808 research outputs found
Sequent Calculi with procedure calls
In this paper, we introduce two focussed sequent calculi, LKp(T) and LK+(T), that are based on Miller-Liang's LKF system for polarised classical logic. The novelty is that those sequent calculi integrate the possibility to call a decision procedure for some background theory T, and the possibility to polarise literals "on the fly" during proof-search. These features are used in our other works to simulate the DPLL(T) procedure as proof-search in the extension of LKp(T) with a cut-rule. In this report we therefore prove cut-elimination in LKp(T). Contrary to what happens in the empty theory, the polarity of literals affects the provability of formulae in presence of a theory T. On the other hand, changing the polarities of connectives does not change the provability of formulae, only the shape of proofs. In order to prove this, we introduce a second sequent calculus, LK+(T) that extends LKp(T) with a relaxed focussing discipline, but we then show an encoding of LK+(T) back into the more restrictive system LK(T). We then prove completeness of LKp(T) (and therefore of LK+(T)) with respect to first-order reasoning modulo the ground propositional lemmas of the background theory T
A sequent calculus with procedure calls
The proof of Cut-elimination is unfortunately bugged. It is repaired in "Sequent Calculi with procedure calls", hal-00779199, v4In this paper, we extend Miller-Liang's system LKF into a calculus LK(T), allowing calls to a decision procedure. We prove cut-elimination of LK(T)
Two loop detection mechanisms: a comparison
In order to compare two loop detection mechanisms we describe two calculi for theorem proving in intuitionistic propositional logic. We call them both MJ Hist, and distinguish between them by description as `Swiss' or `Scottish'. These calculi combine in different ways the ideas on focused proof search of Herbelin and Dyckhoff & Pinto with the work of Heuerding emphet al on loop detection. The Scottish calculus detects loops earlier than the Swiss calculus but at the expense of modest extra storage in the history. A comparison of the two approaches is then given, both on a theoretic and on an implementational level
Sequent Calculus in the Topos of Trees
Nakano's "later" modality, inspired by G\"{o}del-L\"{o}b provability logic,
has been applied in type systems and program logics to capture guarded
recursion. Birkedal et al modelled this modality via the internal logic of the
topos of trees. We show that the semantics of the propositional fragment of
this logic can be given by linear converse-well-founded intuitionistic Kripke
frames, so this logic is a marriage of the intuitionistic modal logic KM and
the intermediate logic LC. We therefore call this logic
. We give a sound and cut-free complete sequent
calculus for via a strategy that decomposes
implication into its static and irreflexive components. Our calculus provides
deterministic and terminating backward proof-search, yields decidability of the
logic and the coNP-completeness of its validity problem. Our calculus and
decision procedure can be restricted to drop linearity and hence capture KM.Comment: Extended version, with full proof details, of a paper accepted to
FoSSaCS 2015 (this version edited to fix some minor typos
An Epistemic Interpretation of Paraconsistent Weak Kleene Logic
This paper extends Fitting's epistemic interpretation of some Kleene logics, to also account for Paraconsistent Weak Kleene logic. To achieve this goal, a dualization of Fitting's "cut-down" operator is discussed, rendering a "track-down" operator later used to represent the idea that no consistent opinion can arise from a set including an inconsistent opinion. It is shown that, if some reasonable assumptions are made, the truth-functions of Paraconsistent Weak Kleene coincide with certain operations defined in this track-down fashion. Finally, further reflections on conjunction and disjunction in the weak Kleene logics accompany this paper, particularly concerning their relation with containment logics. These considerations motivate a special approach to defining sound and complete Gentzen-style sequent calculi for some of their four-valued generalizations
Cut-Simulation and Impredicativity
We investigate cut-elimination and cut-simulation in impredicative
(higher-order) logics. We illustrate that adding simple axioms such as Leibniz
equations to a calculus for an impredicative logic -- in our case a sequent
calculus for classical type theory -- is like adding cut. The phenomenon
equally applies to prominent axioms like Boolean- and functional
extensionality, induction, choice, and description. This calls for the
development of calculi where these principles are built-in instead of being
treated axiomatically.Comment: 21 page
Theories of truth based on four-valued infectious logics
Infectious logics are systems that have a truth-value that is assigned to a compound formula whenever it is assigned to one of its components. This paper studies four-valued infectious logics as the basis of transparent theories of truth. This take is motivated as a way to treat different pathological sentences differently, namely, by allowing some of them to be truth-value gluts and some others to be truth-value gaps and as a way to treat the semantic pathology suffered by at least some of these sentences as infectious. This leads us to consider four distinct four-valued logics: one where truth-value gaps are infectious, but gluts are not; one where truth-value gluts are infectious, but gaps are not; and two logics where both gluts and gaps are infectious, in some sense. Additionally, we focus on the proof theory of these systems, by offering a discussion of two related topics. On the one hand, we prove some limitations regarding the possibility of providing standard Gentzen sequent calculi for these systems, by dualizing and extending some recent results for infectious logics. On the other hand, we provide sound and complete four-sided sequent calculi, arguing that the most important technical and philosophical features taken into account to usually prefer standard calculi are, indeed, enjoyed by the four-sided systems
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