Lecture 2 Logic and reasoning 第 2 讲逻辑及其推理

Lecture 2 Logic and reasoning第2讲逻辑及其推理 2.1 Propositional Logic 命题逻辑 2.2 Predicate Calculus 谓词逻辑

Acknowledgment • robotics.stanford.edu/~latombe/cs121/2003/home.htm 补充读物：Nils J. Nilsson著, 郑扣根庄越挺译 Artificial Intelligence A New Synthesis 机械工业出版社 2000 (中南大学图书馆 TP18 NES) • www.cs.umbc.edu/~ypeng/F02671/lecture-notes/Ch06.ppt

sensors environment ? agent actuators Knowledge base Propositional Logic 命题逻辑 Knowledge-Based Agent

Types of Knowledge • Procedural(过程性的), e.g.: functionsSuch knowledge can only be used in one way -- by executing it • Declarative(描述性的，说明性的), e.g.: constraintsIt can be used to perform many different sorts of inferences

Logic Logic is a declarative language to: • Assert sentences representing facts that hold in a world W (these sentences are given the value true) • Deduce the true/false values to sentences representing other aspects of W

涵蕴 Entail Sentences Sentences ╞ represent represent Conceptualization World W Facts about W Facts about W hold hold Connection World-Representation

1 Battery-OK  Bulbs-OK  Headlights-Work 2 Battery-OK  Starter-OK Empty-Gas-Tank  Engine-Starts 3 Engine-Starts Flat-Tire  Car-OK 4 Headlights-Work 5 Battery-OK 6 Starter-OK 7 Empty-Gas-Tank 8 Car-OK Conceptualization World W ╞ Flat-Tire 1. 如果电池正常且车灯正常，那么车灯亮 2. 如果电池正常，启动器正常且油箱不空，那么引擎可以发动 3. 如果引擎可以发动，且轮胎没有爆胎，那么汽车正常 4. 车灯亮 5. 电池正常 6.启动器正常 7. 油箱不空 8. 汽车不正常诊断结论：轮胎爆胎

Propositional logic: Syntax • Propositional logic is the simplest logic – illustrates basic ideas • The proposition symbols P1, P2 etc are sentences • If S is a sentence, S is a sentence (negation) • If S1 and S2 are sentences, S1 S2 is a sentence (conjunction) • If S1 and S2 are sentences, S1 S2 is a sentence (disjunction) • If S1 and S2 are sentences, S1 S2 is a sentence (implication) • If S1 and S2 are sentences, S1 S2 is a sentence (biconditional)

Propositional logic: Semantics Each model specifies true/false for each proposition symbol E.g. P1,2 P2,2 P3,1 false true false With these symbols, 8 possible models, can be enumerated automatically. Rules for evaluating truth with respect to a model m: S is true iff S is false S1 S2 is true iff S1 is true andS2 is true S1 S2 is true iff S1is true or S2 is true S1 S2 is true iff S1 is false orS2 is true i.e., is false iff S1 is true andS2 is false S1 S2 is true iff S1S2 is true andS2S1 is true

Model • Assignment of a truth value – true or false – to every atomic sentence • Examples: • Let A, B, C, and D be the propositional symbols • m = {A=true, B=false, C=false, D=true} is a model • m’ = {A=true, B=false, C=false} is not a model • With n propositional symbols, one can define 2n models

Model of a KB • Let KBbe a set of sentences • A model m is a model of KB iff it is a model of all sentences in KB, that is, all sentences in KB are true in m • Given a vocabulary A, B, C and D, how many models for A^B  C are there? • for A^B  B?

Model of a A^B  C • A^B  C有多少模型 • {A^B  C， A^B}有多少模型

valid sentenceor tautology Satisfiability of a KB （可满足性） A KB is satisfiable iff it admits(接纳) at least one model; otherwise it is unsatisfiable KB1 = {P, QR} is satisfiableKB2 = {PP} is satisfiable KB3 = {P, P} is unsatisfiable

Logical Entailment （逻辑涵蕴） • KB : set of sentences （知识库） • : arbitrary sentence • KB entails– written KB ╞– iff every model of KB is also a model of  • Alternatively, KB ╞ iff • {KB,} is unsatisfiable (不可满足的) • KB   is valid （永真的）

Model of a A^B  C • 令KB= {A^B  C， A^B} •  = C • 有KB ╞

Validity(永真性) and satisfiability(可满足性) A sentence is valid if it is true in all models, e.g., True, A A, A  A, (A  (A  B))  B Validity is connected to inference via the Deduction Theorem: （演绎定理） KB ╞ α if and only if (KB α) is valid A sentence is satisfiable if it is true in some model e.g., A B, C A sentence is unsatisfiable if it is true in no models e.g., AA Satisfiability is connected to inference via the following: KB ╞ α if and only if (KBα) is unsatisfiable

Inference • I: Set of inference rules • KB: Set of sentences • Inferenceis the process of applying successive inference rules from I to KB, each rule adding its conclusion to KB （推理就是不断将I中的规则应用到KB，并将其结论加入到KB中的过程。）

Inference Problem • Given: • KB: a set of sentence • : a sentence • Answer: • KB  ?

