rkj@aber.ac.uk

CSM6120Introduction to Intelligent

SystemsPropositional and Predicate Logic

Logic and AI Would like our AI to have knowledge about the

world, and logically draw conclusions from it

Search algorithms generate successors and evaluate them, but do not “understand” much about the setting

Example question: is it possible for a chess player to have 8 pawns and 2 queens? Search algorithm could search through tons of

states to see if this ever happens…

Types of knowledge There are four basic sorts of knowledge

(different sources may vary as to number and definitions)

Declarative Facts E.g. Joe has a car

Procedural Knowledge exercised in the performance of some

task E.g. What to check if car won't start

First check a, then b, then c, until car starts

Types of knowledge Meta

Knowledge about knowledge E.g. knowing how an expert system came to its

conclusion

Heuristic Rules of thumb, empirical Knowledge gained from experience, not proven

i.e. could be wrong!

Methods of representation Object-attribute-value

Say you have an oak tree. You want to encode the fact that it is a tree, and that its species is oak

There are three pieces of information in there: It's a tree It has (belongs to) a species The species is oak

The fact that all trees have a species might be obvious to you, but it's not obvious to a computer We have to encode it

How to encode? Maybe something like this:

(species tree oak) species(tree, oak) tree(species, oak)

Or some other similar method that your program can parse and understand

What if you’re not sure? You could include an uncertainty factor

The uncertainty factor is a number which can be taken into account by your program when making its decisions

The final conclusion of any program where uncertainty was used in the input is likely to also have an uncertainty factor

If you're not sure of the facts, how can the program be sure of the result?

Encoding uncertainty To encode uncertainty, we may do something

like this: species(tree, oak, 0.8)

This would mean we were only 80% sure that the tree concerned was an oak tree

Rules A knowledge base (KB) may have rules

IF premises THEN conclusion

May be more than one premise

May contain logical functions AND, OR, NOT

If premises evaluate to TRUE, rule fires IF tree (species, oak) THEN tree (type, deciduous)

More complex rulesIF tree is a beechAND the season is

summerTHEN tree is green

IF tree is a beechAND the season is

summerAND tree-type is red

beechTHEN tree is red

This example illustrates a problem whereby two rules could both fire, but lead to different conclusions

Strategies to resolve would include firing the most specific rule first The second rule is

more specific

What is a Logic? A language with concrete rules

No ambiguity in representation (may be other errors!) Allows unambiguous communication and processing Very unlike natural languages e.g. English

Many ways to translate between languages A statement can be represented in different logics And perhaps differently in same logic

Expressiveness of a logic How much can we say in this language?

Not to be confused with logical reasoning Logics are languages, reasoning is a process (may use logic)

Syntax and semantics Syntax

Rules for constructing legal sentences in the logic Which symbols we can use (English: letters, punctuation) How we are allowed to combine symbols

Semantics How we interpret (read) sentences in the logic Assigns a meaning to each sentence

Example: “All lecturers are seven foot tall” A valid sentence (syntax) And we can understand the meaning (semantics) This sentence happens to be false (there is a

counterexample)

Propositional logic Syntax

Propositions, e.g. “it is wet” Connectives: and, or, not, implies, iff (equivalent)

Brackets (), T (true) and F (false)

Semantics (Classical/Boolean) Define how connectives affect truth

“P and Q” is true if and only if P is true and Q is true Use truth tables to work out the truth of

statements

A story Your Friend comes home; he is completely wet You know the following things:

Your Friend is wet If your Friend is wet, it is because of rain, sprinklers, or both If your Friend is wet because of sprinklers, the sprinklers must be

on If your Friend is wet because of rain, your Friend must not be

carrying the umbrella The umbrella is not in the umbrella holder If the umbrella is not in the umbrella holder, either you must be

carrying the umbrella, or your Friend must be carrying the umbrella

You are not carrying the umbrella

Can you conclude that the sprinklers are on?

Can AI conclude that the sprinklers are on?

Knowledge base for the story FriendWet FriendWet → (FriendWetBecauseOfRain OR

FriendWetBecauseOfSprinklers) FriendWetBecauseOfSprinklers → SprinklersOn FriendWetBecauseOfRain →

NOT(FriendCarryingUmbrella) UmbrellaGone UmbrellaGone → (YouCarryingUmbrella OR

FriendCarryingUmbrella) NOT(YouCarryingUmbrella)

Syntax What do well-formed sentences in the knowledge

base look like?

