Assignment 4

Extend one of the focusing provers (browse or download gzipped tar file) for intuitionistic propositional logic to a decision procedure. There are two principal methods and you may choose either one.

Loop Detection

This means you have to design and implement a condition which allows you to compare a goal to some of its subgoals. You do not need to prove the correctness of your condition, and it does not need to be efficient. However, you should clearly state the criteria you are implementing with your solution. If you want more information, check the paper by Jacob Howe cited below.

Jacob Howe, Two loop-detection mechanisms: a comparison (54KB), In proceedings of TAB'97 (Springer LNCS 1227), 15 pp.

Contraction-Free Derivations

This means that you have to analyze the structure of the antecedent of a passive implication. In the pure sequent calculus (you will have to see how to adapt this to the intuitionistic case) we replace the implication left rule by the following rules. For more information see the paper by Roy Dyckhoff cited below.

Propositions (A, B, C)
A ::= P         atomic propositions
    | A1 & A2   conjunction
    | T         truth
    | A1 v A2   disjunction
    | F         falsehood
    | A1 -> A2  implication

Hypotheses (consider modulo exchange)
Gamma ::= . | Gamma, A

Sequents
Gamma ==> C


Gamma, P, B      ==> C
----------------------- ->L/P
Gamma, P, P -> B ==> C

Gamma, A1 -> (A2 -> B) ==> C
----------------------------- ->L/&
Gamma, (A1 & A2) -> B  ==> C

Gamma, B      ==> C
-------------------- ->L/T
Gamma, T -> B ==> C

Gamma, A1 -> B, A2 -> B ==> C
------------------------------ ->L/v
Gamma, (A1 v A2) -> B   ==> C

Gamma         ==> C
-------------------- ->/F
Gamma, F -> C ==> C

Gamma, A2 -> B         ==> A1 -> A2
Gamma, B               ==> C
------------------------------------ ->L/->
Gamma, (A1 -> A2) -> B ==> C

R. Dyckhoff, Contraction-free sequent calculi for intuitionistic logic, J. Symb. Logic 57, pp 795-807, 1992. Copies can be obtained from my secretary Rosemary Battenfelder in WeH 8124.