
Completeness for Game Logic
Game logic was introduced by Rohit Parikh in the 1980s as a generalisati...
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Display to Labelled Proofs and Back Again for Tense Logics
We introduce translations between display calculus proofs and labelled c...
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Teaching a Formalized Logical Calculus
Classical firstorder logic is in many ways central to work in mathemati...
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PsiCalculi Revisited: Connectivity and Compositionality
Psicalculi is a parametric framework for process calculi similar to pop...
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Completeness Theorems for FirstOrder Logic Analysed in Constructive Type Theory (Extended Version)
We study various formulations of the completeness of firstorder logic p...
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A theory of linear typings as flows on 3valent graphs
Building on recently established enumerative connections between lambda ...
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Focused Proofsearch in the Logic of Bunched Implications
The logic of Bunched Implications (BI) freely combines additive and mult...
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LambekGrishin Calculus: Focusing, Display and Full Polarization
Focused sequent calculi are a refinement of sequent calculi, where additional sideconditions on the applicability of inference rules force the implementation of a proof search strategy. Focused cutfree proofs exhibit a special normal form that is used for defining identity of sequent calculi proofs. We introduce a novel focused display calculus fD.LG and a fully polarized algebraic semantics FP.LG for LambekGrishin logic by generalizing the theory of multitype calculi and their algebraic semantics with heterogenous consequence relations. The calculus fD.LG has strong focalization and it is sound and complete w.r.t. FP.LG. This completeness result is in a sense stronger than completeness with respect to standard polarized algebraic semantics (see e.g. the phase semantics of Bastenhof for LambekGrishin logic or Hamano and Takemura for linear logic), insofar we do not need to quotient over proofs with consecutive applications of shifts over the same formula. We plan to investigate the connections, if any, between this completeness result and the notion of full completeness introduced by Abramsky et al. We also show a number of additional results. fD.LG is sound and complete w.r.t. LGalgebras: this amounts to a semantic proof of the socalled completeness of focusing, given that the standard (display) sequent calculus for LambekGrishin logic is complete w.r.t. LGalgebras. fD.LG and the focused calculus f.LG of Moortgat and Moot are equivalent with respect to proofs, indeed there is an effective translation from f.LGderivations to fD.LGderivations and vice versa: this provides the link with operational semantics, given that every f.LGderivation is in a CurryHoward correspondence with a directional λμμterm.
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