First Order Logic
Néstor Cataño
[email protected]
Faculty of Engineering
Pontificia Universidad Javeriana
First Order Logic – p.1/23
Lecture Plan
1. First Order Logic Syntax
2. First Order Logic Semantics
3. Models and Tautologies
4. Examples
First Order Logic – p.2/23
Lecture Plan
1. First Order Logic Syntax
2. First Order Logic Semantics
3. Models and Tautologies
4. Examples
First Order Logic – p.3/23
First Order Logic (FOL)
Domain of discourse
Variables range over specific domains such Integers or Reals
Relations such as ≤ and functions like × and +
Variables may be quantified universally or existentially
Examples of formulae in FOL:
∀x ∈ Z. ∃y ∈ Z. x + y = 0
∀x ∈ N . x 6= x + 1
First Order Logic – p.4/23
Terms
Terms are expressions formed using function symbols and variables
term := var | const | f unc (term, term, ..., term)
First Order Logic – p.5/23
Terms
Terms are expressions formed using function symbols and variables
term := var | const | f unc (term, term, ..., term)
For instance, add(x, add(y, one)) is a term
First Order Logic – p.5/23
Signatures
Signature G = (V, F, R) with three disjoint sets:
Set of variable symbols V
Function symbols F
Relation symbols R
Each function and relation symbol is associated an arity
For instance add has arity two
Constant symbols such as zero or one are considered to be function
symbols of arity 0
First Order Logic – p.6/23
Formulae
Formulas are formed from terms using relations
simpF orm := rel(term, term, ..., term) | term ≡ term
f orm :=
simpF orm | (f orm ∧ f orm) | (f orm∨f orm) |
(f orm → f orm) | (¬f orm) | ∀x.(f orm)
∃x.(f orm) | true | f alse
First Order Logic – p.7/23
Formulae
Formulas are formed from terms using relations
simpF orm := rel(term, term, ..., term) | term ≡ term
f orm :=
simpF orm | (f orm ∧ f orm) | (f orm∨f orm) |
(f orm → f orm) | (¬f orm) | ∀x.(f orm)
∃x.(f orm) | true | f alse
For instance, ge(one, zero)∨ge(add(one, one), x)) is a formula
First Order Logic – p.7/23
Lecture Plan
1. First Order Logic Syntax
2. First Order Logic Semantics
3. Models and Tautologies
4. Examples
First Order Logic – p.8/23
Lecture Plan
1. First Order Logic Syntax
2. First Order Logic Semantics
3. Models and Tautologies
4. Examples
First Order Logic – p.9/23
Semantics
In order to interpret terms, we first need to assume a given domain D of
values, for instance, integers or natural numbers
A first order structure S = (G, D, F, R, f ) is defined over a signature G
G = (V, F, R)
A domain D of discourse
A set of functions F (including constant symbols)
A set of relations R
And a mapping f : F ∪ R → F ∪ R
First Order Logic – p.10/23
First Order Structure: Example
S = ( ({x, y}, {add, mult}, {ge}), N , {+, ×}, {≥}, f )
f associates add to + (addition over integers)
f associates mult to × (multiplication over integers)
f associates ge to ≥
First Order Logic – p.11/23
Semantic Interpretation: Terms
An assignment a : V → D is a mapping of variables to values in the
domain D
A semantic interpretation Ta : terms(G) → D maps each term to a
variable in the domain
Ta (v) = a(v), for v ∈ V
Ta (f unc (e1 , e2 , ..., en )) = f (f unc)(Ta (e1 ), Ta (e2 ), ..., Ta (en ))
First Order Logic – p.12/23
Semantic Interpretation: Example
D is the domain of Integers
Assignment a = {x 7→ 2, y 7→ 3, z 7→ 4}
f maps add to the addition over Integers
Therefore, Ta (add(add(x, y), z)) = 5 + 4 = 9
First Order Logic – p.13/23
Semantic Interpretation: Formulae
Interpretation Ma : f orms(G) → {TRUE, FALSE} assigns a truth value to
each formula in G
First Order Logic – p.14/23
Semantic Interpretation: Formulae
Interpretation Ma : f orms(G) → {TRUE, FALSE} assigns a truth value to
each formula in G
Ma (rel(e1 , ..., en )) = f (rel)(Ta (e1 ), ..., Ta (en ))
Ma (e1 ≡ e2 ) = (Ta (e1 ) = Ta (e2 ))
Ma (f1 ∧ f2 ) = TRUE iff Ma (f1 ) = TRUE and Ma (f2 ) = TRUE
Ma (f1 ∨f2 ) = TRUE iff Ma (f1 ) = TRUE or Ma (f2 ) = TRUE
Ma (f1 → f2 ) = TRUE iff Ma (f1 ) = FALSE or Ma (f2 ) = TRUE
Ma (¬f1 ) = TRUE iff Ma (f1 ) = FALSE
Ma (true) = TRUE
Ma (f alse) = FALSE
First Order Logic – p.14/23
Semantic Interpretation: Example
