CS 8520: Artificial Intelligence

CSC 8520 Fall, 2008. Paula Matuszek. Slides in part from aima.eecs.berkeley.edu/slides-ppt, chs 7-9 1

CS 8520: Artificial Intelligence

Logical Agents and First Order Logic

Paula Matuszek

Fall, 2008


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


Knowledge bases

• Knowledge base = set of sentences in a formal languageDeclarative 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


A simple knowledge-based agent

• This agent tells the KB what it sees, asks the KB what to do, tells the KB what it has done (or is about to do).

• 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


Wumpus World PEAS Description• Performance measure

– gold +1000, death -1000– -1 per step, -10 for using the arrow

• 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


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 – The wumpus itself is

essentially a natural feature, not another agent


Exploring a wumpus worldDirectly observed:

S: stench

B: breeze

G: glitter

A: agent

V: visited

Inferred (mostly):

OK: safe square

P: pit

W: wumpus

1,1 2,1 3,1 4,1


Exploring a wumpus world

In 1,1 we don’t get B or S, so we know 1,2 and 2,1 are safe. Move to 1,2.

In 1,2 we feel a breeze.



In 1,2 we feel a breeze. So we know there is a pit in 1,3 or 2,2.



So go back to 1,1, then to 1,2, where we smell a stench.



We don't feel a breeze in 2,1, so 2,2 can't be a pit, so 1,3 must be a pit.

We don't smell a stench in 1,2, so 2,2 can't be the wumpus, so 1,3 must be the wumpus.

2,2 has neither pit nor wumpus and is therefore okay.



We move to 2,2. We don’t get any sensory input.



So we know that 2,3 and 3,2 are ok.


Exploring a wumpus worldMove to 3,2, where we observe stench, breeze and glitter!

At this point we could infer the existence of another pit (where?), but since we have found the gold we don't bother. We have won.


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 yx+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


Entailment• Entailment means that one thing follows from

another:• Knowledge base KB entails sentence S if and only

if S 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


Entailment in the wumpus worldSituation after detecting

nothing in [1,1], moving right, breeze in [2,1]

Consider possible models for KB assuming only pits: there are 3 Boolean choices 8 possible models (ignoring sensory data)


Wumpus models for pits


Wumpus models

• KB = wumpus-world rules + observations. Only the three models in red are consistent with KB.


Wumpus ModelsKB plus hypothesis S1 that pit is not in 1.2.

KB is solid red boundary. S1 is dotted yellow boundary. KB is contained within S1, so KB entails S; in every model in which KB is true, so is S. We can conclude that the pit is not in1.2.

KB plus hypothesis S2 that pit is not in 2.2 .

KB is solid red boundary. S2 is dotted brown boundary. KB is not within S1, so KB does not entail S2; nor does S2 entail KB. So we can't conclude anything about the truth of S2 given KB.


Inference• KB entailsi S = sentence S can be derived from KB by procedure i• Soundness: i is sound if it derives only sentences S that are entailed by

KB• Completeness: i is complete if it derives all sentences S that are

entailed by KB.• First-order logic:

– Has a sound and complete inference procedure– Which will answer any question whose answer follows from what is

known by the KB.– And is richly expressive

• We must also be aware of the issue of grounding: the connection between our KB and the real world.– Straightforward for Wumpus or our adventure games– Much more difficult if we are reasoning about real situations– Real problems seldom perfectly grounded, because we ignore details. – Is the connection good enough to get useful answers?


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)


Wumpus world sentencesLet 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)


Inference by enumeration• Depth-first enumeration of all models is sound and complete

• For n symbols, time complexity is O(2n), space complexity is O(n)


Inference-based agents in the wumpus world

A wumpus-world agent using propositional logic:

P1,1

W1,1

Bx,y (Px,y+1 Px,y-1 Px+1,y Px-1,y)

Sx,y (Wx,y+1 Wx,y-1 Wx+1,y Wx-1,y)

W1,1 W1,2 … W4,4

W1,1 W1,2

W1,1 W1,3 …

64 distinct proposition symbols, 155 sentences



• Rapid proliferation of clauses. – For instance, Wumpus KB contains "physics" sentences for every

single square

• Very bushy inference, especially if forward chaining.• Not trivial to express complex relationships; people don't

naturally think in logical terms.• Monotonic: if something is true it stays true• Binary: something is either true or false, never maybe or

unknown

Expressiveness limitation of propositional logic


Pros and cons of propositional logicPropositional logic is declarativePropositional logic allows partial/disjunctive/negated

information (unlike most data structures and databases)

Propositional logic is compositional:– meaning of B1,1 P1,2 is derived from meaning of B1,1 and of P1,2

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


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


First-order logic• Whereas propositional logic assumes the

world contains facts,• first-order logic (like natural language)

assumes the world contains– Objects: people, houses, numbers, colors,

baseball games, wars, …– Relations: red, round, prime, brother of, bigger

than, part of, comes between, …– Functions: father of, best friend, one more than,

plus, …


Syntax of FOL: Basic elements• Constants KingJohn, 2, Villanova,...

