery power source is adequate • If both are satisfied, then when the robot tries to lift a block it is holding, its arm moves. • Representing conditions by binary-valued features • x1 (BAT_OK) • x2 (LIFTABLE) • x3 (MOVES) xi = 0 or 1 (c) 1999-2010 SNU CSE Biointelligence Lab

  10. 13.1 Using Constraints on Feature Values (Cont’d) • Let the robot do reasoning • For this we need • Language – expressing constraints and values of features • Inference mechanisms – performing required reasoning • One possible tool for this: propositional calculus • A descendant of Boolean algebra • Expressing the constraint of the example in the language of the propositional calculus BAT_OK  LIFTABLE  MOVES (c) 1999-2010 SNU CSE Biointelligence Lab

  11. 13.1 Using Constraints on Feature Values (Cont’d) • Logic involves • A language (with a syntax – what is a legal expression) • Inference rule • Semantics for associating elements of the language with elements of some subject matter • Two logical languages that you will learn • propositional calculus • 13.2 Language • 13.3 Rules of Inference, Ch 14. Resolution • 13.5 Semantics • first-order predicate calculus (FOPC) • Ch 15, Ch. 16 (c) 1999-2010 SNU CSE Biointelligence Lab

  12. Note on Notations • In the textbook • Logical expressions: typewriter font • Symbols that stand for logical expressions: lowercase Greek letters • α, β, γ, … (c) 1999-2010 SNU CSE Biointelligence Lab

  13. 13.2 The Language - Elements • Atoms • two distinguished atoms T and F and • the countably infinite set of those strings of characters that begin with a capital letter, • for example, P, Q, R, …, P1, P2, ON_A_B, and so on. • Connectives • , , , and , called “or”, “and”, “implies”, and “not”, respectively. (c) 1999-2010 SNU CSE Biointelligence Lab

  14. 13.2 The Language - Elements • Syntax of well-formed formula (wff), also called sentences • Any atom is a wff. • If ω1 and ω2 are wffs, so are • ω1ω2 (disjunction) • ω1ω2 (conjunction) • ω1ω2(implication) • ω1(negation) • There are no other wffs. • Example: P    is not a wff (c) 1999-2010 SNU CSE Biointelligence Lab

  15. 13.2 The Language (Cont’d) • Literal • atoms and a  sign in front of them • AntecedentandConsequent • In ω1ω2, ω1is called the antecedent of the implication. • ω2 is called the consequent of the implication. • Extra-linguistic separators, ( and ) • group wffs into (sub) wffs according to the recursive definitions. (c) 1999-2010 SNU CSE Biointelligence Lab

  16. 13.3 Rule of Inference • Ways by which additional wffs can be produced from other ones • Commonly used rules • modus ponens: wff ω2 can be inferred from the wffs ω1 and ω1ω2 •  introduction: wff ω1ω2 can be inferred from the two wffs ω1 and ω2 • commutativity : wff ω2ω1 can be inferred from the wff ω1ω2 •  elimination: wff ω1 can be inferred from the ω1ω2 •  introduction: wff ω1ω2 can be inferred from either from the single wff ω1 or from the single wff ω2 •  elimination: wff ω1 can be inferred from the wff  (ω1 ). (c) 1999-2010 SNU CSE Biointelligence Lab

  17. 13.4 Definitions of Proof • Proof • The sequence of wffs {ω1, ω2, …, ωn} is called a proof (ora deduction) of ωn from a set of wffs  iff each ωi is either in  or can be inferred from a wff earlier in the sequence by using one of the rules of inference. • Theorem • If there is a proof of ωn from , ωn is a theorem of the set . •  ㅏ ωn • Denote the set of inference rules by the letter R. •  ㅏRωn • ωn can be proved from  using the inference rules in R (c) 1999-2010 SNU CSE Biointelligence Lab

  18. Example • Given a set, , of wffs: {P, R, P  Q}, • {P, P  Q, Q, R, Q  R} is a proof of Q  R. • The concept of proof can be based on a partial order. Figure 13.1 A Sample Proof Tree (c) 1999-2010 SNU CSE Biointelligence Lab

  19. 13.5 Semantics • Talking about “meanings” • Semantics • Has to do with associating elements of a logical language with elements of a domain of discourse. • Meaning - Such associations • Interpretation • An association of atoms with propositions • Denotation • In a given interpretation, the proposition associated with an atom (c) 1999-2010 SNU CSE Biointelligence Lab

