1. Logics for Data and Knowledge Representation
First Order Logics (FOL)
Originally by Alessandro Agostini and Fausto Giunchiglia
Modified by Fausto Giunchiglia, Rui Zhang and Vincenzo Maltese

2. Outline
Introduction
Syntax
Semantics
Reasoning Services
Changed 2

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Propositions
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All knowledge representation languages deal with sentences
Sentences denote propositions (by definition), namely they express something true or false (the mental image of what you mean when you write the sentence)
Logic languages deal with propositions
Logic languages have different expressiveness, i.e. they have different expressive power.
“All men are mortals” “Obama is the president of the USA”

4. What we can express with PL and ClassL (I)
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“There is a monkey” Monkey
We use Intensional Interpretation: the truth valuation ν of the proposition Monkey determines a truth-value ν(Monkey).
ν tells us if a proposition P “holds” or not.
ClassL
“Fausto is a Professor” Fausto ∈ σ(Professor)
We use Extensional Interpretation: the interpretation function σ of Professor determines a class of objects σ(Professor) that includes Fausto. In other words, given x ∈ P:
“x belongs to P” or “x in P” or “x is an instance of P” 4

5. What we can express with PL and ClassL (II)
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Monkey ∧ Banana
We mean that there is a monkey and there is a banana.
ClassL
Monkey ⊓ Banana
We mean that there is an x ∈ σ(Monkey) ∩ σ(Banana) 5

6. What we cannot express in PL and ClassL
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n-ary relations
They express objects in Dn
Near(Kimba,Simba) LessThan(x,3) Connects(x,y,z)
Functions
They return a value of the domain, Dn → D
Sum(2,3) Multiply(x,y) Exp(3,6)
Universal quantification
∀x Man(x) → Mortal(x) “All men are mortal”
Existential quantification
∃x (Dog(x) ∧ Black(x)) “There exists a dog which is black”

7. The need for greater expressive power
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We need FOL for a greater expressive power. In FOL we have:
constants/individuals (e.g. 2)
variables (e.g. x)
Unary predicates (e.g. Man)
N-ary predicates (eg. Near)
functions (e.g. Sum, Exp)
quantifiers (∀, ∃)
equality symbol = (optional)
NOTE:
P(a) → ∃x P(x)
∀x P(x) → ∃x P(x)
∀x P(x) → P(x) 7

8. Example of what we can express in FOL
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constants
Cita
Monkey
1-ary predicates
n-ary predicates
Eats
Hunts
Kimba
Simba
Lion
Near 8

9. The use of FOL in mathematics
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FOL has being introduced to express mathematical properties
The set of axioms describing the properties of equality between natural numbers (by Peano):
Axioms about equality
∀x1 (x1 = x1) reflexivity
∀x1 ∀x2 (x1 = x2 → x2 = x1) symmetricity
∀x1 ∀x2 ∀x3 (x1 = x2  x2 = x3 → x1 = x3) transitivity
∀x1 ∀x2 (x1 = x2 → S(x1) = S(x2)) successor
NOTE: Other axioms can be given for the properties of the successor, the addition (+) and the multiplication (x). 9

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Alphabet of symbols
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Variables x1, x2, …, y, z
Constants a1, a2, …, b, c
Predicate symbols A11, A12, …, Anm
Function symbols f11, f12, …, fnm
Logical symbols ∧, ∨, , → , ∀, ∃
Auxiliary symbols ( )
Indexes on top are used to denote the number of arguments, called arity, in predicates and functions.
Indexes on the bottom are used to disambiguate between symbols having the same name.
Predicates of arity =1 correspond to properties or concepts

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Terms
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Terms can be defined using the following BNF grammar:
<term> ::= <variable> | <constant> | <function> (<term> {,<term>}*)
A term is called a closed term iff it does not contain variables
x, 2, SQRT(x), Sum(2, 3), Sum(2, x), Average(x1, x2, x3)
Sum(2, 3)

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Well formed formulas
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Well formed formulas (wff) can be defined as follows:
<atomic formula> ::= <predicate> (<term> {,<term>}*) |
<term> = <term>
<wff> ::= <atomic formula> | ¬<wff> | <wff> ∧ <wff> | <wff> ∨ <wff> |
<wff> → <wff> | ∀ <variable> <wff> | ∃ <variable> <wff>
NOTE: <term> = <term> is optional. If it is included, we have a FO language with equality.
NOTE: We can also write ∃x.P(x) or ∃x:P(x) as notation (with ‘.’ or “:”)

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Well formed formulas
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FOL
“x is a man”
Man(x)
“3 is an odd number”
Odd(3)
“Each number is odd or even”
∀x (Odd(x)  Even(x))
“Each even number greater than 2 is the sum of two prime numbers”
∀x ( Even(x)  GreaterThan(x,2) →
∃y ∃z (Prime(y)  Prime(z)  x = Sum(y,z)) )
There is (at least) a white dog
∃x (Dog(x)  White(x))
There are (at least) two dogs
∃x ∃y (Dog(x)  Dog(y)  (x = y)) 13

