Computational logic First Order Logic

SRM INSTITUTE OF SCIENCE AND TECHNOLOGY
RAMAPURAM CAMPUS, CHENNAI-600 089
COMPUTATIONAL LOGIC
Dr.J.Faritha Banu
SRM IST- Ramapuram

Topics Covered in this Presentation are
First Order Logic
Syntax of FL - Alphabet of FL
Well formed Formula
Symbolization of FL
Argument – FL
Parse Tree

First Order Logic
 Every statement in predicate logic or first order logic is divided into 2 parts. 1. Subject 2. Predicate
 Example: x is a integer, In this statement x is a subject and is a integer – predicate
 Predicate: properties of an subject which are neither true nor false until the value of the variable is
specified.
 Example: If a positive integer is a perfect square and less than four, then it must be equal to one.
 We may rewrite it as: For each x, if x is a positive integer less than four, and if there exists a positive
integer y such that x is equal to the square of y, then x is equal to one.
 For symbolizing this sentence, we require quantifiers ‘for each’, and ‘there exists’; predicates ‘is a
positive integer’, ‘is less than’, and ‘is equal to’; function symbol ‘square of’; constants ‘four’ and
‘one’, and the variables x,y.
 The logic obtained by extending PL and including these types of symbols is called first order logic

Syntax of FL - Alphabet of FL
{⊤ ,⊥}, the set of propositional constants,
{ f i
j : i, j ∈ N}, the set of function symbols,
{ f i
j : i, j ∈ N}, ∪{≈}, the set of predicates,
{x0,x1,x2, . . .}, the set of variables,
{¬,∧,∨,→,↔}, the set of connectives,
{∀,∃}, the set of quantifiers, and
{), (, , }, the set of punctuation marks.
The symbol ∀ is called the universal quantifier and the symbol ∃ is called the
existential quantifier.
Any string over the alphabet of FL is an expression (an FL-expression)

Syntax of FL - Alphabet of FL
First-order logic (like natural language) assumes the world
contains: variable Relations, Functions{∀,∃}, the set of
quantifiers, and
Variables : people, houses, numbers, theories, colors, football games,
wars, centuries
Relations: red, round, multistoried , brother of, bigger than, inside,
part of, has color, occurred after, owns, comes between,
Functions: father of, best friend, second half of, one more than,
beginning of

Well formed Formula
The following is an inductive definition of terms.,
𝑡 ∷= 𝑥𝑖|𝑓𝑖
0
|𝑓𝑖
𝑗
𝑡, 𝑡, … 𝑡 (j times t), where t is a generic term.
We will use terms for defining the (well-formed) formulas. Writing X for a generic
formula, x for a generic variable, and t for a generic term, the grammar for formulas
is:
X ∷= ⊤| ⊥ | (s≈ t) | 𝑃𝑖
0
| 𝑃𝑖
𝑚
(t1, t2, . . . , tm) | ¬X | (X ∧ X) | (X ∨ X) | (X → X)
|(X ↔ X) | ∀xiX | ∃xiX
The formulas in the forms ⊤, ⊥, P i
0 , (s ≈ t), and P i
m (t1, t2, . . . , tm) are called
atomic formulas; and other formulas are called compound formulas.

WFF
 The following expressions are formulas:
⊤,
⊥ → ⊤
(f1
0
≈ f5
0
)
P2
1
(f1
1
(x5))
∀x1((P2
1
(f1
1
(x5))
∀x2∃x5(P5
2
(x0, f1
1
(x1)) ↔ P1
3
(x1, x2, x6))
¬∀x1(P5
2
(𝑓1
1
(x2), x3))
 Whereas the following expressions are not formulas:
⊤(x0) - ⊤ , a variable is not allowed to occur in parentheses
𝑓1
0
≈ 𝑓5
0
- formula since ≈ needs a pair of parentheses
𝑓1
0
(𝑓5
0
) - 𝑓1
0
is a 0-ary function symbol and it cannot take an argument.
¬∀x1(P5
2
(𝑓1
0
(x2), x3))
∀x2∃x5(P5
2
(x0, f1
0
(x1)) ↔ P1
3
(x1, x2, x6))
∀x1((P2
1
(f1
2
(x5))
P2
(f1
(x ))

Usage reduction of parenthesis and subscripts
Drop the outer parentheses from formulas.
Drop the superscripts from the predicates and function symbols
Drop writing subscripts with variables, function symbols and predicates whenever
possible
Omit the parentheses and commas in writing the arguments of function symbols and
predicates provided no confusion arises.
Have precedence rules for the connectives and quantiﬁers to reduce parentheses
 precedence rules are same as propositional logic with ¬, ∀ , ∃ are equal precedence

Usage reduction of parenthesis and subscripts
P, Q, R …… as predicates
x, y, z…………….as variables
a , b, c ………… as constants
Example : ∀x2∃x5(P4(x0, f1(x1)) ↔ P1(x2, x5, x1))
∀x∃y(P(z, f(u)) ↔ Q(x, y, u))
∀x∃y(Pzf(u) ↔ Qxyu)
∀x∃y(Pzfu ↔ Qxyu)

Symbolize
Define 𝐿(𝑥) to mean “x is a lecturer”. (unary predicate)
 Alice is a lecturer: 𝐿(Alice)
 Mickey Mouse is not a lecturer: (¬𝐿(Mickey Mouse))
 𝑦 is a lecturer: 𝐿(𝑦)
Define 𝑂(𝑥, 𝑦) to mean “x is older than y”. (binary predicate/relation)
 Alex is older than Sam: 𝑂(Alex, Sam)
 𝑎 is older than 𝑏: 𝑂(𝑎, 𝑏)
Quantifiers
 The universal quantifier ∀: the statement is true for every object in the domain.
 The existential quantifier ∃: the statement is true for one or more objects in the domain.

