On Deriving Nested Calculi for Intuitionistic Logics from Semantic Systems

This paper shows how to derive nested calculi from labelled calculi for propositional intuitionistic logic and first-order intuitionistic logic with constant domains, thus connecting the general results for labelled calculi with the more refined formalism of nested sequents. The extraction of nested calculi from labelled calculi obtains via considerations pertaining to the elimination of structural rules in labelled derivations. Each aspect of the extraction process is motivated and detailed, showing that each nested calculus inherits favorable proof-theoretic properties from its associated labelled calculus

From Display to Labelled Proofs for Tense Logics

We introduce an effective translation from proofs in the display calculus to proofs in the labelled calculus in the context of tense logics. We identify the labelled calculus proofs in the image of this translation as those built from labelled sequents whose underlying directed graph possesses certain properties. For the basic normal tense logic Kt, the image is shown to be the set of all proofs in the labelled calculus G3Kt

Electrode thickness measurement of a Si(Li) detector for the SIXA array

Cathode electrodes of the Si(Li) detector elements of the SIXA X-ray spectrometer array are formed by gold-palladium alloy contact layers. The equivalent thickness of gold in one element was measured by observing the characteristic L-shell X-rays of gold excited by monochromatised synchrotron radiation with photon energies above the L3 absorption edge of gold. The results obtained at 4 different photon energies below the L2 edge yield an average value of 22.4(35) nm which is consistent with the earlier result extracted from detection efficiency measurements. PACS: 29.40.Wk; 85.30.De; 07.85.Nc; 95.55.Ka Keywords: Si(Li) detectors, X-ray spectrometers, X-ray fluorescence, detector calibration, gold electrodes, synchrotron radiationComment: 10 pages, 4 PostScript figures, uses elsart.sty, submitted to Nucl. Instrum. Meth.

A Tableaux Calculus for Reducing Proof Size

A tableau calculus is proposed, based on a compressed representation of clauses, where literals sharing a similar shape may be merged. The inferences applied on these literals are fused when possible, which reduces the size of the proof. It is shown that the obtained proof procedure is sound, refutationally complete and allows to reduce the size of the tableau by an exponential factor. The approach is compatible with all usual refinements of tableaux.Comment: Technical Repor

Proving Craig and Lyndon Interpolation Using Labelled Sequent Calculi

We have recently presented a general method of proving the fundamental logical properties of Craig and Lyndon Interpolation (IPs) by induction on derivations in a wide class of internal sequent calculi, including sequents, hypersequents, and nested sequents. Here we adapt the method to a more general external formalism of labelled sequents and provide sufficient criteria on the Kripke-frame characterization of a logic that guarantee the IPs. In particular, we show that classes of frames definable by quantifier-free Horn formulas correspond to logics with the IPs. These criteria capture the modal cube and the infinite family of transitive Geach logics

Connectionist modal logic: Representing modalities in neural networks

AbstractModal logics are amongst the most successful applied logical systems. Neural networks were proved to be effective learning systems. In this paper, we propose to combine the strengths of modal logics and neural networks by introducing Connectionist Modal Logics (CML). CML belongs to the domain of neural-symbolic integration, which concerns the application of problem-specific symbolic knowledge within the neurocomputing paradigm. In CML, one may represent, reason or learn modal logics using a neural network. This is achieved by a Modalities Algorithm that translates modal logic programs into neural network ensembles. We show that the translation is sound, i.e. the network ensemble computes a fixed-point meaning of the original modal program, acting as a distributed computational model for modal logic. We also show that the fixed-point computation terminates whenever the modal program is well-behaved. Finally, we validate CML as a computational model for integrated knowledge representation and learning by applying it to a well-known testbed for distributed knowledge representation. This paves the way for a range of applications on integrated knowledge representation and learning, from practical reasoning to evolving multi-agent systems

Prenex Separation Logic with One Selector Field

International audienceWe show that infinite satisfiability can be reduced to finite satisfiabil-ity for all prenex formulas of Separation Logic with k ≥ 1 selector fields (SL k). This fact entails the decidability of the finite and infinite satisfiability problems for the class of prenex formulas of SL 1 , by reduction to the first-order theory of a single unary function symbol and an arbitrary number of unary predicate symbols. We also prove that the complexity of this fragment is not elementary recursive, by reduction from the first-order theory of one unary function symbol. Finally, we prove that the Bernays-Schönfinkel-Ramsey fragment of prenex SL 1 formulas with quantifier prefix in the language ∃ * ∀ * is PSPACE-complete

Towards a Proof Theory of G\"odel Modal Logics

Analytic proof calculi are introduced for box and diamond fragments of basic modal fuzzy logics that combine the Kripke semantics of modal logic K with the many-valued semantics of G\"odel logic. The calculi are used to establish completeness and complexity results for these fragments