Models and Related Notions
In predicate calculus we have many of the semantic notions which have the same definition. They are,
- If a wff has true value in its interpretation the it satisfies the wff.
- An interpretation is called a model of wff if it satisfies the wff.
- A wff is valid if it has its value as true in all the, interpretations.
- If two wffs have same truth value in all the interpretations then they are identical.
- A wff without a model is said to be inconsistent or unsatisfiable.
- A set Δ logically entails w(Δ |== w) if a wff w has True in the interpretations in which each of the wff in set Δ has their value True.
Knowledge
- The knowledge known by an agent about the world can be represented by the formulas of predicate-calculus.
- A set of such formulas Δ is known as a knowledge base.
- A user agent adds a formula w to Δ if he "knows" or "believes" W.
- Let us consider a set of formulas which represents the knowledge of possible world.
- On(Q, F) ∧ Clear(R)
- Clear(P) ∧ Clear(R) ⊃ On(Q, F)
- Clear(P)
- Clear(P) ∨ Clear(Q).
- Clear(R)
- Using these formulas the disc-world problem can be constructed which satisfies the formulas as shown in Figure
Fig: Three Discs World Situations
- In the three models the mappings between relation constants and relations in the world are not same.
- In one of the model On is mapped to the relation On = {<P, Q>, <Q, Floor>, <R, Floor>} and Clear is mapping to Clear =, {<P>, <R>}.
- In second model we have mapping,
- On = {<P, F>, <Q, F>, <R, F>
- Clear = {<P>, <Q>, <R>
- In third model mapping,
- On = {<R, Q>, <Q, F>, <P, F>
- Clear =.{<R>, <P>}
- These formulas can also represented by other models which does not have predicate-calculus mnemonics.
- Some of the models may use numbers to assign the blocks, instead of P, Q and R.
- Some of the models may use more number of formulas so that number of models are decreased,
- Example: Adding Clear(Q) in the knowledge base reduces the models to one in the above Figure.
Quantification
- For specifying a property that applies to every object or atleast one object representing all the relations is difficult as there may be infinite set of objects.
- Like Clear(B1) ∧ Clear(B2) ∧ .........
- (or) Clear(B1) ∨ Clear(B2) ∨ ........
- So variable symbols and quantifier symbols which are syntactic entities, are used to overcome the problem.
- The variable symbols are strings starting with lowercase letters which are present at the ending of alphabets like p, q, r, s , pl, p2 and so on. It is a term. Thus f(y, Smith, B16) is a ternary functional expression.
- ∀ and ∃ are quantifier symbols where ∀ is universal quantifier and read as 'For all', ∃ is existential quantifier read as 'There exists'.
- Let a wff is denoted by w and a variable symbol is denoted by r. Then (∀r)w and (∃r)w are wffs. r is called the variable quantified over and w is said to be within the scope of the quantifier.
- Closed wff or closed sentence: If all the variable symbols apart from r in w are quantified over in w, then (Qr)w where Q is ∀ or ∃ is called a closed wff or closed sentence.
- Example: [∀y][P(y) ⊃ R(y)]
- The order of application of universal quantifiers to a wff does not matter.
- Example: (∀x)[(∃y)w] is similar to (∀y)[(∀x)w] so this can be written as [∀xy][w].
- Similarly existential quantifiers, but in their mixture the order of application matters.
- Example: (∀x)[(∃y)w] is not similar to (∃y)[(∀x)w].
- The variables used along with the quantifiers are "dummy variables", they can be replaced by any variable if they are replaced at every place they occurred.
- Like, (∀x)w is similar to (∀y)w if the places where x occurred is replaced by y.
- In first-order predicate calculus the quantifying over variable symbols is achieved. Quantifying over functions and relations is allowed in second and higher order predicate calculus which may be complex. Here we use first-order predicate calculus.