Symbolize
All men like cake and pie .
Mx : x is a men Lxc : x likes cake Lxp : x like pie
∀ x ( Mx → (Lxc ∧ Lxp))
All dogs are blue
Dx : X is a dog Bx : x is blue
∀ x ( Dx → Bx)
Some Dogs are blue
Dx : X is a dog Bx : x is blue
∃x (Dx ∧ Bx)
Some men like cake and pie .
Mx : x is a men Lxc : x likes cake Lxp : x like pie
∃x (Mx (Lxc ∧ Lxp))
Such an assignment l, which associates variables to elements of the universe or domain of an
interpretation is called a valuation (or a variable assignment function).
X isdog

Argument – FL
Bapuji was a saint. Since every saint is an altruist, Therefore was Bapuji was an altruist
Symbolizing in PL
b: Bapuji
Sx: X is an saint
Ax: X is an altruist
Ans : Sb - Bapuji was a saint
 Sb, ∀x (Sx→Ax) ⊨ Ab
Everyone has a father
 Rewrite the sentence as
 Each person, there exists a person who is his father
 Fxy : x is father of y
 Hx: x is a person
 ∀x (Hx → ∃y Fyx)
(for each person x, there exists a person y, y is father of x.)
 ∀x ∃y ( Hx ∧ Fyx) - is not correct way. (Wrong)

Argument – FL
If two persons ﬁght over a third one’s property, then the third one gains.
 Fxyz : x and y ﬁght over z
 Gz : z gains
 p(z) : property of z
 ∀x ∀y ∀z(Fxy p(z) → Gz).

Exercise on Symbolize
1. “Every student in this class has taken a course in Java.
J(x) denoting “x has taken a course in Java” S(x) denoting “x is a student in this class”
translation is : ∀x (S(x)→ J(x))
or ∀x (Sx→ Jx)
2. “Some student in this class has taken a course in Java.”
∃x (S(x) ∧ J(x))
3. “Some student in this class has visited Mexico.”
Let M(x) denote “x has visited Mexico” and S(x) denote “x is a student in this class,” and U
be all people.
∃x (S(x) ∧ M(x))
4. “Every student in this class has visited Canada or Mexico.”
∀x ((Sx)→ (M(x) ∨ C(x) ))

• “Every house is a physical object”
• “Some physical objects are houses”
• ”Every house has an owner” or, equivalently, “every house is owned by somebody”
• Everybody owns a house”
• “Sue owns a house”
• “Peter does not own a house”
• “Somebody does not own a house”

• “Every house is a physical object” - ∀x(H(x) → PO(x)),
• “Some physical objects are houses” ∃x(PO(x) ∧ H(x))
• ”Every house has an owner” or, equivalently, “every house is owned by somebody”
∀x(H(x) → ∃yO(y, x))
• Everybody owns a house” ∀x∃y(O(x, y) ∧ H(y))
• “Sue owns a house” ∃x(O(Sue, x) ∧ H(x))
• “Peter does not own a house” ¬∃x(O(Peter, x) ∧ H(x))
• “Somebody does not own a house” ∃x∀y(O(x, y) → ¬H(y))
• (Reference : why ∀x use →, ∃x use ∧ https://www.youtube.com/watch?v=h5UTvdcgFHw)

Parse Tree
P, Q, R …… as predicates
New elements in the parse tree:
 Quantifiers ∀𝑥 and ∃𝑦 have one subtree, similar to the unary connective negation.
 A predicate symbol 𝑃 (𝑡1, 𝑡2, … , 𝑡𝑛) has a node labelled 𝑃 with a sub-tree for each
of the terms 𝑡1, 𝑡2, … , 𝑡𝑛.
A function symbol 𝑓(𝑡1, 𝑡2, … , 𝑡𝑛) has a node labelled 𝑓 with a sub-tree for each of
the terms 𝑡1, 𝑡2, … , 𝑡𝑛.a , b, c ………… as constants
Parse tree for Example 1: ((∀𝑥 (𝑃 (𝑥) ∧ 𝑄(𝑥))) → (¬𝑃 (𝑓(𝑥, 𝑦)) ∨ 𝑄(𝑦)))

Parse Tree
 Parse tree for Example 1: ((∀𝑥 (𝑃 (𝑥) ∧ 𝑄(𝑥))) → (¬𝑃 (𝑓(𝑥, 𝑦)) ∨ 𝑄(𝑦)))

References
1. Arindama Singh," Logics for Computer Science", PHI Learning Private
Ltd,2nd Edition, 2018
2. Wasilewska & Anita, "Logics for computer science: classical and non-
classical", Springer, 2018
3. Huth M and Ryan M, Logic in Computer Science: Modeling and Reasoning
about systems‖, Cambridge University Press, 2005
4. Dana Richards & Henry Hamburger, "Logic And Language Models For
Computer Science", Third Edition, World Scientific Publishing Co. Pte.
Ltd,2018

Computational logic First Order Logic

Computational logic First Order Logic

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