Maths - Horn Clause

It is useful, for certain types of reasoning, to have the things we know about a situation in the form of implications. For instance:

A -> B   A implies B

of even:

A /\ B /\ C -> D   A and B and C implies D

Deduction Using Implies

So if the following are true:

  • A -> B
  • B -> C
  • C -> D

then we can deduce:

A -> D

for example, by combining the implications (substituting).

Forward and Backward Chaining

Diagrams, such as the one above, give us a graphical guide.

As Meet and Join

We may need to express the implication in terms of meet and join ('or' and 'and') . To see how to do this lets look at a truth table for it:

A B A->B
F F T
F T T
T F F
T T T

Note: I've drawn the truth table as Boolean values but we can use constructive logic instead (law of excluded middle not required).

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


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see also:
  • Stanford - Development of Intuitionistic Logic
  • Stanford - Intuitionistic Logic
  • Youtube video

    Advanced Topics in Programming Languages Series: Parametric Polymorphism and the Girard-Reynolds Isomorphism. This talk is based on a series of papers by Philip Wadler, a principal designer of the Haskell programming language. Featured are a number of double-barreled names in computer science:

    • Hindley-Milner (Strong typing without having to type the types)
    • Wadler-Blott (Making ad-hoc polymorphism less ad-hoc with parametricity)
    • Curry-Howard (Isomorphism between types and theorems, terms and proofs)
    • Girard-Reynolds (Isomorphism between types and terms in the presence of parametricity)
  • or Djinn code see here.
  • Parametricity
  • "Propositions as Types" by Philip Wadler
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