• Predicates Brother, >,...

• Functions Sqrt, LeftLegOf,...

• Variables x, y, a, b,...

• Connectives , , , , • Equality =

• Quantifiers ,


Terms and Atomic sentences• A term is a logical expression that refers to an

object.– Book(Naomi). Naomi's book.

– Textbook(8520). Textbook for 8520.

• An atomic sentence states a fact.– Student(Naomi).

– Student(Naomi, Paula).

– Student(Naomi, AI).

Note that the interpretation of these is different; it depends on how we consider them to be grounded.


Complex sentences• Complex sentences are made from atomic

sentences using connectivesS, S1 S2, S1 S2, S1 S2, S1 S2,

E.g. Sibling(KingJohn,Richard) Sibling(Richard,KingJohn)


Truth in first-order logic• Sentences are true with respect to a model and an

interpretation• Model contains objects (domain elements) and

relations among them• Interpretation specifies referents for

constant symbols → objectspredicate symbols → relationsfunction symbols → functional relations

• An atomic sentence predicate(term1,...,termn) is true iff the objects referred to by term1,...,termn

are in the relation referred to by predicate


Knowledge base for the wumpus world

• Perception t,s,b Percept([s,b,Glitter],t) Glitter(t)

• Reflex t Glitter(t) BestAction(Grab,t)


Deducing hidden properties x,y,a,b Adjacent([x,y],[a,b])

[a,b] {[x+1,y], [x-1,y],[x,y+1],[x,y-1]}Properties of squares: s,t At(Agent,s,t) Breeze(t) Breezy(s)Squares are breezy near a pit:Diagnostic rule---infer cause from effect

s Breezy(s) r Adjacent(r,s) Pit(r)

Causal rule---infer effect from causer Pit(r) [s Adjacent(r,s) Breezy(s)]


Knowledge engineering in FOL1. Identify the task2. Assemble the relevant knowledge3. Decide on a vocabulary of predicates, functions,

and constants4. Encode general knowledge about the domain5. Encode a description of the specific problem

instance6. Pose queries to the inference procedure and get

answers7. Debug the knowledge base


Summary• First-order logic:

– objects and relations are semantic primitives– syntax: constants, functions, predicates,

equality, quantifiers

• Increased expressive power: sufficient to define wumpus world


Inference• When we have all this knowledge we want

to DO SOMETHING with it

• Typically, we want to infer new knowledge– An appropriate action to take– Additional information for the Knowledge Base

• Some typical forms of inference include– Forward chaining– Backward chaining– Resolution


Example knowledge base• The law says that it is a crime for an American to sell

weapons to hostile nations. The country Nono, an enemy of America, has some missiles, and all of its missiles were sold to it by Colonel West, who is American.

• Prove that Col. West is a criminal


Inference• We need to DO SOMETHING with our

knowledge.– Forward chaining– Backward chaining– Resolution


Example knowledge base contd.... it is a crime for an American to sell weapons to hostile nations:

American(x) Weapon(y) Sells(x,y,z) Hostile(z) Criminal(x)Nono … has some missiles, i.e., x Owns(Nono,x) Missile(x):

Owns(Nono,M1) and Missile(M1)… all of its missiles were sold to it by Colonel West

Missile(x) Owns(Nono,x) Sells(West,x,Nono)Missiles are weapons:

Missile(x) Weapon(x)An enemy of America counts as "hostile“:

Enemy(x,America) Hostile(x)West, who is American …

American(West)The country Nono, an enemy of America …

Enemy(Nono,America)

–


Forward chaining algorithm


Forward chaining proof






Properties of forward chaining• Sound and complete for first-order definite clauses

• Datalog = first-order definite clauses + no functions

• FC terminates for Datalog in finite number of iterations

• May not terminate in general if α is not entailed

• This is unavoidable: entailment with definite clauses is semidecidable


Efficiency of forward chainingIncremental forward chaining: no need to match a rule on

iteration k if a premise wasn't added on iteration k-1 match each rule whose premise contains a newly added positive

literal

Matching itself can be expensive:Database indexing allows O(1) retrieval of known facts

– e.g., query Missile(x) retrieves Missile(M1)

Forward chaining is widely used in deductive databases


Backward chaining algorithm


Backward chaining example
















Properties of backward chaining• Depth-first recursive proof search: space is linear

in size of proof• Incomplete due to infinite loops

fix by checking current goal against every goal on stack

• Inefficient due to repeated subgoals (both success and failure) fix using caching of previous results (extra space)

• Widely used for logic programming


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

• Choice may depend on whether you are likely to have many goals or lots of data.

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CS 8520: Artificial Intelligence

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