75 research outputs found
Coalgebraic completeness-via-canonicity for distributive substructural logics
We prove strong completeness of a range of substructural logics with respect
to a natural poset-based relational semantics using a coalgebraic version of
completeness-via-canonicity. By formalizing the problem in the language of
coalgebraic logics, we develop a modular theory which covers a wide variety of
different logics under a single framework, and lends itself to further
extensions. Moreover, we believe that the coalgebraic framework provides a
systematic and principled way to study the relationship between resource models
on the semantics side, and substructural logics on the syntactic side.Comment: 36 page
The ubiquity of conservative translations
We study the notion of conservative translation between logics introduced by
Feitosa and D'Ottaviano. We show that classical propositional logic (CPC) is
universal in the sense that every finitary consequence relation over a
countable set of formulas can be conservatively translated into CPC. The
translation is computable if the consequence relation is decidable. More
generally, we show that one can take instead of CPC a broad class of logics
(extensions of a certain fragment of full Lambek calculus FL) including most
nonclassical logics studied in the literature, hence in a sense, (almost) any
two reasonable deductive systems can be conservatively translated into each
other. We also provide some counterexamples, in particular the paraconsistent
logic LP is not universal.Comment: 15 pages; to appear in Review of Symbolic Logi
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? and Schmitz (2015), we show that the deducibility problem for full Lambek calculus with exchange and weakening (FL_{ew}) 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 FL_{ew}. 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)
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
Non-associative, Non-commutative Multi-modal Linear Logic
Adding multi-modalities (called subexponentials) to linear logic enhances its power as a logical framework, which has been extensively used in the specification of e.g. proof systems, programming languages and bigraphs. Initially, subexponentials allowed for classical, linear, affine or relevant behaviors. Recently, this framework was enhanced so to allow for commutativity as well. In this work, we close the cycle by considering associativity. We show that the resulting system (acLLĪ£ ) admits the (multi)cut rule, and we prove two undecidability results for fragments/variations of acLLĪ£
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