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Monoid based semantics for linear formulas. (English) Zbl 1024.03063

Authors’ abstract: Each Girard quantale (i.e. commutative quantale with a selected dualizing elemnt) provides a support for a semantics for linear propositional formulas (but not for linear derivations). Several constructions of Girard quantales are known. We give two more constructions, one using an arbitrary partially ordered monoid and one using a partially ordered group (both commutative). In both cases the semantics can be controlled by a relation between pairs of elements of the support and formulas. This gives us a neat way of handling duality.

MSC:

03F52 Proof-theoretic aspects of linear logic and other substructural logics
06F07 Quantales
03G25 Other algebras related to logic
06F15 Ordered groups
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References:

[1] CSLI lectures notes (1992)
[2] DOI: 10.1090/conm/092/1003210 · doi:10.1090/conm/092/1003210
[3] Theoretical Computer Science pp 1– (1987) · Zbl 0734.68002
[4] Rendiconti del Circolo Matematico di Palermo. Ser II Supplemento 12 pp 99– (1986)
[5] London mathematical society lecture notes series 222 (1995)
[6] Pitman research notes in mathematics (1990)
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