  20. 13.5 Semantics (Cont’d) • Under a given interpretation, atoms have values – True or False. • Special Atom • T : always has value True • F : always has value False • An interpretation by assigning values directly to the atoms in a language can be specified – regardless of which proposition about the world each atom denotes. (c) 1999-2010 SNU CSE Biointelligence Lab

  21. Propositional Truth Table • Given the values of atoms under some interpretation, use a truth table to compute a value for any wff under that same interpretation. • Let ω1 and ω2 be wffs. • (ω1ω2) has True if both ω1 and ω2 have value True. • (ω1ω2) has True if one or both ω1 or ω2 have value True. • ω1 has value True if ω1 has value False. • The semantics of  is defined in terms of  and . Specifically, (ω1ω2) is an alternative and equivalent form of ( ω1ω2) . (c) 1999-2010 SNU CSE Biointelligence Lab

  22. Propositional Truth Table (Cont’d) • If an agent describes its world using n features and these features are represented in the agent’s model of the world by a corresponding set of n atoms, then there are 2n different ways its world can be. • Given values for the n atoms, the agent can use the truth table to find the values of any wffs. • Suppose the values of wffs in a set of wffs are given. • Do those values induce a unique interpretation? • Usually “No.” • Instead, there may be many interpretations that give each wff in a set of wffs the value True . (c) 1999-2010 SNU CSE Biointelligence Lab

  23. Satisfiability • An interpretation satisfies a wff if the wff is assigned the value True under that interpretation. • Model • An interpretation that satisfies a wff • In general, the more wffs that describe the world, the fewer models. • Inconsistent or Unsatisfiable • When no interpretation satisfies a wff, the wff is inconsistent or unsatisfiable. • e.g. F or P  P BAT_OK  LIFTABLE  MOVES (c) 1999-2010 SNU CSE Biointelligence Lab

  24. Validity • A wff is said to be valid • It has value Trueunderall interpretations of its constituent atoms. • e.g. • P  P • T •  ( P  P ) • Q  T • [(P  Q)  P]  P • P  (Q  P) • Use of the truth table to determine the validity of a wff takes time exponential in the number of atoms (c) 1999-2010 SNU CSE Biointelligence Lab

  25. Equivalence • Two wffs are said to be equivalent iff their truth values are identical under all interpretations. • DeMorgan’s laws (ω1ω2)  ω1ω2 (ω1ω2)  ω1ω2 • Law of the contrapositive (ω1ω2)  (ω2 ω1) • If ω1 and ω2 are equivalent, then the following formula is valid: (ω1ω2)  (ω2 ω1)  abbreviated as ω1 ω2 (c) 1999-2010 SNU CSE Biointelligence Lab

  26. Entailment • If a wff ω has value True under all of interpretations for which each of the wffs in a set  has value True,  logically entailsω and ωlogically follows from  and ωis a logical consequence of . • e.g. • {P}ㅑP • {P, P  Q} ㅑQ • F ㅑω(any wff) • P  QㅑP {BAT_OK, MOVES, BAT_OK  LIFTABLE  MOVES} ㅑLIFTABLE (c) 1999-2010 SNU CSE Biointelligence Lab

  27. Entailment & Inference • Importance of entailment in AI • provides a very strong way of showing that if certain propositions are true about a world, then some other propositions of interest must also be true • Powerful tool for determining the truth or falsity of propositions about the world • Study of the following themes are important • How to represent information as wffs • How to produce entailed wffs efficiently • We can always to this using the truth table method • Inference is an attractive substitute for entailment • They are linked by the concepts of soundness and completeness (c) 1999-2010 SNU CSE Biointelligence Lab

  28. 13.6 Soundness and Completeness • If, for any set of wffs, , and wff, ω, ㅏRω implies ㅑω, the set of inference rules, R, issound. • If, for any set of wffs, , and wff, ω, it is the case that whenever ㅑω, there exist a proof of ω from  using the set of inference rules, we say that R is complete. • When inference rules aresound and complete, we can determine whether one wff follows from a set of wffs by searching for a proof (instead of by using the truth table). (c) 1999-2010 SNU CSE Biointelligence Lab

  29. 13.6 Soundness and Completeness (Cont’d) • When the inference rules are sound, if we can find a proof of ω from , ω logically follows from . • When the inference rules are complete, we will eventually be able to confirm that ω follows from  by using a complete search procedure to search for a proof. • Substituting proof methods for truth table methods usually gives great computational advantage • To determine whether or not a wff logically follows from a set of wffs or can be proved from a set of wffs is, in general, an NP-hard problem. (c) 1999-2010 SNU CSE Biointelligence Lab

