283 research outputs found
Intuitionistic G\"odel-L\"ob logic, \`a la Simpson: labelled systems and birelational semantics
We derive an intuitionistic version of G\"odel-L\"ob modal logic ()
in the style of Simpson, via proof theoretic techniques. We recover a labelled
system, , by restricting a non-wellfounded labelled system for
to have only one formula on the right. The latter is obtained using
techniques from cyclic proof theory, sidestepping the barrier that 's
usual frame condition (converse well-foundedness) is not first-order definable.
While existing intuitionistic versions of are typically defined over
only the box (and not the diamond), our presentation includes both modalities.
Our main result is that coincides with a corresponding
semantic condition in birelational semantics: the composition of the modal
relation and the intuitionistic relation is conversely well-founded. We call
the resulting logic . While the soundness direction is proved using
standard ideas, the completeness direction is more complex and necessitates a
detour through several intermediate characterisations of .Comment: 25 pages including 8 pages appendix, 4 figure
Tower-Complete Problems in Contraction-Free Substructural Logics
We investigate the non-elementary computational complexity of a family of
substructural logics without contraction. With the aid of the technique
pioneered by Lazi\'c and Schmitz (2015), we show that the deducibility problem
for full Lambek calculus with exchange and weakening
() is not in Elementary (i.e., the class of decision
problems that can be decided in time bounded by an elementary recursive
function), but is in PR (i.e., the class of decision problems that can be
decided in time bounded by a primitive recursive function). More precisely, we
show that this problem is complete for Tower, which is a non-elementary
complexity class forming a part of the fast-growing complexity hierarchy
introduced by Schmitz (2016). The same complexity result holds even for
deducibility in BCK-logic, i.e., the implicational fragment of
. We furthermore show the Tower-completeness of the
provability problem for elementary affine logic, which was proved to be
decidable by Dal Lago and Martini (2004).Comment: The full version of the paper accepted to CSL 202
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