Complete Calculus Research Articles

Fluorescence diagnostics of periodontitis

This study is aimed at the development of a non-invasive method for detecting supraand subgingival calculi whose formation leads to the development of the most widespread stomatological disease, i.e., periodontitis. In practice, diagnostics and treatment of this disease is commonly performed visually and tactilely. This does not allow complete scaling or dental calculi are removed together with hard tooth tissues. It was experimentally shown that the presence or absence of dental calculi can be rapidly detected by measuring fluorescence spectra in the visible wavelength region, not damaging hard tooth tissues. Fluorescence diagnostics can be performed simultaneously with scaling and allows real-time identification of calculus and the instant of its complete removal without damaging hard tooth tissues.

Resolution with Order and Selection for Hybrid Logics

We investigate labeled resolution calculi for hybrid logics with inference rules restricted via selection functions and orders. We start by providing a sound and refutationally complete calculus for the hybrid logic $\mathcal{H}(@,{\downarrow},\mathsf{A})$ , even under restrictions by selection functions and orders. Then, by imposing further restrictions in the original calculus, we develop a sound, complete and terminating calculus for the $\mathcal{H}(@)$ sublanguage. The proof scheme we use to show refutational completeness of these calculi is an adaptation of a standard completeness proof for saturation-based calculi for first-order logic that guarantees completeness even under redundancy elimination. In fact, one of the contributions of this article is to show that the general framework of saturation-based proving for first-order logic with equality can be naturally adapted to saturation-based calculi for other languages, in particular modal and hybrid logics.

Implementing a fair monodic temporal logic prover

Monodic first-order temporal logic is a fragment of first-order temporal logic for which sound and complete calculi have been devised. One such calculus is ordered fine-grained resolution with selection, which is implemented in the theorem prover TeMP. However, the architecture of TeMP cannot guarantee the fairness of its derivations. In this paper we present an architecture for a resolution-based monodic first-order temporal logic prover that can ensure fair derivations and we describe the implementation of this fair architecture in the theorem prover TSPASS.

Modal logics with Belnapian truth values

Various four- and three-valued modal propositional logics are studied. The basic systems are modal extensions BK and BS4 of Belnap and Dunn's four-valued logic of firstdegree entailment. Three-valued extensions of BK and BS4 are considered as well. These logics are introduced semantically by means of relational models with two distinct evaluation relations, one for verification (support of truth) and the other for falsification (support of falsity). Axiom systems are defined and shown to be sound and complete with respect to the relational semantics and with respect to twist structures over modal algebras. Sound and complete tableau calculi are presented as well. Moreover, a number of constructive non-modal logics with strong negation are faithfully embedded into BS4, into its three-valued extension B3S4, or into temporal BS4, BtS4. These logics include David Nelson's three-valued logic N3, the four-valued logic N4 bottom, the connexive logic C, and several extensions of bi-intuitionistic logic by strong negation.

Equivalence and quantifier rules for logic with imperfect information

In this paper, we present a prenex form theorem for a version of Independence Friendly logic, a logic with imperfect information. Lifting classical results to such logics turns out not to be straightforward, because independence conditions make the formulas sensitive to signalling phenomena. In particular, nested quantification over the same variable is shown to cause problems. For instance, renaming of bound variables may change the interpretations of a formula, there are only restricted quantifier extraction theorems, and slashed connectives cannot be so easily removed. Thus we correct some claims from Hintikka [8], Caicedo & Krynicki [3] and Hodges [11]. We refine definitions, in particular the notion of equivalence, and sharpen preconditions, allowing us to restore (restricted versions of) those claims, including the prenex form theorem of Caicedo & Krynicki [3], and, as a side result, we obtain an application to Skolem forms of classical formulas. It is a known fact that a complete calculus for IF-logic is impossible, but with our results we establish several quantifier rules that form a partial calculus of equivalence for a general version of IF-logic reflecting general properties of information flow in games.

A term-graph clausal logic: completeness and incompleteness results ★

A clausal logic allowing to handle term-graphs is defined. Term-graphs are a generalization of terms (in the usual sense) possibly containing shared subterms and cycles. The satisfiability problem for this logic is shown to be undecidable (not even semi-decidable), but some fragments are identified for which it is semi-decidable. A complete (w.r.t validity) calculus for these fragments is proposed. Some simple examples give a taste of this calculus at work.

On Positive Relational Calculi

We discuss the question of inclusions between positive relational terms and some of its aspects, using the form of a dialogue. Two possible approaches to the problem are emphasized: natural deduction and graph manipulations. Both provide sound and complete calculi for proving the valid inclusions, supporting nice strategies to obtain proofs in normal form, but the latter appears to present several advantages, which are discussed.

