Propositional Logic, First Order Logic, Proof Methods, and Knowledge-based Agents

Transcription

1 Propositional Logic, First Order Logic, Proof Methods, and Knowledge-based Agents Prof. M. Walker Natural Language and Dialogue Systems Lab

2 How can we represent human knowledge about the world and how to do things? Natural Language and Dialogue Systems Lab

3 The Problem with Natural Language Ambiguous I know more intelligent people than him. I saw the Grand Canyon flying to New York. I saw a 747 flying to New York. => We need a formal way of representing the knowledge usually represented in language

4 Two schools of thought in AI for the last 50 years Symbolic: represent knowledge using logic and rules that define inference procedures on logic Statistical: represent knowledge as probabilities based on observations of the world Both of these traditions had their roots in the same paper: McCulloch and Pitts 1943

5 McCulloch Pitts Neuron INPUT X1 W1 INPUT X2 INPUT Xn W2 Wn Threshhold unit Does weighted sum of inputs pass threshhold? OUTPUT Total input = weight on line 1 x input on 1 + weight on line 2 x input on 2 + weight on line n x input on n (for all n) Basic model: performs weighted sum of inputs, compares this to internal threshold level, and turns on if this level exceeded.

6 Propositional Logic Calculus The symbols of propositional logic are: P, Q, R, S These can stand for any statement. E.g. it is raining, I am at work The truth symbols: T true, F false The connectives: AND OR NOT ~ IMPLIES => EQUALS

7 Propositional Logic Syntax of Sentences True, False, P, Q, R are sentences The negation of a sentence is a sentence NOT P ( it is not raining ) The conjunction of two sentences is a sentence, P AND Q ( it is raining and I am at work ) P, Q are called the conjuncts The disjunction of two sentences is a sentence, P OR Q ( it is raining or I am at work ) P, Q are called the disjuncts

8 Propositional Logic Syntax of Sentences The implication of one sentence from another P IMPLIES Q ( it is sunny implies I am outdoors ) P is called the premise/antecedent Q is called the conclusion/consequent The equivalence of two sentences is a sentence P OR Q EQUALS R Legal sentences are also called well-formed formulae

9 ((P AND Q) IMPLIES R) EQUALS NOT P OR NOT Q OR R Is well-formed: P, Q, R are propositions P AND Q is a sentence (P AND Q) IMPLIES R is a sentence NOT P, NOT Q are sentences NOT P OR NOT Q is a sentence NOT P OR NOT Q OR R is a sentence ((P AND Q) IMPLIES R) EQUALS NOT P OR NOT Q OR R

10 Efficient propositional inference Two families of efficient algorithms for propositional inference: Complete backtracking search algorithms DPLL algorithm (Davis, Putnam, Logemann, Loveland) Incomplete local search algorithms WalkSAT algorithm

11 Pros and cons of propositional logic Propositional logic is declarative Propositional logic allows partial/disjunctive/negated information (unlike most data structures and databases) Propositional logic is compositional: meaning of B 1,1 P 1,2 is derived from meaning of B 1,1 and of P 1,2 Meaning in propositional logic is context-independent (unlike natural language, where meaning depends on context) Propositional logic has very limited expressive power (unlike natural language) E.g., cannot say "pits cause breezes in adjacent squares except by writing one sentence for each square

12 McCulloch Pitts Neuron for AND X +1 Y +1 X + Y -2 X AND Y l -2

13 Logical Agents Natural Language and Dialogue Systems Lab Chapter 7

14 Outline Knowledge-based agents Wumpus world Logic in general - models and entailment Propositional (Boolean) logic Equivalence, validity, satisfiability Inference rules and theorem proving forward chaining backward chaining resolution

15 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

16 A simple knowledge-based agent The agent must be able to: Represent states, actions, etc. (Tell (KB, Make-percept.) Incorporate new percepts Update internal representations of the world (Tell (KB, Make-action..) Deduce hidden properties of the world Deduce appropriate actions

17 Wumpus World PEAS description Performance measure: gold +1000, death -1000, -1 per step, -10 for using arrow Environment Squares adjacent to wumpus are smelly (STENCH) Squares adjacent to pit are breezy Glitter iff gold is in the same square Shooting kills wumpus if you are facing it Shooting uses up the only arrow Grabbing picks up gold if in same square Releasing drops the gold in same square Sensors: Five Item Ordered Vector Stench, Breeze: if Wumpus or Pit Adjacent Glitter: if Gold in square Bump: When agent hits a wall Scream: when Wumpus dies perceived everywhere Actuators: Left turn, Right turn, Forward, Grab, Release, Shoot