14. Scope and index of logical operators
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Given two wff α and β
Unary operators
In ¬α, ∀xα and∃xα,
α is the scope and x is the index of the operator
Binary operators
In α  β, α  β and α  β,
α and β are the scope of the operator
NOTE: in the formula ∀x1 A(x2), x1 is the index but x1 is not in the scope, therefore the formula can be simplified to A(x2). 14

15. Free and bound variables
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A variable x is bound in a formula γ if it is γ = ∀x α(x) or ∃x α(x) that is x is both in the index and in the scope of the operator.
A variable is free otherwise.
A formula with no free variables is said to be a sentence or closed formula.
A FO theory is any set of FO-sentences.
NOTE: we can substitute the bound variables without changing the meaning of the formula, while it is in general not true for free variables. 15

16. Interpretation function
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An interpretation I for a FO language L over a domain D is a function such that:
I(ai) = di ∈D for each constant ai
I(An)  Dn for each predicate A of arity n
I(fn) is a function f: Dn  D  Dn +1 for each function f of arity n 16

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Assignment
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An assignment for the variables {x1, …, xn} of a FO language L over a domain D is a mapping function a: {x1, …, xn}  Dn
a(xi) = di ∈ D
NOTE: In countable domains (finite and enumerable) the elements of the domain D are given in an ordered sequence <d1,…,dn> such that the assignment of the variables xi follows the sequence.
NOTE: the assignment a can be defined on free variables only. 17

18. Interpretation over an assignment a
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An interpretation Ia for a FO language L over an assignment a and a domain D is an extended interpretation where:
Ia(c) = I(c) for each constant c
Ia(x) = a(x) for each variable x
Ia(fn(t1,…, tn)) = I(fn)(Ia(t1),…, Ia(tn)) for each function f of arity n
NOTE: Ia is defined on terms only 18

19. Interpretation over an assignment a
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D = the set N of natural numbers
Succ(x) = the function that returns the successor x+1
zero = we could assign the value 0 to it.
Ia(xi) = a(xi) = i ∈ N
Ia(zero) = I(zero) = 0 ∈ N
Ia(Succ(x)) = I(Succ)(Ia(x)) = S(a(x)) such that S(a(x)) = a(x)+1
Ia(Succ(xi)) = I(Succ)(Ia(xi)) = S(i) = i+1 19

20. Satisfaction relation
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We are now ready to provide the notion of satisfaction relation:
M ⊨ γ [a]
(to be read: M satisfies γ under a or γ is true in M under a)
where:
M is an interpretation function I over D
M is a mathematical structure <D, I>
a is an assignment {x1, …, xn}  Dn
γ is a FO-formula
NOTE: if γ is a sentence with no free variables, we can simply write: M ⊨ γ (without the assignment a) 20

21. Satisfaction relation for well formed formulas
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γ atomic formula:
γ: t1= t2 M ⊨ (t1= t2) [a] iff Ia(t1) = Ia(t2)
γ: An(t1,…, tn) M ⊨ An(t1,…, tn) [a] iff (Ia(t1), …, Ia(tn)) ∈ I(An) 21

22. Satisfaction relation for well formed formulas
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γ well formed formula:
γ:  α M ⊨  α [a] iff M ⊭ α [a]
γ: α  β M ⊨ α  β [a] iff M ⊨ α [a] and M ⊨ β [a]
γ: α  β M ⊨ α  β [a] iff M ⊨ α [a] or M ⊨ β [a]
γ: α  β M ⊨ α  β [a] iff M ⊭ α [a] or M ⊨ β [a]
γ: ∀xiα M ⊨ ∀xiα [a] iff M ⊨ α [s] for all assignments
s = <d1,…, d’i,…, dn> where s varies from a only
for the i-th element (s is called an i-th variant of a)
γ: ∃xiα M ⊨ ∃xiα [a] iff M ⊨ α [s] for some assignment s = <d1,…, d’i,…, dn> i-th variant of a 22

23. Satisfaction relation for a set of formulas
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We say that a formula γ is true (w.r.t. an interpretation I) iff every assignment
a = <d1,…, dn> satisfies γ, i.e. M ⊨ γ [a] for all a.
NOTE: under this definition, a formula γ might be neither true nor false w.r.t. an interpretation I (it depends on the assignment)
If γ is true under I we say that I is a model for γ.
Given a set of formulas Γ, M satisfies Γ iff M ⊨ γ for all γ in Γ 23

24. Satisfiability and Validity
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We say that a formula γ is satisfiable iff there is a structure
M = <D, I> and an assignment a such that M ⊨ γ [a]
We say that a set of formulas Γ is satisfiable iff there is a structure M = <D, I> and an assignment a such that
M ⊨ γ [a] for all γ in Γ
We say that a formula γ is valid iff it is true for any structure and assignment, in symbols ⊨ γ
A set of formulas Γ is valid iff all formulas in Γ are valid. 24

