Logische Grundlagen des Software Engineerings

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Logische Grundlagen des Software Engineerings

• Prof. Dr. Jakob Rehof
• Lehrstuhl 14, Software Engineering

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Intuitionistic logic

• Classical logic is about truth
• Intuitionistic logic is about constructibility
• Intuitionistic logic is therefore also called
  constructive logic
• All statements are classically regarded as either
  true or false p v p holds (tertium non datur)
• there is 7 sevens somewhere in the decimal
  representation of the number
• there are irrational p and q such that pq is
  rational
• Intuitionistic logic, in contrast, requires that
  we can construct the objects whose existence we
  assert and hence rejects the tertium non datur
  principle

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Intuitionistic logic

• L.E.J.Brouwer (1881-1966)
• Arend Heyting (1898-1980)
• Glivenko
• Kolmogorov
• Gentzen
• Gödel
• ...

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BHK-interpretation(informal constructive
semantics)
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BHK-interpretation(informal constructive
semantics)
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BHK-interpretation(informal constructive
semantics)

• Note This informal interpretation can be
  formalized in
• recursion theory Kleenes notion of
  realizability
• lambda calculus the Curry-Howard isomorphism

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Natural deduction
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Proof trees
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Examples of classical tautologies
Not all of these are intuitionistic theorems!
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Example
--- p ? -p
Write p as p?0. We are to prove (((p?0)?0)?0) ?
(p?0) ((p?0)?0)?0, p, p?0 - p?0
((p?0)?0)?0, p, p?0 - p ---------------------
--------------------------------------------------
-------------------------
((p?0)?0)?0, p, p?0 - 0
-----------------------------------------------
((p?0)?0)?0, p -
(p?0)?0 ((p?0)?0)?0, p - ((p?0)?0)?0
---------------------------------
--------------------------------------------------
--------------
((p?0)?0)?0, p - 0

------------------------------------

((p?0)?0)?0 - p?0
------------------------
---------------
- (((p?0)?0)?0) ? (p?0)
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Algebraic semantics of classical logic

• Boolean semantics
• Set semantics
• Boolean algebra

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Boolean semantics of classical logic
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Set semantics of classical logic
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Set semantics of classical logic
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Semantics of classical logic
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Boolean algebra
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Algebraic semantics of intuitionistic logic

• Lindenbaum algebra
• Heyting algebra
• Topological semantics
• Algebraic semantics
• Kripke semantics

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Lindenbaum algebra
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Lindenbaum algebra
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Lindenbaum algebra
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Lindenbaum algebra
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Lindenbaum algebra
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The problem with complement
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Heyting algebra
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Heyting algebra
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Topological semantics
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Algebraic semantics of intuitionistic logic
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Algebraic semantics
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Algebraic semantics
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Example!
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Example!
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Characterization
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Characterization
Property 1. shows that propositional
intuitionistic logic is decidable (in doubly
exponential space). In fact, as we will see
later, it can be decided in polynomial space.
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Kripke model
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Kripke model
Implied rule for negation
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Example
Kripke model
p
c1
c0
q
c2

• In this Kripke model we have
• c0 - - - (p v q)
• c0 - (p?q)?q
• not c0 - (p v q)

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Kripke model
Soundness and completeness of Kripe semantics
Proof By turning every Heyting algebra into a
Kripke model, using prime filter construction.
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Disjunction property
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Implicational fragment
Note that this requires a proof (right-to-left
implication)!
Conservativity over the implicational fragment