Logic

Propositional Logic

Logic as a Knowledge Representation Language

• A Logic is a formal language, with precisely defined syntax and semantics, which supports sound inference. Independent of domain of application.

• Different logics exist, which allow you to represent different kinds of things, and which allow more or less efficient inference.– ..

– propositional logic, predicate logic, temporal logic, modal logic, description logic..

• But representing some things in logic may not be very natural, and inferences may not be efficient. More specialised languages may be better

Formal Languages

• Intelligent systems require that we have– Knowledge formally represented– New inferences/conclusions possible.

• Formal languages have been developed to support knowledge representation.

• One important one is the use of logic - very general purpose way to formally represent truths about the world, and draw sound conclusions from these.

Propositional logic

• In general a logic is defined by– syntax: what expressions are allowed in the

language.– Semantics: what they mean, in terms of a

mapping to real world – proof theory: how we can draw new

conclusions from existing statements in the logic.

• Propositional logic is the simplest..

What is a proposition• Proposition = Statement that may be

either true or false. • John is in the classroom. • Mary is enrolled in 270A.• If A is true, and A implies B, then B is

true.• If some A are B, and some B are C, then

some A are C.• If some women are students, and some

students are men, then ….

Propositional Logic: Syntax

• Symbols (e.g., letters, words) are used to represent facts about the world, e.g.,– “P” represents the fact “Andrew likes chocolate”– “Q” represents the fact “Andrew has chocolate”

• These are called atomic propositions

• True and false are also atomic propositions

• Logical connectives are used to represent and: , or: , if-then: , not: .

• Statements or sentences in the language are constructed from atomic propositions and logical connectives.– P Q “Andrew likes chocolate and he doesn’t have any.”

– P Q “If Andrew likes chocolate then Andrew has chocolate”

Propositional Logic: Semantics

• What does it all mean?

• Sentences in propositional logic tell you about what is true or false.– P Q means that both P and Q are true.– P Q means that either P or Q is true (or

both)

Concerns• What does it mean to say a statement is

true?

• What are sound rules for reasoning

• What can we represent in propositional logic?

• What is the efficiency?

• Can we guarantee to infer all true statements?

Propositional Logic: Semantics– P Q means that if P is true, so is Q.

• This is all formally defined using truth tables.

X Y X v YT T TT F TF T TF F F

We now know exactly what is meant in terms of the truth of the elementary propositions when we get a sentence in the language (e.g., P => Q v R).

Truth Tables

• The truth tables for Propositional Calculus are as follows

Negation

Conjunction

Disjunction

Implication

Exclusive Or

IFF Equivalence

Proof Theory

• How do we draw new conclusions from existing supplied facts?• We can define inference rules, which are guaranteed to give true

conclusions given true premises.• For propositional logic useful one is modus ponens:

• If A is true and A=> B is true, then conclude B is true.

A, A B—————————

B

Proof Theory and Inference

• So, let P mean “It is raining”, Q mean “I carry my umbrella”.

• If we know that P is true, and P => Q is true..

• We can conclude that Q is true.• Note that certain expressions are

equivalent – think about P => Q and P v Q.

More complex rules of inference

• Other rules of inference can be used, e.g.,:

• This is essentially the resolution rule of inference, used in Prolog.

A v B, B v C———————————————

A v C

Consider

• What can we conclude?

sunny v raining raining v umbrella

Semantics• Model = possible world• x+y = 4 is true in the world x=3, y=1.• x+y = 4 is false in the world x=3, y = 1.• Entailment S1,S2,..Sn |= S means in

every world where S1…Sn are true, S is true.

• Careful: No mention of proof – just checTaoiseach all the worlds.

• Some cognitive scientists argue that this is the way people reason.

Reasoning or Inference Systems

• Proof is a syntactic property.

• Rules for deriving new sentences from old ones.

• Sound: any derived sentence is true.

• Complete: any true sentence is derivable.

• NOTE: Logical Inference is monotonic. Can’t change your mind.