  30. 13.7 The PSAT Problem • Propositional satisfiability (PSAT) problem: the problem of finding a model for a formula. • Clause - a disjunction of literals • Conjunctive Normal Form (CNF) • A formula written as a conjunction of clauses • An exhaustive procedure for solving the CNF PSAT problem is to try systematically all of the ways to assign True and False to the atoms in the formula. • If there are n atoms in the formula, there are 2n different assignments • For large n, it is computationally infeasible ex) BAT_OK  LIFTABLE ex) (BAT_OK MOVES)  (LIFTABLE  MOVES) (c) 1999-2010 SNU CSE Biointelligence Lab

  31. 13.7 The PSAT Problem (Cont’d) • Interesting special cases of the PSAT problem • 2SAT and 3SAT • kSAT problem • To find a model for a conjunction of clauses, the longest of which contains exactly k literals • 2SAT • Polynomial complexity • 3SAT • NP-complete • Many problems take only polynomial expected time. (c) 1999-2010 SNU CSE Biointelligence Lab

  32. 13.7 The PSAT Problem (Cont’d) • GSAT (Greedy SAT) • Nonexhaustive, greedy, hill-climbing type of search procedure • Begin by selecting a random set of values for all of the atoms in the formula. • The number of clauses having value True under this interpretation is noted. • Next, go through the list of atoms and calculate, for each one, the increase in the number of clauses whose values would be True if the value of that atom were to be changed. • Change the value of that atom giving the largest increase • Terminated after some fixed number of changes • May terminate at a local maximum (c) 1999-2010 SNU CSE Biointelligence Lab

  33. 13.8 Other Important Topics13.8.1 Language Distinctions • The propositional calculus is a formal language that an artificial agent uses to describe its world. • Possibility of confusing the informal languages of mathematics and of English with the formal language of the propositional calculus itself. • {P, P  Q} ㅏ Q • ㅏ is not a symbol in the language of propositional calculus • It is a symbol in language used to talk about the propositional calculus • ㅏ, ㅑ: metalinguistic symbols • Never be confused with the symbol  (c) 1999-2010 SNU CSE Biointelligence Lab

  34. 13.8.2 Metatheorems • Theorems about the propositional calculus • Important Theorems • Deductive theorem • If {ω 1, ω2, …, ωn}ㅑω, (ω 1 ω2 … ωn) ωis valid. • Vice versa • Reductio ad absurdum (Latin: "reduction to the absurd") • If the set  has a model but   {ω} does not, then  ㅑ ω. (c) 1999-2010 SNU CSE Biointelligence Lab

  35. 13.8.3 Associative Laws and Distributive Laws • Associative Laws (ω1ω2) ω3 ω1 ( ω2ω3) (ω1ω2) ω3 ω1 ( ω2ω3) • Distributive Laws ω1 (ω2ω3)  (ω1ω2)(ω1ω3) ω1 (ω2ω3)  (ω1ω2)(ω1ω3) (c) 1999-2010 SNU CSE Biointelligence Lab

  36. Appendix1: Review of the NP-family • NP-hard (non-deterministic polynomial-time hard) • As hard as the hardest problems in NP. Such problems need not be in NP; indeed, they may not even be decision problems. • NP-complete • These are the hardest problems in NP. Such a problem is NP-hard and in NP. • NP-easy • At most as hard as NP, but not necessarily in NP, since they may not be decision problems. • NP-equivalent • Exactly as difficult as the hardest problems in NP, but not necessarily in NP. NP is the set of all decision problems for which the 'yes'-answers have efficiently verifiable proofs of the fact that the answer is indeed 'yes'. source: http://en.wikipedia.org/wiki/NP-hard (c) 1999-2010 SNU CSE Biointelligence Lab

  37. Appendix 2: GSAT Example • Flip most-improving variable • What if first guess is A=1, B=1, C=1? • 2 clauses satisfied • Flip A to 0  3 clauses satisfied • Flip B to 0  all 4 clauses satisfied (pick this!) • Flip C to 0  3 clauses satisfied • But what if first guess is A=0, B=1, C=1? • 3 clauses satisfied • Flip A to 1  3 clauses satisfied • Flip B to 0  3 clauses satisfied • Flip C to 0  3 clauses satisfied • Pick one anyway … (picking A wins on next step) A v C A v B ~C v ~B ~B v ~A (c) 1999-2010 SNU CSE Biointelligence Lab

  38. (c) 1999-2010 SNU CSE Biointelligence Lab

More Related