25. Example of satisfiability
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D = {0, 1, 2, 3}
constants = {zero} variables = {x}
predicate symbols = {even, odd} functions = {succ}
γ1 : ∀x (even(x)  odd(succ(x)))
γ2 : even(zero)
γ3 : odd(zero)
Is there any M and a such that M ⊨ γ1  γ2 [a]?
Ia(zero) = I(zero) = 0
Ia(succ(x)) = I(succ)(Ia(x)) = S(n) = {1 if n=0, n+1 otherwise}
I(even) = {0, 2} I(odd) = {1, 3}
Is there any M and a such that M ⊨ γ1  γ3 [a]?
Yes! Just invert the interpretation of even and odd. 25

26. Example of valid formulas
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∀x1 (P(x1)  Q(x1))  ∀x1 P(x1)  ∀x1 Q(x1)
∃x1 (P(x1)  Q(x1))  ∃x1 P(x1)  ∃x1 Q(x1)
∀x1 P(x1)  ∃x1 P(x1)
∀x1 ∃x1 P(x1) ∃x1 P(x1)
∃x1 ∀x1 P(x1)  ∀x1 P(x1) 26

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Entailment
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Let be  a set of FO- formulas, γ a FO- formula, we say that
 ⊨ γ
(to be read  entails γ)
iff for all the interpretations M and assignments a,
if M ⊨  [a] then M ⊨ γ [a]. 27

28. Reasoning Services: EVAL
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Model Checking (EVAL)
Is a FO-formula γ true under a structure M = <D, I> and an assignment a?
Check M ⊨ γ [a]
EVAL
γ , M, a
Yes No
Undecidable in infinite domains (in general)
Decidable in finite domains 28

29. Reasoning Services: SAT
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Satisfiability (SAT)
Given a FO-formula γ, is there any structure M = <D, I> and an assignment a such that M ⊨ γ [a]?
Theorem (Church, 1936): SAT is undecidable
However, it is decidable in finite domains.
SAT γ
M, a No 29

30. Reasoning Services: VAL
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Validity (VAL)
Given a FO-formula γ, is γ true for all the interpretations M and assignments a, i.e. ⊨ γ?
Theorem (Church, 1936): VAL is undecidable
However, it is decidable in finite domains.
VAL γ
Yes No 30

31. How to reason on finite domains (I)
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⊨ ∀x P(x) [a] D = {a, b, c}
we have only 3 possible assignments a(x) = a, a(x) = b, a(x) = c
we translate in ⊨ P(a)  P(b)  P(c)
⊨ ∃x P(x) [a] D = {a, b, c}
we translate in ⊨ P(a)  P(b)  P(c) 31

32. How to reason on finite domains (II)
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⊨ ∀x ∃y R(x,y) [a] D = {a, b, c}
we have 9 possible assignments, e.g. a(x) = a, a(y) = b
we translate in ⊨ ∃y R(a,y)  ∃y R(b,y)  ∃y R(c,y)
and then in ⊨ (R(a,a)  R(a,b)  R(a,c) ) 
(R(b,a)  R(b,b)  R(b,c) ) 
(R(c,a)  R(c,b)  R(c,c) ) 32

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Calculus
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Semantic tableau is a decision procedure to determine the satisfiability of finite sets of formulas.
There are rules for handling each of the logical connectives thus generating a truth tree.
A branch in the tree is closed if a contradiction is present along the path (i.e. of an atomic formula, e.g. B and B)
If all branches close, the proof is complete and the set of formulas are unsatisfiable, otherwise are satisfiable.
With refutation tableaux the objective is to show that the negation of a formula is unsatisfiable.
Γ = {(A B), B} B
A  B A B
closed 33

34. Rules of the semantic tableaux in FOL (I)
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The initial set of formulas are considered in conjunction and are put in the same branch of the tree
Conjunctions lie on the same branch
Disjunctions generate new branches
Propositional rules:
() A  B () A  B
A A | B B
()   A A
A   A
(A B)  B B
(A  B)
A  B A B
closed 34

35. Rules of the semantic tableaux in FOL (II)
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Universal quantifiers are instantiated with a terminal t.
Existential quantifiers are instantiated with a new constant symbol c.
(∀) ∀x.P(x)
P(t)
(∃) ∃x.P(x)
P(c)
NOTE: a constant is a terminal
NOTE: Multiple applications of the ∀ rule might be necessary (as in the example)
{∀x.P(x), ∃x.(¬P(x)  ¬P(f(x)))
∀x.P(x)
∃x.(¬P(x)  ¬P(f(x))
¬P(c)  ¬P(f(c))
(∃)
P(c)
(∀)
¬P(c)
¬P(f(c))
()
closed
P(f(c))
(∀)
closed 35

36. Rules of the semantic tableaux in FOL (III)
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Correctness: the two rules for universal and existential quantifiers together with the propositional rules are correct:
closed tableau → unsatisfiable set of formulas
Completeness: it can also be proved:
unsatisfiable set of formulas → ∃ closed tableau
However, the problem is finding such a closed tableau. An unsatisfiable set can in fact generate an infinite-growing tableau.
∀ rule is non-deterministic: it does not specify which term to instantiate with. It also may require multiple applications.
In tableaux with unification, the choice of how to instantiate the ∀ is delayed until all the other formulas have been expanded. At that point the ∀ is applied to all unbounded formulas. 36