Inference Rule （推理规则） • An inference rule {, } consists of 2 sentence patterns  and  called the conditions and one sentence pattern  called the conclusion • If  and  match two sentences of KB then the corresponding  can be inferred according to the rule 

{  , }   {, }  Example: Modus Ponens （假言推理） Example: Battery-OK  Bulbs-OK  Headlights-Work Battery-OK  Bulbs-OK Headlights-Work

Sound Inference Rules (deductive rules) • Here are some examples of sound rules of inference. • Each can be shown to be sound using a truth table -- a rule is sound if it’s conclusion is true whenever the premise is true. RULEPREMISE CONCLUSION Modus Ponens（肯定前件） A, A => B B Modus Tollens （否定后件）~B, A => B ~A And Introduction A, B A ^ B And Elimination A ^ B A Or Introduction A A v B Double Negation ~~A A Chaining A => B, B => C A => C

 Connective symbol (implication 蕴含)╞ Logical entailment（逻辑涵蕴） Inference（推理） KB ╞ iff KB   is valid 

Soundness （可靠性） • An inference rule is sound if it generates only entailed sentences （如果一个推理规则只推出蕴含的句子，则为可靠的） • All inference rules previously given are sound, e.g.:modus ponens: {   , }  • The following rule:{   , }  is unsound, which does not mean it is useless （溯因推理，abduction）  

Completeness （完备性） • A set of inference rules is complete if every entailed sentences can be obtained by applying some finite succession of these rules （所有的被蕴含的句子都可被推出，则该规则集为完备的） • Modus ponens alone is not complete, e.g.:from A  B and B, we cannot get A

Use inference rules: generate new sentences X from a one or more existing sentences S. S is called the premise and X the conclusion of the rule. • Proof procedure: a set of inference rules and a procedure of how to use these rules • If X can be generated from S by proof procedure i, we say X is derived from S by i, denoted S|iX, or S | X. • Soundness. An inference procedure is sound if every sentence X it produces from a set of sentences S logically follows from S. (No contradiction is created). if S | X then S |= X • Completeness. A inference procedure is complete, if it is able to produce every sentence that logically follows from any give S. if S |= X then S | X

Proof methods • Proof methods divide into (roughly) two kinds: • Application of inference rules • Legitimate (sound) generation of new sentences from old • Proof = a sequence of inference rule applications Can use inference rules as operators in a standard search algorithm • Typically require transformation of sentences into a normal form • Model checking • truth table enumeration (always exponential in n) • improved backtracking, e.g., Davis--Putnam-Logemann-Loveland (DPLL) • heuristic search in model space (sound but incomplete) e.g., min-conflicts-like hill-climbing algorithms

Proof The proof of a sentence  from a set of sentences KB is the derivation of by applying a series of sound inference rules

Proof The proof of a sentence  from a set of sentences KB is the derivation of by applying a series of sound inference rules Battery-OK  Bulbs-OK  Headlights-Work Battery-OK  Starter-OK Empty-Gas-Tank  Engine-Starts Engine-Starts Flat-Tire  Car-OK Headlights-Work Battery-OK Starter-OK Empty-Gas-Tank Car-OK Battery-OK  Starter-OK  (5+6) Battery-OK  Starter-OK Empty-Gas-Tank  (9+7) Engine-Starts  (2+10) Engine-Starts  Flat-Tire  (3+8) Flat-Tire  (11+12)

一种新的推理规则： Resolution 消解，归结

Complementary Literals • A literal is a either an atomic sentence or the negated atomic sentence, e.g.: P or P • Two literals are complementary if one is the negation of the other, e.g.: P and P

Normal forms of PL sentences • Disjunctive normal form (DNF) • Any sentence can be written as a disjunction of conjunctions of literals. • Examples: P ^ Q ^ ~R; A^B v C^D v P^Q^R; P • Widely used in logical circuit design (simplification) • Conjunctive normal form (CNF) • Any sentence can be written as a conjunction of disjunctions of literals. • Examples: P v Q v ~R; (A v B) ^ (C v D) ^ (P v Q v R); P • Normal forms can be obtained by applying equivalence laws [(A v B) => (C v D)] => P  ~[~(A v B) v (C v D)] v P // law for implication  [~~(A v B) ^ ~(C v D)] v P // de Morgan’s law  [(A v B)^(~C ^ ~D)] v P // double negation and de Morgan’s law  (A v B v P)^(~C^~D v P) // distribution law (A v B v P)^(~C v P)^(~D v P) // a CNF

Unit Resolution（消解，归结）Rule • Given two sentences:L1… Lp and Mwhere Li,…, Lp and M are all literals, and M and Li are complementary literals • Infer:L1…Li-1Li+1… Lp

Engine-Starts  Car-OK Examples From:Engine-Starts  Car-OK Engine-Starts Infer:Car-OK Modus ponens From:Engine-Starts  Car-OK Car-OK Infer:Engine-Starts Modus tolens

Shortcoming of Unit Resolution From: • Engine-Starts  Flat-Tire Car-OK • Engine-Starts Empty-Gas-Tank we can infer nothing!

