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Syntax , Semantics and Entailment

Logic in general

Logic's are formal languages for representing information such that conclusion can be drawn.

Syntax defines the sentences in the language.

Semantics define the "meaning" of sentence
 - i.e., define truth of a sentence in a world

E.g., the language of arithmetic
- x+2 ≥ y is a sentence; x2 +y > {} is not a sentence 
- x+2 ≥ y is true if the number x+2 is no less than the number y
- x+2 ≥ y is true in a world where x =7, y = 1
- x+2 ≥y is false in a world where x = 0, y = 6 


Entailment means that one thing follows from another:
          KB |=α
Knowledge base KB entails sentence α if and only if α is true in all worlds where KB is true
- E.g., small dog and cats.... -d +c
- E.g., x+y = 4 entails 4 = x+y
- Entailment is a relationship between sentence (i.e., syntax) that is based on semantics.
- a relationship between two sentence such that if the First is true, the second must also be true, as in 
Her son drives her to work every day and Her son knows how to drive.

Inference, Soundness, Completeness
  • KB |α , sentence α can be derived from KB by procedure i
  • Soundness: i is sound if whenever KB |=α, it is also true that KB|=α
  • Completeness: i is complete if whenever KB |=α, it is also true that KB |=α
  • Preview :we will define a logic (first-order logic) which is expressive enough to say almost anything of interest, and for which there exists a sound and complete inference procedure. That is, the procedure will answer any question whose answer follows from what is known by the KB

Logic as a KR language

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