A BNF grammar: Symbol := P, Q, R, …, FriendWet, … Sentence := True | False | Symbol | NOT(Sentence) |

(Sentence AND Sentence) | (Sentence OR Sentence) | (Sentence → Sentence)

We will drop parentheses sometimes, but formally they really should always be there

Semantics A model specifies which of the propositional

symbols are true and which are false Given a model, I should be able to tell you whether a

sentence is true or false Truth table defines semantics of operators:

• Given a model, can compute truth of sentence recursively with these

a b NOT(a) a AND b a OR b a → b

false false true false false true

false true true false true true

true false false false true false

true true false true true true

Tautologies A sentence is a tautology if it is true for any

setting of its propositional symbols

• (P OR Q) OR (NOT(P) AND NOT(Q)) is a tautology

P Q P OR Q NOT(P) AND NOT(Q)

(P OR Q) OR (NOT(P) AND NOT(Q))

false false false true true

false true true false true

true false true false true

true true true false true

Is this a tautology? (P → Q) OR (Q → P)

Logical equivalences Two sentences are logically equivalent if they have

the same truth value for every setting of their propositional variables

• P OR Q and NOT(NOT(P) AND NOT(Q)) are logically equivalent. Tautology = logically equivalent to True

P Q P OR Q NOT(NOT(P) AND NOT(Q))

false false false false

false true true true

true false true true

true true true true

Famous logical equivalences (a OR b) ≡ (b OR a) commutatitvity (a AND b) ≡ (b AND a) commutatitvity ((a AND b) AND c) ≡ (a AND (b AND c)) associativity ((a OR b) OR c) ≡ (a OR (b OR c)) associativity NOT(NOT(a)) ≡ a double-negation elimination (a → b) ≡ (NOT(b) → NOT(a)) contraposition (a → b) ≡ (NOT(a) OR b) implication elimination NOT(a AND b) ≡ (NOT(a) OR NOT(b)) De Morgan NOT(a OR b) ≡ (NOT(a) AND NOT(b)) De Morgan (a AND (b OR c)) ≡ ((a AND b) OR (a AND c)) distributivity (a OR (b AND c)) ≡ ((a OR b) AND (a OR c)) distributivity

Entailment A set of premises A logically entails a

conclusion B (written as A |= B) if and only if every model/interpretation that satisfies the premises also satisfies the conclusion, i.e. A → B i.e. B is a valid consequence of A

{p} |= (p ∨ q)

{p} |# (p ∧ q) (e.g. p=T, q=F)

{p, q} |= (p ∧ q)

Entailment We have a knowledge base (KB) of things that we know are

true FriendWetBecauseOfSprinklers FriendWetBecauseOfSprinklers → SprinklersOn

Can we conclude that SprinklersOn? We say SprinklersOn is entailed by the KB if, for every setting

of the propositional variables for which the KB is true, SprinklersOn is also true

SprinklersOn is entailed! (KB |= SprinklersOn)

FWBOS SprinklersOn Knowledge base

false false false

false true false

true false false

true true true

Inconsistent knowledge bases Suppose we were careless in how we specified our

knowledge base:PetOfFriendIsABird → PetOfFriendCanFlyPetOfFriendIsAPenguin → PetOfFriendIsABirdPetOfFriendIsAPenguin → NOT(PetOfFriendCanFly)PetOfFriendIsAPenguin

No setting of the propositional variables makes all of these true Therefore, technically, this knowledge base

implies/entails anything… TheMoonIsMadeOfCheese

Satisfiability A sentence is satisfiable if it is true in some

model If a sentence α is true in model m, then m satisfies α, or

we say m is the model of α A KB is satisfiable if a model m satisfies all clauses in it

How do you verify if a sentence is satisfiable? Enumerate the possible models, until one is found to

satisfy Can order the models so as to evaluate the most

promising ones first

Many problems in Computer Science can be reduced to a satisfiability problem E.g. CSP is all about verifying if the constraints are

satisfiable by some assignments

Inference Inference is the process of deriving a specific

sentence from a KB (where the sentence must be entailed by the KB) KB |-i a = sentence a can be derived from KB by

procedure i

“KBs are a haystack” Entailment = needle in haystack Inference = finding it

If KB is true in the real world, then any sentence derived from KB by a sound inference procedure is also true in the real world

Simple algorithm for inference Want to find out if sentence a is entailed by

knowledge base…

For every possible setting of the propositional variables, If knowledge base is true and a is false, return false

Return true

Not very efficient: 2#propositional variables settings There can be many propositional variables among

premises that are irrelevant to the conclusion. Much wasted work!