18 Wumpus world characterization Fully Observable No only local perception Deterministic Yes outcomes exactly specified Episodic No sequential at the level of actions Static Yes Wumpus and Pits do not move Discrete Yes Single-agent? Yes Wumpus is essentially a natural feature

19 Exploring a wumpus world (randomly generated) PERCEPT = [ none, none, none, none, none]

20 Exploring a wumpus world

21 Exploring a wumpus world PERCEPT = [ none, Breeze, none, none, none]

22 Exploring a wumpus world Go back to start state where its safe.

23 Exploring a wumpus world Go to square 1,2, you know its okay PERCEPT = [ Stench, none, none, none, none] Because there is no breeze, Agent knows that square 2,2 does not have a pit Because there was no stench in square1,2, agent knows there is no W in square 2,2

24 Exploring a wumpus world Go to square 2,2, you know its okay PERCEPT = [ none, none, none, none, none] Because there is no breeze Agent knows that square 2,3 and 3,1 does not have a pit

25 Exploring a wumpus world Infer that 2,3 and 3,2 are also okay

26 Exploring a wumpus world

27 Entailment Entailment means that one thing follows from another: KB α Knowledge base KB entails sentence if and only if is true in all worlds where KB is true E.g., the KB containing the Giants won and the Reds won entails Either the Giants won or the Reds won E.g., x+y = 4 entails 4 = x+y Entailment is a relationship between sentences (i.e., syntax) that is based on semantics

28 Models Logicians typically think in terms of models, which are formally structured worlds with respect to which truth can be evaluated We say m is a model of a sentence if is true in m M( ) is the set of all models of Then KB iff M(KB) M( ) E.g. KB = Giants won and Reds won = Giants won

29 Entailment in the wumpus world Situation after detecting nothing in [1,1], moving right, breeze in [2,1] Consider possible models for KB assuming only pits 3 Boolean choices 8 possible models

30 Wumpus models

31 Wumpus models: what is possibly true given percepts KB = wumpus-world rules + observations

32 Wumpus models KB = wumpus-world rules + observations 1 = "[1,2] is safe", KB 1, proved by model checking

33 Wumpus models KB = wumpus-world rules + observations

34 Wumpus models KB = wumpus-world rules + observations 2 = "[2,2] is safe", KB 2

35 Inference KB i = sentence can be derived from KB by procedure i Soundness: i is sound if whenever KB i it is also true that KB Completeness: i is complete if whenever KB it is also true that KB i Preview: we will define a logic (first-order logic) which is expressive enough to say almost anything of interest, and for which there exists a sound and complete inference procedure. That is, the procedure will answer any question whose answer follows from what is known by the KB.

36 Wumpus world sentences Let P i,j be true if there is a pit in [i, j]. Let B i,j be true if there is a breeze in [i, j]. P 1,1 B 1,1 B 2,1 "Pits cause breezes in adjacent squares" B 1,1 (P 1,2 P 2,1 ) B 2,1 (P 1,1 P 2,2 P 3,1 )

37 Inference by enumeration Depth-first enumeration of all models is sound and complete Soundness: if the inference procedure says it is entailed, it is Completeness: if it is entailed, the inference procedure can prove it For n symbols, Time complexity is O(2 n ) (have to check all possible assignments) Space complexity is O(n)

38 Logical equivalence Two sentences are logically equivalent} iff true in same models: ß iff β and β

39 Validity and satisfiability 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: KB if and only if (KB ) 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 if and only if (KB ) is unsatisfiable

40 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

41 Resolution Conjunctive Normal Form (CNF) conjunction of disjunctions of literals clauses E.g., (A B) (B C D) Resolution inference rule (for CNF): l i l k, m 1 m n l i l i-1 l i+1 l k m 1 m j-1 m j+1... m n where l i and m j are complementary literals. E.g., P 1,3 P 2,2, P 2,2 P 1,3 Resolution is sound and complete for propositional logic

