15-816 Linear Logic
Lecture 8: Decision Problems
We begin with another example: the encoding of validity for quantified
boolean formulas (QBF) in linear logic. This yields the PSPACE-hardness
of the decision problem for MALL (multiplicative, additive, linear
logic). This fragment contains linear implication, simultaneous
conjunction and unit, alternative conjunction and truth, and disjunction
and impossibility. Since the decision problem can also easily be seen
to lie in PSPACE, it is PSPACE-complete.
The multiplicative fragment (MLL, containing linear implication,
simultaneous conjunction and unit) is NP-complete (although we do not
show the proof). The decidability of the multiplicative, exponential
fragment (MELL, containing MLL and the ``of course'' modality) is
unknown, although Petri-net reachability provides and EXPSPACE
lower bound. Finally, MAELL, that is, all of propositional linear
logic, is undecidable (again a result we do not show in this
lecture).
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Frank Pfenning
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