1 Logical Agents Russell & Norvig Chapter 7

2 Programming Language Moment We often say a programming language is procedural or declarative. What is meant by that?

3 Declarative Statements in Prolog (variables begin with capital letter) fun(X) :- /* an item is fun if */ red(X), /* the item is red */ car(X). /* and it is a car */ fun(X) :- /* or an item is fun if */ blue(X), /* the item is blue */ bike(X). /* and it is a bike */ fun(ice_cream). /* ice cream is also fun. */

4 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

5 Terms percept - information I sense or “perceive” Tell – way of updating the contents of the knowledge base Ask – ask KB to tell you want to do

6 A simple knowledge-based agent The agent must be able to: –Represent states, actions, etc. –Incorporate new percepts –Update internal representations of the world –Deduce hidden properties of the world –Deduce appropriate actions

7 Wumpus World PEAS (performance, environment, sensors, actuators) description Performance measure –gold +1000, death –-1 per step, -10 for shooting Environment –Squares adjacent to wumpus are smelly –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: Stench, Breeze, Glitter, Bump, Scream Actuators: Left turn, Right turn, Forward, Grab, Release, Shoot

8 Categorization of Environments Accessible vs inaccessible – have complete information about environment state? Deterministic vs Nondeterministic outcomes exactly specified? Episodic vs history sensitive (sequential) – actions in episode have no relationship to actions in other episodes Static vs dynamic changes only by actions of agent? Discrete vs continuous- fixed, finite number of actions/percepts? Single-agent vs multi-agent? How many agents? How would you categorize wumpus world?

9 Wumpus world characterization Fully Observable/Accessible No – only local perception Deterministic Yes – outcomes exactly specified Episodic Yes – environment doesn’t respond to runs, only state. Static Yes – Wumpus and Pits do not move Discrete Yes Single-agent? Yes – Wumpus is essentially a natural feature (not an agent)

10 How would you direct the agent what to do? What does this remind you of?

11 Exploring a wumpus world Upper Left indicates precepts B: Breeze S: Stench G: Glitter Upper Right Status OK: Safe, no pit or wumpus ?: Unknown P?: Could be Pit In Center [A]: Agent has been there P: is a Pit W: is a Wumpus

12 Exploring a wumpus world Sense a breeze, so what do I know?

13 Exploring a wumpus world So what should I do?

14 Exploring a wumpus world

15 Exploring a wumpus world

16 Exploring a wumpus world

17 Exploring a wumpus world What is sensor in 2,2? Do sensors note Pit or Wumpus in diagonal square? Do I really know 2,3 and 3,2 are okay?

18 Exploring a wumpus world Will I always know exactly what to do? Give an example?

19 Logic in general Logics are formal languages for representing information such that conclusions can be drawn Syntax defines the sentences in the language Semantics define the "meaning" of sentences; –i.e., define truth of a sentence in a world E.g., the language of arithmetic –x+2 ≥ y is a sentence; x2+y > {} is not a sentence –x+2 ≥ y is true iff the number x+2 is no less than the number y –x+2 ≥ y is true in a world where x = 7, y = 1 –x+2 ≥ y is false in a world where x = 0, y = 6 Sometimes use the term “possible worlds”

20 Entailment Entailment means that one thing follows from another: KB ╞ α Knowledge base KB entails sentence α if and only if α is true in all worlds where KB is true –Playing mud football “entails” getting muddy –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 –E.g. x y =1 entails what? –Entailment is a relationship between sentences (i.e., syntax) that is based on semantics Entailment doesn’t say you can prove it – only that it is true!

21 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 α if α is true in m M(α) is the set of all models of α – or all possible worlds where α is true Then KB ╞ α iff M(KB)  M(α) In other words, the KB doesn’t contain any models in which α isn’t true. E.g. –KB = Giants won and Reds won –α = Giants won

22 Example  : fruits are low calorie A model of the sentence is a world with tomatoes. M(  ) – all models where  is true. M(KB) - all models legal in our knowledge base All Fruity worlds dried fruits mangobanana cherries apple papaya grapefruit

23 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 Can’t go diagonal, so only consider choices from [A] squares. 3 Boolean choices for ? squares  8 possible models

24 All Possible Wumpus models (Pits only – show all “possible”)

25 Wumpus models KB (red)= wumpus-world rules + observations Blue can’t happen as inconsistent with percepts

26 Wumpus models Remember KB ╞ α iff M(KB)  M(α) KB = wumpus-world rules + observations α 1 = "[1,2] is safe", KB ╞ α 1, proved by model checking Notice set of α 1 includes some models not allowed in KB In all possible models in KB, [1,2] is safe

27 Wumpus models Can we prove [2,2] is safe?

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

29 Inference KB ├ i α = sentence α can be derived from KB by procedure (set of rules) i Soundness: i is sound if whenever KB ├ i α, it is also true that KB╞ α (only true things are derived) Completeness: i is complete if whenever KB╞ α, it is also true that KB ├ i α (if α is true, we can derive it) 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.