42 Conversion to CNF ;FIX B 1,1 (P 1,2 P 2,1 ) 1. Eliminate, replacing with ( ) ( ). (B 1,1 (P 1,2 P 2,1 )) ((P 1,2 P 2,1 ) B 1,1 ) 2. Eliminate, replacing with. ( B 1,1 P 1,2 P 2,1 ) ( (P 1,2 P 2,1 ) B 1,1 ) 3. Move inwards using de Morgan's rules and double-negation: ( B 1,1 P 1,2 P 2,1 ) (( P 1,2 P 2,1 ) B 1,1 ) 4. Apply distributivity law ( over ) and flatten: ( B 1,1 P 1,2 P 2,1 ) ( P 1,2 B 1,1 ) ( P 2,1 B 1,1 )

43 Resolution algorithm Proof by contradiction, i.e., show KB unsatisfiable

44 Resolution example KB = (B 1,1 (P 1,2 P 2,1 )) B 1,1 = P 1,2 = what we want to prove, no pit in 1,2,, ie. P 1,2

45 Forward and backward chaining Horn Form (restricted) KB = conjunction of Horn clauses Horn clause = proposition symbol; or (conjunction of symbols) symbol E.g., C (B A) (C D B) Modus Ponens (for Horn Form): complete for Horn KBs 1,, n, 1 n β β Can be used with forward chaining or backward chaining. These algorithms are very natural and run in linear time

46 Forward chaining Idea: fire any rule whose premises are satisfied in the KB, add its conclusion to the KB, until query is found

47 Forward chaining algorithm Forward chaining is sound and complete for Horn KB

48 Forward chaining example

49 Forward chaining example

50 Forward chaining example

51 Forward chaining example

52 Forward chaining example

53 Forward chaining example

54 Forward chaining example

55 Forward chaining example

56 Proof of completeness FC derives every atomic sentence that is entailed by KB 1. FC reaches a fixed point where no new atomic sentences are derived 2. Consider the final state as a model m, assigning true/false to symbols 3. Every clause in the original KB is true in m a 1 a k b 4. Hence m is a model of KB 5. If KB q, q is true in every model of KB, including m

57 Backward chaining Idea: work backwards from the query q: to prove q by BC, check if q is known already, or prove by BC all premises of some rule concluding q Avoid loops: check if new subgoal is already on the goal stack Avoid repeated work: check if new subgoal 1. has already been proved true, or 2. has already failed

58 Backward chaining example

59 Backward chaining example

60 Backward chaining example

61 Backward chaining example

62 Backward chaining example

63 Backward chaining example

64 Backward chaining example

65 Backward chaining example

66 Backward chaining example

67 Backward chaining example

68 Forward vs. backward chaining FC is data-driven, automatic, unconscious processing, e.g., object recognition, routine decisions May do lots of work that is irrelevant to the goal BC is goal-driven, appropriate for problem-solving, e.g., Where are my keys? How do I get into a PhD program? Complexity of BC can be much less than linear in size of KB

69 Efficient propositional inference Two families of efficient algorithms for propositional inference: Complete backtracking search algorithms DPLL algorithm (Davis, Putnam, Logemann, Loveland) Incomplete local search algorithms WalkSAT algorithm

70 The DPLL algorithm EVERY FOL KB can be converted to a PL KB Determine if an input propositional logic sentence (in CNF) is satisfiable. Improvements over truth table enumeration: 1. Early termination A clause is true if any literal is true. A sentence is false if any clause is false. 2. Pure symbol heuristic Pure symbol: always appears with the same "sign" in all clauses. e.g., In the three clauses (A B), ( B C), (C A), A and B are pure, C is impure. Make a pure symbol literal true. 3. Unit clause heuristic Unit clause: only one literal in the clause The only literal in a unit clause must be true.

71 The DPLL algorithm

72 The WalkSAT algorithm Incomplete, local search algorithm Evaluation function: The min-conflict heuristic of minimizing the number of unsatisfied clauses Balance between greediness and randomness

73 The WalkSAT algorithm

74 Today s Thought Question What kind of knowledge cannot easily be represented in propositional logic?

75 Summary Logical agents apply inference to a knowledge base to derive new information and make decisions Basic concepts of logic: syntax: formal structure of sentences semantics: truth of sentences wrt models entailment: necessary truth of one sentence given another inference: deriving sentences from other sentences soundness: derivations produce only entailed sentences completeness: derivations can produce all entailed sentences Wumpus world requires the ability to represent partial and negated information, reason by cases, etc. Resolution is complete for propositional logic Forward, backward chaining are linear-time, complete for Horn clauses Propositional logic lacks expressive power