Inference by enumeration

Proof methods Proof methods divide into (roughly) two kinds: Model checking (inference by enumeration)

Truth table enumeration (always exponential in n) Improved backtracking, e.g., Davis-Putnam-Logemann-

Loveland (DPLL) Heuristic search in model space (sound but incomplete)

Application of inference rules/reasoning patterns 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

Reasoning patterns Obtain new sentences directly from some other

sentences in knowledge base according to reasoning patterns

E.g., if we have sentences a and a → b, we can correctly conclude the new sentence b This is called modus ponens

E.g., if we have a AND b, we can correctly conclude a

All of the logical equivalences from before also give reasoning patterns

Formal proof that the sprinklers are on Knowledge base:

1. FriendWet

2. FriendWet → (FriendWetBecauseOfRain OR FriendWetBecauseOfSprinklers)

3. FriendWetBecauseOfSprinklers → SprinklersOn

4. FriendWetBecauseOfRain → NOT(FriendCarryingUmbrella)

5. UmbrellaGone

6. UmbrellaGone → (YouCarryingUmbrella OR FriendCarryingUmbrella)

7. NOT(YouCarryingUmbrella)

Formal proof that the sprinklers are on8. YouCarryingUmbrella OR FriendCarryingUmbrella (modus ponens on 5

and 6)

9. NOT(YouCarryingUmbrella) → FriendCarryingUmbrella (equivalent to 8)

10. FriendCarryingUmbrella (modus ponens on 7 and 9)

11. NOT(NOT(FriendCarryingUmbrella) (equivalent to 10)

12. NOT(NOT(FriendCarryingUmbrella)) → NOT(FriendWetBecauseOfRain) (equivalent to 4 by contraposition)

13. NOT(FriendWetBecauseOfRain) (modus ponens on 11 and 12)

14. FriendWetBecauseOfRain OR FriendWetBecauseOfSprinklers (modus ponens on 1 and 2)

15. NOT(FriendWetBecauseOfRain) → FriendWetBecauseOfSprinklers (equivalent to 14)

16. FriendWetBecauseOfSprinklers (modus ponens on 13 and 15)

17. SprinklersOn (modus ponens on 16 and 3)

Reasoning about penguins Knowledge base:1. PetOfFriendIsABird → PetOfFriendCanFly

2. PetOfFriendIsAPenguin → PetOfFriendIsABird

3. PetOfFriendIsAPenguin → NOT(PetOfFriendCanFly)

4. PetOfFriendIsAPenguin

5. PetOfFriendIsABird (modus ponens on 4 and 2)

6. PetOfFriendCanFly (modus ponens on 5 and 1)

7. NOT(PetOfFriendCanFly) (modus ponens on 4 and 3)

8. NOT(PetOfFriendCanFly) → FALSE (equivalent to 6)

9. FALSE (modus ponens on 7 and 8)

10. FALSE → TheMoonIsMadeOfCheese (tautology)

11. TheMoonIsMadeOfCheese (modus ponens on 9 and 10)

Evaluation of deductive inference Sound

Yes, because the inference rules themselves are sound (This can be proven using a truth table argument)

Complete If we allow all possible inference rules, we’re searching in

an infinite space, hence not complete If we limit inference rules, we run the risk of leaving out

the necessary one…

Monotonic If we have a proof, adding information to the KB will not

invalidate the proof

Getting more systematic Any knowledge base can be written as a single formula in

conjunctive normal form (CNF) CNF formula: (… OR … OR …) AND (… OR …) AND … … can be a symbol x, or NOT(x) (these are called literals) Multiple facts in knowledge base are effectively ANDed together

FriendWet → (FriendWetBecauseOfRain OR FriendWetBecauseOfSprinklers)

becomes

(NOT(FriendWet) OR FriendWetBecauseOfRain OR FriendWetBecauseOfSprinklers)

Converting story to CNF FriendWet

FriendWet

FriendWet → (FriendWetBecauseOfRain OR FriendWetBecauseOfSprinklers) NOT(FriendWet) OR FriendWetBecauseOfRain OR