30 Propositional logic: Syntax Propositional logic is the simplest logic – illustrates basic ideas Each symbol can be true or false The proposition symbols P 1, P 2 etc are sentences –If S is a sentence,  S is a sentence (negation) –If S 1 and S 2 are sentences, S 1  S 2 is a sentence (conjunction) –If S 1 and S 2 are sentences, S 1  S 2 is a sentence (disjunction) –If S 1 and S 2 are sentences, S 1  S 2 is a sentence (implication) –If S 1 and S 2 are sentences, S 1  S 2 is a sentence (biconditional) Is this notation familiar?

31 Propositional (boolean) logic: Semantics Each model specifies true/false for each proposition symbol E.g. P 1,2 P 2,2 P 3,1 falsetruefalse With these symbols, 8 possible models, can be enumerated automatically. Rules for evaluating truth with respect to a model m:  Sis true iff S is false S 1  S 2 is true iff S 1 is true and S 2 is true S 1  S 2 is true iff S 1 is true or S 2 is true S 1  S 2 is true iffS 1 is false orS 2 is true i.e., is false iffS 1 is true andS 2 is false S 1  S 2 is true iffS 1  S 2 is true andS 2  S 1 is true Simple recursive process evaluates an arbitrary sentence, e.g.,  P 1,2  (P 2,2  P 3,1 ) = true  (true  false) = true  true = true

32 Truth tables for connectives

33 Exploring a wumpus world What Predicates are known?

34 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 ) B 2,2  (P 1 2  P 3 2  P 2 3  P 2,1 )

35 Idea Consider all possible values of propositions Throw out ones made invalid by KB Examine  in remaining possibilities Would work, but is exponential, right?

36 Truth tables for inference  P1,1  B1,1 B2,1 (actually different than world of previous slide)  P2,1 (as you are there and feel breeze)  is  P1,2

37 Inference by enumeration – consider all models of KB Depth-first enumeration of all models is sound and complete For n symbols, time complexity is O(2 n ), space complexity is O(n) Notice – try each possible value for symbol PL-True?(sentence,model) returns true if sentence holds within model

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

39 Deduction theorem Known to the ancient Greeks: For any sentences  and , (  ╞  ) if and only if the sentence (    ) is valid.

40 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 ╞ α if and only if (KB  α) is valid example:  : Ali will be accepted to USU A sentence is satisfiable if it is true in some model e.g., A  B, C I can make it true; some way of satisfying example:  : Ali will get an A in 6100 Many problems in computer science are really satisfiability problems. For example, constraint satisfaction. With appropriate transformations, search problems can be solved by constraint satisfaction. For example, “Find the shortest path to WalMart” becomes “Is there a path to WalMart of less than two miles?”

41 Validity and satisfiability A sentence is unsatisfiable if it is true in no models e.g., A  A example:  : Ali will get an A in 6411 (No such class exists for him to take.) We are familiar with this as proof by contradiction. Satisfiability is connected to inference via the following: KB ╞ α if and only if (KB  α) is unsatisfiable

42 Proof methods Proof methods divide into (roughly) two kinds: Every known inference algorithm for propositional logic has a worst case complexity that is exponential in the size of the input –Model checking (as we saw) truth table enumeration (always exponential in n) improved backtracking, e.g., Davis--Putnam-Logemann-Loveland (DPLL) –– Can I find an assignment of variables which makes this true? heuristic search in model space (sound but incomplete: doesn’t lie, but may not find a solution when one exists) e.g., min-conflicts (guess something that makes the fewest clauses unhappy) hill-climbing algorithms –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

43 Reasoning patterns Modus Ponens ,  ______________  And Elimination  ______________  Can be used wherever they apply without the need for enumerating models. A series of inferences (a proof) is an alternative to enumerating models. In the worst case, searching for proofs is no more efficient than enumerating models. However, finding a proof can be highly efficient The horizontal line separates the “if” part from the “then” part.

44 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 This single rule, Resolution, coupled with a search algorithm yields a sound and complete inference algorithm. Think, if B is true, what do I know? If B is false, what do I know? Since B must be true or false, what do I now know?

45 Refutation Completeness Given A is true, we cannot use resolution to generate that A  B is true We can use resolution to argue whether A  B is true. This is termed refutation completeness. but first, we need to get our rules in a form that resolution works on

46 Conversion to CNF (conjunctive normal form, conjunction of disjunctions) B 1,1  (P 1,2  P 2,1 ) 1.Eliminate , replacing α  β with (α  β)  (β  α). (B 1,1  (P 1,2  P 2,1 ))  ((P 1,2  P 2,1 )  B 1,1 ) 2. Eliminate , replacing α  β with  α  β. (  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 distributive 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 )

47 Resolution Algorithms In order to show KB ╞ , we show (KB  ) is unsatisfiable. We do this by proving a contradiction What will a contradiction look like in resolution What if we have So what does deriving NOTHING mean?