Translation into Propositional Logic

• If it rains, then the game will be cancelled.• If the game is cancelled, then we clean house.• Can we conclude?

– If it rains, then we clean house.

• p = it rains, q = game cancelled r = we clean house.

• If p then q. not p or q• If q then r. not q or r• if p then r. not p or r (resolution)

What can’t we say?• Quantification: every student has a father.• Relations: If X is married to Y, then Y is

married to X.• Probability: There is an 80% chance of rain.• Combine Evidence: This car is better than

that one because…• Uncertainty: Maybe John is playing golf.

Advantages of propositional logic

Propositional 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)

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Disadvantages

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

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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, …

And

• Functions: father of, best friend, one more than, plus, …

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Syntax of FOL: Basic elements

• Constants TaoiseachJohn, 2, DIT,...

• Predicates Brother, >,...

• Functions Sqrt, LeftLegOf,...

• Variables x, y, a, b,...

• Connectives , , , , • Equality =

• Quantifiers ,

Atomic sentences

Atomic sentence = predicate (term1,...,termn) or term1 = term2

Term = function (term1,...,termn) or constant or variable

• E.g., Brother(TaoiseachJohn,RichardTheLionheart) > (Length(LeftLegOf(Richard)), Length(LeftLegOf(TaoiseachJohn)))

Complex sentences

• Complex sentences are made from atomic sentences using connectives

S, S1 S2, S1 S2, S1 S2, S1 S2,

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

>(1,2) ≤ (1,2)

>(1,2) >(1,2)

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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 forconstant symbols → objectspredicate symbols → relationsfunction symbols → functional relations

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

are in the relation referred to by predicate

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Universal quantification <variables> <sentence>

Everyone at DIT is smart:x At(x,DIT) Smart(x)

x P is true in a model m iff P is true with x being each possible object in the model

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• Roughly speaTaoiseach, equivalent to the conjunction of instantiations of P

At(TaoiseachJohn,DIT) Smart(TaoiseachJohn) At(Richard,DIT) Smart(Richard) At(DIT,DIT) Smart(DIT) ...

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A common mistake to avoid

• Typically, is the main connective with • Common mistake: using as the main

connective with :x At(x,DIT) Smart(x)

means “Everyone is at DIT and everyone is smart”

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Existential quantification <variables> <sentence>

• Someone at DIT is smart: x At(x,DIT) Smart(x)$

x P is true in a model m iff P is true with x being some possible object in the model

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• Roughly speaTaoiseach, equivalent to the disjunction of instantiations of P

At(TaoiseachJohn,DIT) Smart(TaoiseachJohn) At(Richard,DIT) Smart(Richard) At(DIT,DIT) Smart(DIT) ...

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Another common mistake to avoid

• Typically, is the main connective with

• Common mistake: using as the main connective with :

x At(x,DIT) Smart(x)

is true if there is anyone who is not at DIT!

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Properties of quantifiers x y is the same as y x x y is the same as y x

x y is not the same as y x x y Loves(x,y)

– “There is a person who loves everyone in the world” y x Loves(x,y)

– “Everyone in the world is loved by at least one person”

• Quantifier duality: each can be expressed using the other x Likes(x,IceCream) x Likes(x,IceCream) x Likes(x,Broccoli) x Likes(x,Broccoli)

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Equality

• term1 = term2 is true under a given interpretation if and only if term1 and term2 refer to the same object

• E.g., definition of Sibling in terms of Parent:x,y Sibling(x,y) [(x = y) m,f (m = f)

Parent(m,x) Parent(f,x) Parent(m,y) Parent(f,y)]•

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Using FOL

The kinship domain:• Brothers are siblings

x,y Brother(x,y) Sibling(x,y)

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• One's mother is one's female parentm,c Mother(c) = m (Female(m) Parent(m,c))

• “Sibling” is symmetricx,y Sibling(x,y) Sibling(y,x)

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Knowledge engineering in FOL

1. 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

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