74.419 Artificial Intelligence 2005 FirstOrder Predicate Logic

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74.419 Artificial Intelligence 2005 -
First-Order Predicate Logic -

• First-Order Predicate Logic (FOL or FOPL), also
  called First-Order Predicate Calculus
• Formal Language
• Semantics through Interpretation Function
• Axioms
• Inference System

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FOPL- Formal Language / Syntax -
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Formal Language
A Formal Language is specified as L (NT, T, P,
S) NT Set of Non-Terminal Symbols T Set of
Terminal Symbols P Set of Production or Grammar
Rules S Start Symbol (top-level node in syntax
tree / parse tree) A formal language specifies
the syntactically correct or well-formed
expressions of a language.
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Terminals and Non-Terminals
NT Non-Terminals wff (well-formed formula),
atomic-formula Predicate, Term, Function,
Constant, Variable Quantifier, Connective T
Terminals Predicate (Symbols) P, Q, married, ...,
Function (Symbols) f, g, father-of,
... Variables x, y, z, ... Constants Sally,
block-1, c Connectives ?, ?, ?, ? Negation
Symbol ? Quantifiers ?, ? Equality
Symbol Parentheses ( , ) Other Symbols
Domain Specifc
General
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Production / Grammar Rules
Non-terminal Rules wff atomic-formula
(wff) ? wff wff Connective wff
Quantifier Variable wff atomic-formula
Predicate (Term, ...) Term Term Term
Function (Term, ...) Variable Constant
Terminal Rules Connective ? ? ? ?
Quantifier ? ? Note n-ary functions and
predicates go with n terms
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Domain-Specific Terminal Rules
Terminal Rules for the specific Domain Predicate
on(_,_) near(_,_) ... Function
distance(_,_) location(_) ... Variable x
y ... Constant Flakey John-Bear Karen
Alan-Alder The-File Kurt
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Quantifiers and Binding

• A variable in a formula can be bound by a
  quantifier.
• bound variable ?x married (Sally, x)
• open formula a variable in the formula is not
  bound by a quantifier ?x married
  (Sally, x) ? happy (y)
• closed formula all variables in the formula are
  bound by quantifiers ?x ?y married
  (x, y)
• Most authors regard quantified formulas only as
  wffs if
• all quantified variables appear in the formula.
• Some authors regard quantified formulas only as
  wffs if
• all variables are bound by quantifiers.

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FOPL- Semantics / Interpretation -
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Semantics - Overview

• Define the Semantics of FOPL expressions
  (formulae)
• Interpretation Maps symbols of the formal
  language (predicates, functions, variables,
  constants) onto objects, relations, and functions
  of the world (formally Domain, relational
  Structure, or Universe)
• Valuation - Assigns domain objects to
  variables
• Constructive Semantics Determines the
  semantics of complex expressions inductively,
  starting with basic expressions
• The Valuation function can be used for
  describing value assignments and constraints in
  case of nested quantifiers.
• The Valuation function otherwise determines the
  satisfaction of a formula only in case of open
  formulae.

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Semantics Domain, Interpretation I
Domain, relational Structure, Universe D finite
set of Objects d1, d2, ... , dn R,... Relations
over D R ? Dn F,... Functions over D F Dn
?D Basic Interpretation Mapping constant I c
d Object function I f F
Function predicate I P R Relation
Valuation V variable V(x) d?D
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Semantics Interpretation
Next, determine the semantics for complex terms
and formulae constructively, based on the basic
interpretation mapping and the valuation function
above.
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Semantics - Interpretation II
Terms with variables I f(t1,...,tn)) I f
(I t1,..., I tn) F(I t1,..., I tn) ?D
where Iti V(ti) if ti is a variable
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Semantics - Interpretation III
atomic Formula I P(t1,...,tn) true if (I
t1,..., I tn) ? I P R negated Formula
I ?? true if I ? is not true complex
Formula I??? true if I ? or I ? true I
??? true if I ? and I ? true I ??? if
I ? not true or I ? true
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Semantics - Interpretation III
quantified Formula (relative to Valuation
function) I ?x? true if ? is true with
V(x)d for some d?D where V is otherwise
identical to the prior V. I ?x? true if ? is
true with V(x)d for all d?D and where V is
otherwise identical to the prior V. Note ?x?y?
is different from ?y?x? In the first case
?x?y? , we go through all value assignments
V'(x), and for each value assignment V'(x) of x,
we have to find a suitable value V'(y) for y.
In the second case ?y?x?, we have to find one
value V'(y) for y, such that all value
assignments V'(x) make ? true.
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Semantics - Quantifiers - Examples
Arithmetics What could this be? - What are f, z,
x, y ? ?z?x?y f(x,y)z What is this? (It's
easier.) ?x?y?z f(x,y)z This one's different.
?x?y f(x)y ?x?y?z f(x)y ? f(x)z ? yz
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Semantics - Satisfiability
Given is an interpretation I into a domain D with
a valuation V, and a formula ?. We say that ?
is satisfied in this interpretation or this
interpretation is a model of ? iff I? is
true. That means the interpretation function I
into the domain D (with valuation V) makes the
formula ? true.
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Logical Consequence - Entailment
Logical Consequence (Entailment) Given a set of
formulae ? and a formula a. a is a logical
consequence of ? iff a is true in every model
in which ? is true. Notation ? a That
means that for every model (interpretation into a
domain) in which ? is true, a must also be true.
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FOPL- Inference System -Axioms Inference Rules
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FOPL Axioms
A1 ? ? ? ? ? A2 ? ? ? ? ? A3 ? ? ? ? ? ? ?
A4 (? ? ?) ? ((? ? ?) ? (? ? ?)) A5 ?x ?(x) ?
?(y) A6 ?(x) ? ?y ?(y)
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Formal Inference - Overview