Resolution rule Unit Resolution A v B, ~A B Resolution A v B, ~B v C A v C • Operates on two disjunctions of literals • The pair of two opposite literals cancel each other, all other literals from the two disjuncts are combined to form a new disjunct as the inferred sentence • Resolution rule can replace the following inference rules Modus Ponens A, ~A v B B Modus Tollens ~B, ~A v B ~A Chaining ~A v B, ~B v C ~A v C

Full Resolution Rule • Given two sentences:L1… Lp and M1… Mqwhere L1,…, Lp, M1,…, Mq are all literals, and Li and Mj are complementary literals • Infer:L1…Li-1Li+1…LkM1…Mj-1Mj+1…Mkin which only one copy of each literal is retained (factoring)

Example From: Engine-Starts  Flat-Tire Car-OK Engine-Starts Empty-Gas-Tank Infer: Empty-Gas-Tank  Flat-Tire Car-OK

一种新的推理规则： Resolution 消解，归结是可靠的，但不是完备的。 A ╞ A v B (利用消解规则推导不出)

Resolution Refutation ： 消解反演，归结反驳是可靠的，且是完备的。

Deduction vs. Satisfiability Test KB  iff {KB,} is unsatisfiable • Hence: • Deciding whether a set of sentences entails another sentence, or not • Testing whether a set of sentences is satisfiable, or not • are closely related problems

Sentence  Clause Form Example: (A B)  (C  D) 1. Eliminate (A B)  (C  D)2. Reduce scope of  (A  B)  (C  D)3. Distribute  over (A  (C  D))  (B  (C  D)) (A  C)  (A  D)  (B  C)  (B  D) Set of clauses: {A  C , A  D , B  C , B  D}

Resolution Refutation Algorithm RESOLUTION-REFUTATION(KB,a) clauses set of clauses obtained from KB and a Repeat: new  {} For each C, C’ in clauses dores  RESOLVE(C,C’) If res contains the empty clause then return yes new  new U resIf newclauses then return no clauses  clauses U new

1 Battery-OK  Bulbs-OK  Headlights-Work 2 Battery-OK  Starter-OK Empty-Gas-Tank  Engine-Starts 3 Engine-Starts Flat-Tire  Car-OK 4 Headlights-Work 5 Battery-OK 6 Starter-OK 7 Empty-Gas-Tank 8 Car-OK

5 2 6 Starter-OK  Empty-Gas-Tank  Engine-Starts 7 Empty-Gas-Tank  Engine-Starts 3 Engine-Starts 8 Flat-Tire  Car-OK Flat-Tire Flat-Tire NIL (矛盾) 1 Battery-OK Bulbs-OK  Headlights-Work 2 Battery-OK Starter-OK  Empty-Gas-Tank Engine-Starts 3  Engine-Starts  Flat-Tire  Car-OK 4 Headlights-Work 5 Battery-OK 6 Starter-OK 7 Empty-Gas-Tank 8 Car-OK

Summary • 命题逻辑：语法、语义 • 模型，可满足，逻辑涵蕴 • 推理，可靠性，完备性 • 归结/消解 • 归结反驳/消解反演 • Next lecture: 谓词逻辑

语法、语义、语用

PL is Too Weak a Representational Language • Consider the problem of representing the following information: • Every person is mortal. (S1) • Confucius is a person. (S2) • Confucius is mortal. (S3) • S3 is clearly a logical consequence of S1 and S2. But how can these sentences be represented using PL so that we can infer the third sentence from the first two?

Weakness of PL • Hard to identify "individuals." E.g., Mary, 3 • Individuals cannot be PL sentences themselves. • Difficult to directly and clearly talk about properties of individuals or relations between individuals (hard to connect individuals to class properties). • E.g., property of being a human implies property of being mortal • Generalizations, patterns, regularities can't easily be represented. • All members of a class have this property • Some member of a class have this property • A better representation is needed to capture the relationship (and distinction) between objects and classes, including properties belonging to classes and individuals

First-Order Logic (Predicate Calculus abbreviated FOL or FOPC) is expressive enough to concisely represent this kind of situation by separating classes and individuals • Explicit representation of individuals and classes, x, Mary, 3, persons. • Adds relations, variables, and quantifiers, e.g., • “Every person is mortal”Forall X: person(X) => mortal(X) • “There is a white alligator”There exists some X: Alligator(X) ^ white(X)

Predicate Calculus

Lecture 2 Logic and reasoning 第 2 讲逻辑及其推理