48 Resolution algorithm Proof by contradiction, i.e., show KB  α unsatisfiable So when we return true, we mean the clause IS unsatisfiable Notice the contradiction so getting an empty means we got a contradiction and KB does imply 

49 Resolution example We want to show that  B 1,1  P 1,2 KB = (B 1,1  (P 1,2  P 2,1 ))  B 1,1 α =  P 1,2 Suppose  is true and show contradiction (  B 1,1  P 1,2  P 2,1 )  (  P 1,2  B 1,1 )  (  P 2,1  B 1,1 )

50 The Ground Resolution Theorem If a set of clauses is unsatisfiable, then the resolution closure of those clauses contains the empty clause. Let’s take a break from this chapter and go back into our text!

51 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) –Sometimes we say Horn Clauses have at most ONE positive. This is because α 1  …  α n  β is the same as  (α 1  …  α n ) V β which is  α 1 V… V  α n V β Modus Ponens (for Horn Form): complete for Horn KBs α 1, …,α n,α 1  …  α n  β β Can be used with forward chaining or backward chaining. These algorithms are very natural and run in linear time

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

53 Forward chaining algorithm Forward chaining is sound and complete for Horn KB Head[c] is the clause that is implied count[c] is number of unsatisfied premises

54 Forward chaining example Red indicates number of unsatisfied premises.

55 Forward chaining example

56 Forward chaining example L was proved in one of two ways

57 Forward chaining example

58 Forward chaining example

59 Forward chaining example

60 Forward chaining example

61 Forward chaining example

62 Proof of completeness Forward Chaining (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

63 Backward chaining (BC) 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

64 Backward chaining example Green is our goal. Red is what we already know

65 Backward chaining example Solid green means we don’t need to worry about this proof as long as we can prove all the other “open” green predicates. It’s trueness or falseness will be the result of other goals.

66 Backward chaining example

67 Backward chaining example

68 Backward chaining example

69 Backward chaining example

70 Backward chaining example

71 Backward chaining example

72 Backward chaining example

73 Backward chaining example

74 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

75 Efficient propositional inference Two families of efficient algorithms for propositional inference: Backtracking search algorithms DPLL algorithm (Davis, Putnam, Logemann, Loveland) Incomplete local search algorithms –WalkSAT algorithm

76 The DPLL algorithm Determine if an input propositional logic sentence (in CNF) is satisfiable. Improvements over truth table enumeration: 1.Early termination (short circuit evaluation) 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 – as no clause is inconvenienced if it isn’t. 3.Unit clause heuristic Unit clause: only one literal in the clause The only literal in a unit clause must be true. Remember our clauses are ANDED together

77 The DPLL algorithm

78 The WalkSAT algorithm Incomplete, local search algorithm Guesses at a solution and then tries to make it better. Makes better by finding one clause that fails and changing a value to make it succeed. Evaluation function to decide which value in the clause to flip: The min-conflict heuristic of minimizing the number of unsatisfied clauses Balance between greediness and randomness

79 The WalkSAT algorithm Note, else is associated with the “with probability p” – so you either randomly change a value or pick the one which causes less grief.

80 Hard satisfiability problems WalkSat works best if there are LOTS of solutions, as it is easier to land on one. For example – each of you randomly pick values for A,B,C,D,E

81 Hard satisfiability problems For your values, is the following CNF true?: (A  B)  (C  D  E)

82 Hard satisfiability problems In general, what makes a problem hard? Consider random 3-CNF sentences. e.g., (  D   B  C)  (B   A   C)  (  C   B  E)  (E   D  B)  (B  E   C) m = number of clauses n = number of symbols –Hard problems seem to cluster near m/n = 4.3 (critical point)

83 Hard satisfiability problems what is probability they can be satisfied?

84 Hard satisfiability problems Median runtime for 100 satisfiable random 3- CNF sentences, n = 50

85 Inference-based agents in the 4x4 wumpus world A wumpus-world agent using propositional logic:  P 1,1 First cell can’t be pit  W 1,1 First cell can’t be wumpus B x,y  (P x,y+1  P x,y-1  P x+1,y  P x-1,y ) Breeze means wumpus close S x,y  (W x,y+1  W x,y-1  W x+1,y  W x-1,y ) Stench means wumpus close W 1,1  W 1,2  …  W 4,4 Wumpus is somewhere  W 1,1   W 1,2 Wumpus can’t be in neighboring cells as only 1  W 1,1   W 1,3 …  64 distinct proposition symbols, 155 sentences

86 Wumpus Algorithm At seats: In the following algorithm, what does each piece do?

87

88 KB contains "physics" sentences for every single square – can’t easily say “This principle is true everywhere!” For every time t and every location [x,y], L x,y  FacingRight t  Forward t  L x+1,y Rapid proliferation of clauses Expressiveness limitation of propositional logic tt

89 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