• Derive new formulae by syntactic manipulation of
  existing formulae
• given set of formulae ?
• ? describes your KB, or a Theory, ... (FOPL
  axioms your own "proper" axioms)
• apply inference rule (based on ? )
• new formula a is derived
• add new formula to KB or Theory
• new KB or Theory is ??a

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Formal Inference
Formal Inference, Theorem Given a set of formulae
? and a set of inference rules IR. A new formula
a can be generated based on ? using inference
rules in IR. We say that a is inferred or
derived from ? or a is a Theorem (of
?) Notation ? a
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IR Modus Ponens

• Modus Ponens
• ? ? ?, ?
• ?
• States that ? can be concluded provided we know
  that the formulae ? ? ? and ? are true in our
  knowledge base.

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IR Universal Instantiation

• Universal Instantiation
• ?x ?(x)
• ?(c)
• where ?(x) is any formula containing the variable
  x, and ?(c) is the formula ?(x) where every
  occurrence of the quantified variable x is
  substituted with the arbitrary constant c.

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IR Existential Generalization

• Existential Generalization
• ?(c)
• ?x ?(x)
• where ?(c) is any formula containing the
  arbitrary constant c, and ?(x) is the same
  formula as ?(c) but with every occurrence of the
  constant c replaced by a variable x.

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IR Replacement Rules
Replacement Rules ? ? ? ?? ? ? ?? ? ?
? ? ? ? ? ? ?(?? ? ??) ?(?? ?
??) ? ? ?
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FOPL Inference System

• The Axioms and the Inference Rules above
  constitute a formal inference system for FOPL.
• This system - we call it FS1 - is complete and
  sound.

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Soundness and Completeness
Soundness An Inference System is sound iff ?
a ? ? a every formula which can be derived
by formal inference from ? is a also logical
consequence of ?. Completeness An Inference
System is complete iff ? a ? ? a every
formula which is a logical consequence of ? can
be derived by formal inference from ? .
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FOPL - Sound and Complete 2
The above inference system for FOPL is sound and
complete. Thus, every formula which can be
derived in FOPL using FS1 (? a) is also a
logical consequence of the given axioms (? a)
? a iff ? a Thus, there is a
correspondence between formal Inference and
logical Consequence.
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Semantics - Example A1
Predicate Logic Language constants Bill-1,
John-3, Sally-1, Mary-1, Mary-2 predicates happy-t
ogether, hate-each-other Structure D objects
Uncle-Bill, Uncle-John, Aunt-Sally,
The-woman-I-don't-like, Mary relations Married,
Divorced (Uncle-Bill, Aunt-Sally) ? Married,
(Uncle-John, Mary) ? Married (Uncle-John,
The-woman-I-don't-like) ? Divorced
Interpretation I(Bill-1)Uncle-Bill,
I(John-3)Uncle-John, I(Sally-1)Aunt-Sally,
I(Mary-1)The-woman-I-don't-like,
I(Mary-2)Mary I(happy-together)Married,
I(hate-each-other)Divorced True or false?
hate-each-other (Bill-1, John-3)
happy-together(Bill-1, Sally-1) hate-each-other(
John-3, Mary-1) happy-together(John-3, Mary-2)
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Semantics -Example A2
Structure D objects Uncle-Bill, Uncle-John,
Aunt-Sally, The-woman-I-don't-like,
Mary relations Married, Divorced (Uncle-John,
The-woman-I-don't-like) ? Divorced (Uncle-Bill,
Aunt-Sally) ? Married, (Uncle-John, Mary) ?
Married (or (Uncle-Bill, Aunt-Sally),
(Uncle-John, Mary) Married) Interpretation
I I(Bill-1) Uncle-Bill, I(John-3) Uncle-John,
I(Sally-1) Aunt-Sally, I(Mary-1) Mary,
I(Mary-2) The-woman-I-don't-like
I(happy-together) Married, I(hate-each-other)
Divorced True or false? hate-each-other
(Bill-1, John-3) ? hate-each-other (John-3,
Mary-1) happy-together (Bill-1, Sally-1) ?
happy-together (John-3, Mary-2) ?x
happy-together(Uncle-Bill, x)) ?x,y,z
happy-together(x,y) ? hate-each-other (x,z) What
if you want to add a formula? ?x,y
happy-together(x,y) ? happy-together(y,x)