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Logic

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Logic

Propositional Logic

- 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..

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.

- 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..

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 ….

- 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”

- 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)

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?

- P Q means that if P is true, so is Q.

X Y X v Y

T T TT F T

F 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).

- The truth tables for Propositional Calculus are as follows

- 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

- 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.

- 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

- What can we conclude?

sunny v raining

raining v umbrella

- 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.

- 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.

- 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)

- 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.

Propositional logic is declarative

Propositional 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

- Whereas propositional logic assumes the world contains facts,
- first-order logic (like natural language) assumes the world containsObjects: 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, …

- ConstantsTaoiseachJohn, 2, DIT,...
- PredicatesBrother, >,...
- FunctionsSqrt, LeftLegOf,...
- Variablesx, y, a, b,...
- Connectives, , , ,
- Equality=
- Quantifiers ,

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 are made from atomic sentences using connectives
S, S1 S2, S1 S2, S1 S2, S1S2,

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

>(1,2) ≤ (1,2)

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

- Sentences are true with respect to a model and an interpretation
- Model contains objects (domainelements) and relations among them
- Interpretation specifies referents for
constantsymbols→objects

predicatesymbols→relations

functionsymbols→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

- <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

- 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)

...

- 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”

- <variables> <sentence>
- Someone at DIT is smart:
- x At(x,DIT) Smart(x)$
- xP is true in a model m iff P is true with x being some possible object in the model

- 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)

...

- 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!

- x y is the same as yx
- x y is the same as yx
- x y is not the same as yx
- x y Loves(x,y)
- “There is a person who loves everyone in the world”

- yx 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) xLikes(x,Broccoli)

- term1 = term2is true under a given interpretation if and only if term1and term2refer to the same object
- E.g., definition of Sibling in terms of Parent:
x,ySibling(x,y) [(x = y) m,f (m = f) Parent(m,x) Parent(f,x) Parent(m,y) Parent(f,y)]

The kinship domain:

- Brothers are siblings
x,y Brother(x,y) Sibling(x,y)

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

- “Sibling” is symmetric
x,y Sibling(x,y) Sibling(y,x)

- Identify the task
- Assemble the relevant knowledge
- Decide on a vocabulary of predicates, functions, and constants
- Encode general knowledge about the domain
- Encode a description of the specific problem instance
- Pose queries to the inference procedure and get answers
- Debug the knowledge base

- First-order logic:
- objects and relations are semantic primitives
- syntax: constants, functions, predicates, equality, quantifiers