Interpreting the formula φ = ge(add(add(x, y), z), y) ∧ ge(z, zero) under the
assignment a = {x 7→ 2, y 7→ 3, z 7→ 4} is done as follows:
Ta (add(add(x, y), z)) = 9 as in the example before
Ta (y) = 3
Ta (z) = 4
Ta (zero) = 0
Ma (ge(add(add(x, y), z), y)) = f (ge)(Ta (add(add(x, y), z)), Ta (y)) = 9 >
3 = TRUE
Ma (ge(z, zero)) = f (ge)(Ta (z), Ta (zero)) = 4 ≥ 0 = TRUE
Ma (φ) = TRUE
First Order Logic – p.15/23
Semantic Interpretation
Let a be an assignment, v a variable, and d ∈ D
A variant a[d/v] is defined as
a[d/v](u) =

a(u) if u 6= v
d if u = v
Ma (∀v(φ)) = TRUE iff for each d ∈ D, Ma[d/v] (φ) = TRUE
Ma (∃v(φ)) = TRUE iff there exists d ∈ D, Ma[d/v] (φ) = TRUE
First Order Logic – p.16/23
Lecture Plan
1. First Order Logic Syntax
2. First Order Logic Semantics
3. Models and Tautologies
4. Examples
First Order Logic – p.17/23
Lecture Plan
1. First Order Logic Syntax
2. First Order Logic Semantics
3. Models and Tautologies
4. Examples
First Order Logic – p.18/23
Models and Tautologies
Notations
S
a |=φ is read as a satisfies φ under the structure S
S
|=φ is read S is a model of φ
|=φ is read φ is a tautology
First Order Logic – p.19/23
Models and Tautologies
Notations
S
a |=φ is read as a satisfies φ under the structure S
S
|=φ is read S is a model of φ
|=φ is read φ is a tautology
Formal Definition
S
a |= φ if Ma (φ) = TRUE
S
S
|= φ if a |= φ for each assignment a
S
|= φ if |= φ for each structure S
First Order Logic – p.19/23
Examples
1. x ≡ y × 2 ?
2. x × 2 ≡ x + x ?
3. (x ≡ y ∧ y ≡ z) → x ≡ z ?
First Order Logic – p.20/23
Examples
1. x ≡ y × 2 ?
2. x × 2 ≡ x + x ?
3. (x ≡ y ∧ y ≡ z) → x ≡ z ?
S
a |= x ≡ y × 2, where S is the the structure that includes the domain of
integers, and the functions × and + are interpreted as the usual
multiplication and addition over integers. This holds when a assigns x to
6 and y to 3
First Order Logic – p.20/23
Examples
1. x ≡ y × 2 ?
2. x × 2 ≡ x + x ?
3. (x ≡ y ∧ y ≡ z) → x ≡ z ?
S
a |= x ≡ y × 2, where S is the the structure that includes the domain of
integers, and the functions × and + are interpreted as the usual
multiplication and addition over integers. This holds when a assigns x to
6 and y to 3
S
|= x × 2 ≡ x + x where the structure S includes the domain of integers,
and × and + are interpreted as usual. This property holds in S
regardless of the actual assignment of the variable x
First Order Logic – p.20/23
Examples
1. x ≡ y × 2 ?
2. x × 2 ≡ x + x ?
3. (x ≡ y ∧ y ≡ z) → x ≡ z ?
S
a |= x ≡ y × 2, where S is the the structure that includes the domain of
integers, and the functions × and + are interpreted as the usual
multiplication and addition over integers. This holds when a assigns x to
6 and y to 3
S
|= x × 2 ≡ x + x where the structure S includes the domain of integers,
and × and + are interpreted as usual. This property holds in S
regardless of the actual assignment of the variable x
|=(x ≡ y ∧ y ≡ z) → x ≡ z
First Order Logic – p.20/23
Substitution
For a formula φ, term e, and variable v, φ[e/v] denotes
Substitution of free occurrences of variable v in φ by e
A free occurrence of v is not allowed to be replaced by a term that
contains variables quantified in φ
First Order Logic – p.21/23
Substitution
(λx. λy. y > x → y > z)(y − 1) requires y to be renamed by some w
such that w 6≡ y, w 6≡ z
In ∀y.(y > x → y > z), variable x is not allowed to be replaced by the
term y − 1
First Order Logic – p.22/23
First Order Logic Expressiveness
Consider a signature that includes a binary relation R representing a graph
First Order Logic – p.23/23
First Order Logic Expressiveness
Consider a signature that includes a binary relation R representing a graph
The graph is undirected, ∀x. ∀y. (xRy → yRx)
There are no isolated nodes, ∀x. ∃y. (xRy ∨ yRx)
First Order Logic – p.23/23
First Order Logic Expressiveness
Consider a signature that includes a binary relation R representing a graph
The graph is undirected, ∀x. ∀y. (xRy → yRx)
There are no isolated nodes, ∀x. ∃y. (xRy ∨ yRx)
The graph contains cycles ?
The graph is finite ?
First Order Logic – p.23/23