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).

[LMSS92] Patrick Lincoln, John Mitchell, Andre Scedrov, and Natarajan Shankar.
Decision Problems for Propositional Linear Logic.
Annals of Pure and Applied Logic, 56:239-311, 1992.

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Frank Pfenning