Prof. Dr. Jakob Rehof Lehrstuhl XIV, Software
Engineering
LMSE
Logische Methoden des Software Engineerings
Diese Vorlesung
• Klassische Logik und CHI
• Lesen: LCHI, Kap. 8
• Übung:
– Zeigen Sie, wie die Reduktionsrelationen für die
Kontrolloperatoren auf Folie 7-9 abgeleitet werden können.
– Beweisen Sie die Aussagen auf den Folien 20, 21 und 22 zu alternativen Präsentationen der klassischen Logik. Konstruieren Sie dabei relevante Beweisterme des klassischen Kalküls.
Classical logic and control operators
• Until about 1990 it was widely beleived that the CHI only makes sense for constructive logic
• Indeed, we have informally explained the CHI with reference to the BHK-interpretation
• In 1990 Tim Griffin changed that picture:
T.G. Griffin: A formulae-as-types notion of control. ACM Principles of Programming Languages POPL 1990
• Griffin wanted to find a type system for typing Felleisen‘s control operators and was led to using a double-negation elimination rule
CHI for classical logic
...... ......
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Classical propositional logic (implicational fragment)
Minimal (intuitionistic) logic
Double-negation elimination
Intuitionistic vs. classical notions of
negation (absurdity)
Variations
Variations
Variations
Variations
Variations
Negation (primitive vs. defined)
Remark
Full system
Abbreviations
Note: Disjunction and conjunction cannot
be eliminated by abbreviation in intuitionistic logic!
-calculus
[Rehof & Sørensen, 1994]
A new binding operator to assign proof term to applications of (--E) rule
-calculus
Classical proof normalization
Variations
Form of normal forms
... and consistency:
Definability
• We know that we can define conjunction and disjunction in terms of implication and negation, in CL
• Therefore, under CHI, we should be able to define pairs and sums at the term level
