Logic
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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.

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Logic

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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 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).


    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 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)


    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


    First-order logic

    • 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, …


    And

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


    Syntax of FOL: Basic elements

    • ConstantsTaoiseachJohn, 2, DIT,...

    • PredicatesBrother, >,...

    • FunctionsSqrt, LeftLegOf,...

    • Variablesx, 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)


    Truth in first-order logic

    • 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


    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


    • 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)

       ...


    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”


    Existential quantification

    • <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)

       ...


    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!


    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)


    Equality

    • 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)]


    Using FOL

    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)


    Knowledge engineering in FOL

    • 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


    Summary

    • First-order logic:

      • objects and relations are semantic primitives

      • syntax: constants, functions, predicates, equality, quantifiers


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