1. 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

2. FOPL - Formal Language / Syntax -

3. 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.

4. 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

5. 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

6. 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

7. 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.

8. FOPL - Semantics / Interpretation -

9. 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.

10. 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

11. Semantics –Interpretation
Next, determine the semantics for complex terms and formulae constructively, based on the basic interpretation mapping and the valuation function above.

12. Semantics - Interpretation II
Terms with variables
I [f(t1,...,tn)) = I [f] (I [t1],..., I [tn]) =
F(I [t1],..., I [tn]) D
where I[ti] = V(ti) if ti is a variable

13. 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

14. 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.

15. 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  y=z

16. 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.

17. Logical Consequence - Entailment
Given a set of formulae  and a formula α.
α is a logical consequence of  iff
α is true in every model in which  is true.
Notation:
 |= α
That means that for every model (interpretation into a domain) in which  is true, α must also be true.

18. FOPL - Inference System - Axioms & Inference Rules

19. FOPL Axioms A1      A2      A3       
A5 x: (x)  (y)
A6 (x)  y: (y)

20. 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 α is derived
add new formula to KB or Theory
new KB or Theory is α

21. Formal Inference Formal Inference, Theorem
Given a set of formulae  and a set of inference rules IR. A new formula α can be generated based on  using inference rules in IR.
We say that
α is inferred or derived from  or
α is a Theorem (of )
Notation:  |– α

22. IR Modus Ponens Modus Ponens   ,  
States that  can be concluded provided we know that the formulae    and  are true in our knowledge base.

23. IR 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.

24. IR 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.

25. IR Replacement Rules Replacement Rules            
     
     
   (  )
(  )   

26. 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.

27. Soundness and Completeness
An Inference System is sound iff
 |– α   |= α
every formula which can be derived by formal inference from  is a also logical consequence of .
Completeness
An Inference System is complete iff
 |= α   |– α
every formula which is a logical consequence of  can be derived by formal inference from  .

28. 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 ( |– α) is also a logical consequence of the given axioms ( |= α) :
 |– α iff  |= α
Thus, there is a correspondence between formal Inference and logical Consequence.

29. Semantics - Example A1 Predicate Logic Language
constants Bill-1, John-3, Sally-1, Mary-1, Mary-2
predicates happy-together, 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)

30. 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)