The proof equivalence problem for multiplicative
linear logic is pspace-complete
Abstract
Willem Heijltjes and Robin Houston
October 1, 2013
mll proof equivalence is the problem of deciding whether one proof in multiplicative linear logic may be turned into another by a series of commuting conversions. It is also the word problem for ?-autonomous categories (Barr, 1991):
the problem of deciding whether two term representations denote the same morphism in any ?-autonomous category. In mll− , without the two units, proof
equivalence corresponds to syntactic equality on proof nets (Girard, 1987; Danos
and Regnier, 1989), and is linear-time decidable. For full mll, thanks to many
years of work on proof nets (Trimble, 1994; Blute et al., 1996; Straßburger and
Lamarche, 2004; Hughes, 2012) we know that the proof equivalence problem
corresponds to a reasonably simple graph-rewiring problem.
On the other side of the fence, thanks to many years of work on combinatorial
games and complexity theory (Flake and Baum, 2002; Hearn and Demaine, 2005,
2009; Gopalan et al., 2006; Ito et al., 2011) we know many examples of rewiring
problems on graphs that are pspace-complete. We will give a reduction from the
configuration-to-configuration problem for Nondeterministic Constraint Logic
(Hearn and Demaine, 2005, 2009), a graphical formalism specifically designed
for easy problem reduction.
The pspace-completeness result rules out a satisfactory notion of proof net
for mll with units, and in particular for this reason it may come as a surprise.
However, pspace-completeness is not unusual for graph rewiring problems, nor
for reconfiguration problems (Ito et al., 2011) of the kind to which mll proof
equivalence belongs. A reconfiguration problem is the problem of finding a path
across related solutions to a given decision problem—in this case mll proof
search. Reconfiguration problems whose associated decision problem is npcomplete, as is mll proof search, frequently are pspace-complete; an important
example, satisfiability-reconfiguration (Gopalan et al., 2006) is the problem of
finding a path across satisfying assignments to a boolean formula by changing
the value of one atom at a time.
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