1 Logical Agents Russell & Norvig Chapter 7. 2 Programming Language Moment We often say a programming language is procedural or declarative. What is meant

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Logical Agents

Russell & Norvig Chapter 7

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Programming Language Moment

• We often say a programming language is procedural or declarative.

• What is meant by that?

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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. */

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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

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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

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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

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Wumpus World PEAS (performance, environment, sensors, actuators) description

• Performance measure– gold +1000, death -1000– -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

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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?•

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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)

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How would you direct the agent what to do?

What does this remind you of?

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Exploring a wumpus world

Upper Left indicates preceptsB: BreezeS: StenchG: Glitter

Upper Right StatusOK: Safe, no pit or wumpus?: UnknownP?: Could be Pit

In Center[A]: Agent has been thereP: is a PitW: is a Wumpus

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Sense a breeze, so what do I know?

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So what should I do?

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Exploring a wumpus worldWhat 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?

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Will I always know exactly what to do?

Give an example?

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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”

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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. xy=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!

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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

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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 worldsdried fruits

mango banana

cherries

apple

papaya

grapefruit

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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

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

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Wumpus models

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

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Wumpus models

• Remember KB ╞ α iff M(KB) M(α)• KB = wumpus-world rules + observations• α1 = "[1,2] is safe", KB ╞ α1, proved by model checking• Notice set of α1 includes some models not allowed in KB• In all possible models in KB, [1,2] is safe

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Wumpus models

• Can we prove [2,2] is safe?

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Wumpus models

• KB = wumpus-world rules + observations• α2 = "[2,2] is safe", KB ╞ α2

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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.

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Propositional logic: Syntax

• Propositional logic is the simplest logic – illustrates basic ideas

• Each symbol can be true or false

• 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)

Is this notation familiar?––––

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Propositional (boolean) logic: Semantics

Each model specifies true/false for each proposition symbolE.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 and S2 is trueS1 S2 is true iff S1is true or S2 is trueS1 S2 is true iff S1 is false or S2 is true i.e., is false iff S1 is true and S2 is falseS1 S2 is true iff S1S2 is true andS2S1 is true

Simple recursive process evaluates an arbitrary sentence, e.g.,

P1,2 (P2,2 P3,1) = true (true false) = true true = true

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Truth tables for connectives

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Exploring a wumpus worldWhat Predicates are known?

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Wumpus world sentences

Let Pi,j be true if there is a pit in [i, j].

Let Bi,j be true if there is a breeze in [i, j]. P1,1

B1,1

B2,1

• "Pits cause breezes in adjacent squares"B1,1 (P1,2 P2,1)

B2,1 (P1,1 P2,2 P3,1)

B2,2 (P1 2 P3 2 P2 3 P2,1)

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Idea

• Consider all possible values of propositions

• Throw out ones made invalid by KB

• Examine in remaining possibilities

• Would work, but is exponential, right?

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Truth tables for inference

P1,1B1,1B2,1 (actually different than world of previous slide) P2,1 (as you are there and feel breeze) is P1,2

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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(2n), space complexity is O(n)• Notice – try each possible value for symbol• PL-True?(sentence,model) returns true if sentence holds within model

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Logical equivalence

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

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Deduction theorem

• Known to the ancient Greeks:

• For any sentences and ,

( ╞ ) if and only if the sentence

( ) is valid.

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Validity and satisfiabilityA 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 validexample: : Ali will be accepted to USU

A sentence is satisfiable if it is true in some modele.g., A B, CI can make it true; some way of satisfyingexample: : Ali will get an A in 6100Many 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?”

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Validity and satisfiability

A sentence is unsatisfiable if it is true in no modelse.g., AAexample: : 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

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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

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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.

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ResolutionConjunctive Normal Form (CNF)

conjunction of disjunctions of literalsclauses

E.g., (A B) (B C D)

• Resolution inference rule (for CNF):li … lk, m1 … mn

li … li-1 li+1 … lk m1 … mj-1 mj+1 ... mn

where li and mj are complementary literals. E.g., P1,3 P2,2, P2,2

P1,3

This single rule, Resolution, coupled with a search algorithm yields a sound and completeinference algorithm.