FriendWetBecauseOfSprinklers

FriendWetBecauseOfSprinklers → SprinklersOn NOT(FriendWetBecauseOfSprinklers) OR SprinklersOn

FriendWetBecauseOfRain → NOT(FriendCarryingUmbrella) NOT(FriendWetBecauseOfRain) OR NOT(FriendCarryingUmbrella)

UmbrellaGone UmbrellaGone

UmbrellaGone → (YouCarryingUmbrella OR FriendCarryingUmbrella) NOT(UmbrellaGone) OR YouCarryingUmbrella OR FriendCarryingUmbrella

NOT(YouCarryingUmbrella) NOT(YouCarryingUmbrella)

Unit resolution Unit resolution: if we have

l1 OR l2 OR … li … OR lkand

NOT(li)

we can concludel1 OR l2 OR … li -1 OR li +1 OR … OR lk

Basically modus ponens

Applying resolution to story problem1. FriendWet

2. NOT(FriendWet) OR FriendWetBecauseOfRain OR FriendWetBecauseOfSprinklers

3. NOT(FriendWetBecauseOfSprinklers) OR SprinklersOn

4. NOT(FriendWetBecauseOfRain) OR NOT(FriendCarryingUmbrella)

5. UmbrellaGone

6. NOT(UmbrellaGone) OR YouCarryingUmbrella OR FriendCarryingUmbrella

7. NOT(YouCarryingUmbrella)

8. NOT(UmbrellaGone) OR FriendCarryingUmbrella (6,7)

9. FriendCarryingUmbrella (5,8)

10. NOT(FriendWetBecauseOfRain) (4,9)

11. NOT(FriendWet) OR FriendWetBecauseOfSprinklers (2,10)

12. FriendWetBecauseOfSprinklers (1,11)

13. SprinklersOn (3,12)

Resolution (general) General resolution allows a complete inference

mechanism (search-based) using only one rule of inference

Resolution rule: Given: P1 P2 P3 … Pn and P1 Q1 … Qm

Conclude: P2 P3 … Pn Q1 … Qm

Complementary literals P1 and P1 “cancel out”

Why it works: Consider 2 cases: P1 is true, and P1 is false

Applying resolution1. FriendWet

2. NOT(FriendWet) OR FriendWetBecauseOfRain OR FriendWetBecauseOfSprinklers

3. NOT(FriendWetBecauseOfSprinklers) OR SprinklersOn

4. NOT(FriendWetBecauseOfRain) OR NOT(FriendCarryingUmbrella)

5. UmbrellaGone

6. NOT(UmbrellaGone) OR YouCarryingUmbrella OR FriendCarryingUmbrella

7. NOT(YouCarryingUmbrella)

8. NOT(FriendWet) OR FriendWetBecauseOfRain OR SprinklersOn (2,3)

9. NOT(FriendCarryingUmbrella) OR NOT(FriendWet) OR SprinklersOn (4,8)

10. NOT(UmbrellaGone) OR YouCarryingUmbrella OR NOT(FriendWet) OR SprinklersOn (6,9)

11. YouCarryingUmbrella OR NOT(FriendWet) OR SprinklersOn (5,10)

12. NOT(FriendWet) OR SprinklersOn (7,11)

13. SprinklersOn (1,12)

Systematic inference… General strategy: if we want to see if sentence a is

entailed, add NOT(a) to the knowledge base and see if it becomes inconsistent (we can derive a contradiction) = proof by contradiction

CNF formula for modified knowledge base is satisfiable if and only if sentence a is not entailed Satisfiable = there exists a model that makes the

modified knowledge base true = modified knowledge base is consistent

Resolution algorithm Given formula in conjunctive normal form,

repeat: Find two clauses with complementary literals, Apply resolution, Add resulting clause (if not already there)

If the empty clause results, formula is not satisfiable Must have been obtained from P and NOT(P)

Otherwise, if we get stuck (and we will eventually), the formula is guaranteed to be satisfiable

Example Our knowledge base:

1) FriendWetBecauseOfSprinklers2) NOT(FriendWetBecauseOfSprinklers) OR SprinklersOn

Can we infer SprinklersOn? We add:3) NOT(SprinklersOn)

From 2) and 3), get4) NOT(FriendWetBecauseOfSprinklers)

From 4) and 1), get empty clause = SpinklersOn entailed

Exercises In twos/threes, please!