Predicate Calculus. First-order logic Whereas propositional logic assumes the world contains facts,...
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Transcript of Predicate Calculus. First-order logic Whereas propositional logic assumes the world contains facts,...
Predicate Calculus
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, …
–
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)
•
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
–•
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)$
x P 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 = 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)]•
•
Using FOL
The kinship domain:• Brothers are siblings
x,y Brother(x,y) Sibling(x,y)
–
•
• 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)
–•
–
•
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
–
•
Semantics for Predicate Calculus
• An interpretation over D is an assignment of the entities of D to each of the constant, variable, predicate and function symbols of a predicate calculus expression such that:
• 1: Each constant is assigned an element of D• 2: Each variable is assigned a non-empty subset
of D;(these are the allowable substitutions for that variable)
• 3: Each predicate of arity n is defined on n arguments from D and defines a mapping from Dn into {T,F}
• 4: Each function of arity n is defined on n arguments from D and defines a mapping from Dn into D
The meaning of an expression
• Given an interpretation, the meaning of an expression is a truth value assignment over the interpretation.
Truth Value of Predicate Calculus expressions
• Assume an expression E and an interpretation I for E over a non empty domain D. The truth value for E is determined by:
• The value of a constant is the element of D assigned to by I
• The value of a variable is the set of elements assigned to it by I
More truth values
• The value of a function expression is that element of D obtained by evaluating the function for the argument values assigned by the interpretation
• The value of the truth symbol “true” is T
• The value of the symbol “false” is F
• The value of an atomic sentence is either T or F as determined by the interpretation I
Similarity with Propositional logic truth values
• The value of the negation of a sentence is F if the value of the sentence is T and F otherwise
• The values for conjunction, disjunction ,implication and equivalence are analogous to their propositional logic counterparts
Universal Quantifier
• The value for
• Is T if S is T for all assignments to X under I, and F otherwise
Existential Quantifier
• The value for
• Is T if S is T for any assignment to X under I, and F otherwise
Some Definitions
• A predicate calculus expressions S1 is satisfied.
• Definition If there exists an Interpretation I and a variable assignment under I which returns a value T for S1 then S1 is said to be satisfied under I.
• S is Satisfiable if there exists an interpretation and variable assignment that satisfies it: Otherwise it is unsatisfiable
Some Definitions
• A set of predicate calculus expressions S is satisfied.
• Definition For any interpretation I and variable assignment where a value T is returned for every element in S the the set S is said to be satisfied,
• A set of expressions is satisfiable if and only if there exist an intrepretation and variable assignment that satisfy every element
• If a set of expressions is not satisfiable, it is said to be inconsistent
• If S has a value T for all possible interpretations , it is said to be valid
Some Definitions• A predicate calculus expressions S1 is satisfied.• Definition If there exists an Interpretation I and a variable
assignment under I which returns a value T for S1 then S1 is said to be satisfied under I.
• A set of predicate calculus expressions S is satisfied.• • Definition For any interpretation I and variable assignment
where a value T is returned for every element in S the the set S is said to be satisfied,
• An inference rule is complete.• Definition If all predicate calculus expressions X that logically
follow from a set of expressions, S can be produced using the inference rule , then the inference rule is said to be complete.
• A predicate calculus expression X logically follows from a set S of predicate calculus expressions .
• For any interpretation I and variable assignment where S is satisfied, if X is also satisfied under the same interpretation and variable assignment then X logically follows from S.
• Logically follows is sometimes called entailment
Soundness
• An inference rule is sound.
• If all predicate calculus expressions X produced using the inference rule from a set of expressions, S logically follow from S then the inference rule is said to be sound.
Completeness
• An inference Rule is complete if given a set S of predicate calculus expressions, it can infer every expression that logically follows from S
Equivalence
• Recall that :• See attached word
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