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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?

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Refutation Completeness

• Given A is true, we cannot use resolution to generate that AB is true

• We can use resolution to argue whether AB is true. This is termed refutation completeness.

• but first, we need to get our rules in a form that resolution works on

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Conversion to CNF (conjunctive normal form, conjunction of

disjunctions)B1,1 (P1,2 P2,1)

1. Eliminate , replacing α β with (α β)(β α).(B1,1 (P1,2 P2,1)) ((P1,2 P2,1) B1,1)

2. Eliminate , replacing α β with α β.(B1,1 P1,2 P2,1) ((P1,2 P2,1) B1,1)

3. Move inwards using de Morgan's rules and double-negation:(B1,1 P1,2 P2,1) ((P1,2 P2,1) B1,1)

4. Apply distributive law ( over ) and flatten:(B1,1 P1,2 P2,1) (P1,2 B1,1) (P2,1 B1,1)

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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

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So what does deriving NOTHING mean?

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Resolution algorithm

• Proof by contradiction, i.e., show KBα unsatisfiable

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So when we return true, we mean the clause IS unsatisfiable

Notice the contradiction

so getting an empty meanswe got a contradiction and KB does imply

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Resolution example• We want to show that B1,1 P1,2

• KB = (B1,1 (P1,2 P2,1)) B1,1

• α = P1,2

• Suppose is true and show contradiction(B1,1 P1,2 P2,1) (P1,2 B1,1) (P2,1 B1,1)

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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!

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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 α1 … αn β is the same as (α1 … αn) V β which is α1 V… V αn V β

• Modus Ponens (for Horn Form): complete for Horn KBsα1, … ,αn, α1 … αn β

β

• Can be used with forward chaining or backward chaining.• These algorithms are very natural and run in linear time

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Forward chaining

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

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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

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Forward chaining example

Red indicates number of unsatisfied premises.

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L was proved in one of twoways

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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 a1 … ak b

4. Hence m is a model of KB5. If KB╞ q, q is true in every model of KB, including m

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Backward chaining (BC)

Idea: work backwards from the query q:to prove q by BC,

check if q is known already, orprove 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 subgoal1. has already been proved true, or2. has already failed

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Backward chaining example

Green is our goal.

Red is what we already know

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Solid green means wedon’t need to worryabout this proof as longas we can prove all theother “open” green predicates.It’s trueness or falseness will bethe result of other goals.

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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

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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

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The DPLL algorithmDetermine 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 heuristicPure 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 heuristicUnit clause: only one literal in the clauseThe only literal in a unit clause must be true.Remember our clauses are ANDED together

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The DPLL algorithm

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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

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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.

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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

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Hard satisfiability problems• For your values, is the following CNF

true?:

(A B) (C D E)

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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)

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Hard satisfiability problemswhat is probability they can be satisfied?

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Hard satisfiability problems

• Median runtime for 100 satisfiable random 3-CNF sentences, n = 50

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Inference-based agents in the 4x4 wumpus world

A wumpus-world agent using propositional logic:

P1,1 First cell can’t be pit

W1,1 First cell can’t be wumpus

Bx,y (Px,y+1 Px,y-1 Px+1,y Px-1,y) Breeze means wumpus close

Sx,y (Wx,y+1 Wx,y-1 Wx+1,y Wx-1,y) Stench means wumpus close

W1,1 W1,2 … W4,4 Wumpus is somewhere

W1,1 W1,2 Wumpus can’t be in neighboring cells as only 1

W1,1 W1,3 …

64 distinct proposition symbols, 155 sentences

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Wumpus Algorithm

• At seats: In the following algorithm, what does each piece do?

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• 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],

Lx,y FacingRightt Forwardt Lx+1,y

• Rapid proliferation of clauses

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Expressiveness limitation of propositional logic

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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 logicForward, backward chaining are linear-time, complete for Horn clauses

• Propositional logic lacks expressive power

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Documents

1 Logical Agents Russell & Norvig Chapter 7. 2 Programming Language Moment We often say a programming language is procedural or declarative. What is meant