First order logic in Artificial intelligence | First order logic | Artificial intelligence

Introduction

                 First order logic is the different way of knowledge representation. In propositional logic we seen that how to solve mathematically an statement is true or false but here us the problem that the propositional logic just can solve the statement which is true or false but it can not solve the complex sentence in the way to solve the complex statement we introduced by a term known as first order logic or it can also say first order predicated logic.

                Propositional logic have so limited expression power so it not use much, for solving the advance level problem first order logic is very helpful in Artificial intelligence.

Some fact of First order logic or first order predicted

                 First order logic is the different way of knowledge representation. It can say it is a upgrade from of propositional logic.

• First order logic is help to convert a normal language in concise from.

• First-order logic (like natural language) does not only assume that the world contains facts like propositional logic but also assumes the following things in the world:

1.  Objects: A, B, people, numbers, colors, wars, theories, squares, pits,name,place etc
2.  Relations: It can be unary relation such as: red, round, is adjacent, or n-any relation such as the sister of, brother of, has color, comes between.
3.  Function: Father of, best friend, third inning of, end of, and so on.

• As a nature language the first order logic also have the two important parts like,

1.  Syntax
2.  Semantic   

• Syntax of first order logic in artificial intelligence

                     Syntax of First Order Logic determines which collection of symbols is a logical expression in first-order logic. The basic syntactic elements of first-order logic are symbols. We write statements in short-hand notation in First Order Logic.

Basic Elements of First-order logic

There are same basic elements of first order logic

• Constants -- numbers like 2,3,4, and name like John, Ram etc.
• Variable -- alphabet like a, b, c , x, u , etc.
• Predicates -- brother, sister, <,>, etc
• Function -- sprt, a leftlegof , Etc.
• Connectivity -- ﹃, <, ^,
• Equality-- ==.
• Qualifier --∀.

Subject:

      Subject is the main part of the statement.

Predicate:

            A predicate can be defined as a relation, which binds two atoms together in a statement.Consider the statement: "x is an integer.", it consists of two parts, the first part x is the subject of the statement and second part "is an integer," is known as a predicate.

Quantifiers in First-order logic:

                A quantifier is a language element which generates quantification, and quantification specifies the quantity of specimen in the universe of discourse.

               These are the symbols that permit to determine or identify the range and scope of the variable in the logical expression.

 There are two types of quantifier:

1. Universal Quantifier, (for all, everyone, everything)
2. Existential quantifier, (for some, at least one).

• Universal Quantifier:

         Universal quantifier is a symbol of logical representation, which specifies that the statement within its range is true for everything or every instance of a particular thing.

The Universal quantifier is represented by a symbol ∀, which resembles an inverted A.

Note: In universal quantifier we use implication "→".

If x is a variable, then ∀x is read as:

ie, for all x,for each x,for every x.

Example:

All man drink coffee.

Let a variable x which refers to a cat so all x can be represented in UOD as below:

First-Order Logic in Artificial intelligence

∀x man(x) → drink (x, coffee).

It will be read as: There are all x where x is a man who drink coffee.

• Existential Quantifier:

           Existential quantifiers are the type of quantifiers, which express that the statement within its scope is true for at least one instance of something.

It is denoted by the logical operator ∃, which resembles as inverted E. When it is used with a predicate variable then it is called as an existential quantifier.

If x is a variable, then existential quantifier will be ∃x or ∃(x). And it will be read as,

There exists a 'x.', For some 'x.', For at least one 'x.'.

Example:

Some boys are intelligent.

First-Order Logic in Artificial intelligence

∃x: boys(x) ∧ intelligent(x)

It will be read as: There are some x where x is a boy who is intelligent.

Points to remember:

• The main connective for universal quantifier ∀ is implication →.
• The main connective for existential quantifier ∃ is and ∧.

Properties of Quantifiers:

• In universal quantifier, ∀x∀y is similar to ∀y∀x.
• In Existential quantifier, ∃x∃y is similar to ∃y∃x.
• ∃x∀y is not similar to ∀y∃x.

Some Examples of First Order Logic using quantifier:

1.  All birds fly.

        In this question the predicate is "fly(bird)."

And since there are all birds who fly so it will be represented as follows.

              ∀x bird(x) →fly(x).

    2. Every man respects his parent.

        In this question, the predicate is "respect(x, y)," where x=man, and y= parent.

Since there is every man so will use ∀, and it will be represented as follows:

              ∀x man(x) → respects (x, parent).

    3. Some boys play cricket.

                  

          In this question, the predicate is "play(x, y)," where x= boys, and y= game. Since there are some boys so we will use ∃, and it will be represented as:

              ∃x boys(x) → play(x, cricket).

   4. Not all students like both Mathematics and Science.

         

         In this question, the predicate is "like(x, y)," where x= student, and y= subject.

Since there are not all students, so we will use ∀ with negation, so following representation for this:

              ¬∀ (x) [ student(x) → like(x, Mathematics) ∧ like(x, Science)].

    5. Only one student failed in Mathematics.

        In this question, the predicate is "failed(x, y)," where x= student, and y= subject.

Since there is only one student who failed in Mathematics, so we will use following representation for this:

              ∃(x) [ student(x) → failed (x, Mathematics) ∧∀ (y) [¬(x==y) ∧ student(y) → ¬failed (x, Mathematics)].

Free and Bound Variables:

               The quantifiers interact with variables which appear in a suitable way. There are two types of variables in First-order logic which are given below:

• Free Variable: A variable is said to be a free variable in a formula if it occurs outside the scope of the quantifier.

          Example: ∀x ∃(y)[P (x, y, z)], where z is a free variable.

• Bound Variable: A variable is said to be a bound variable in a formula if it occurs within the scope of the quantifier.

          Example: ∀x [A (x) B( y)], here x and y are the bound variables.

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