COMMON-SENSE SPATIAL REASONING FOR INFORMATION CORRELATION IN PERVASIVE COMPUTING

The current technological trend depicts a scenario in which space, and more generally the environment in which the computation takes place, represents a key aspect that must be considered in order to improve systems' context awareness, even if the kind of information processed is not only of spatial nature. This article focuses on the notions of “place” and “conceptual spatial relation” to present a formal model of space supporting common-sense spatial reasoning. The model can be viewed as the semantic specification for a hybrid logic, whose formulas represent spatially qualified information. Interesting classes of common-sense spatial models are identified according to the properties of their characteristic relations and an axiomatization of the associated hybrid language is given; a sound and complete tableau-based calculus for these classes of models is provided.

In VitroEvaluation of Nd:YAG Laser Radiation at Three Different Wavelengths (1064, 532, and 355 nm) on Calculus Removal in Comparison with Ultrasonic Scaling

The aim of this study was to investigate the effectiveness of laser treatment for calculus removal at different wavelengths and compare it with that of ultrasonic scaling (US). Conventional methods of treatment of periodontal tissues can be limited by morphology, root anatomy, and the periodontal pocket. In the search for an effective therapy to achieve a biocompatible root surface, laser treatment appeared in the field of periodontology. Radiation from a Nd:YAG laser operating at the fundamental (1064 nm), second harmonic (532 nm) and third harmonic (355 nm) wavelengths was used for calculus removal. Environment scanning electron microscopy (ESEM) was used for the morphological analysis of the treated surfaces. The wavelength dependence of the ablation thresholds for calculus and cementum was evaluated by applying a photoacoustic technique. US achieved complete calculus removal, leaving a smooth and regular surface in a more efficient way than laser treatment. ESEM examination and photoacoustic methods confirmed a nonselective removal of material that entails partial cementum ablation of the root surface with the laser treatment. UV radiation at 355 nm can constitute an adjunctive therapy to US, by eliminating toxic agents and the contaminated layer of cementum.

Making knowledge explicit: How hard it is

Artemov's logic of proofs LP is a complete calculus of propositions and proofs, which is now becoming a foundation for the evidence-based approach to reasoning about knowledge. Additional atoms in LP have form t : F , read as “ t is a proof of F” (or, more generally, as “ t is an evidence for F”) for an appropriate system of terms t called proof polynomials. In this paper, we answer two well-known questions in this area. One of the main features of LP is its ability to realize modalities in any S 4 -derivation by proof polynomials thus revealing a statement about explicit evidences encoded in that derivation. We show that the original Artemov's algorithm of building such realizations can produce proof polynomials of exponential length in the size of the initial S 4 -derivation. We modify the realization algorithm to produce proof polynomials of at most quadratic length. We also found a modal formula, any realization of which necessarily requires self-referential constants of type c : A ( c ) . This demonstrates that the evidence-based reasoning encoded by the modal logic S 4 is inherently self-referential.

A Minimal Hybrid Logic for Intervals

Taking our inspiration from van Benthem’s treatment of temporal interval structures, and Halpern and Shoham’s work on intervals, we introduce an interval hybrid temporal logic with two binary relations, precedence and inclusion, for talking about interval temporal structures. This paper can be seen as an continuation of the work began in an earlier paper, in which we undertook a purely modal treatment of interval temporal structures. By introducing an interval hybrid temporal logic, we enrich the logic with nominals, and thereby increase the expressivity of the logic. We study the interval hybrid temporal logic in its full generality and identify two important classes of interval temporal structures: the class of minimal interval structures, and the class of van Benthem minimal interval structures. We present sound and complete tableau calculi for both classes of structures. We prove that the logic of minimal interval structures is decidable, by developing a novel bulldozing technique that handles both the presence of nominals and the interaction between the two relations. We go on to show that the satisfiability problem is EXPTIME-complete. We conclude the paper with the remark that the decidability (or otherwise) and complexity of the logic of van Benthem minimal interval structures remains an interesting open problem.

Reasoning about static and dynamic properties in alloy

We study a number of restrictions associated with the first-order relational specification language Alloy. The main shortcomings we address are:---the lack of a complete calculus for deduction in Alloy's underlying formalism, the so called relational logic,---the inappropriateness of the Alloy language for describing (and analyzing) properties regarding execution traces.The first of these points was not regarded as an important issue during the genesis of Alloy, and therefore has not been taken into account in the design of the relational logic. The second point is a consequence of the static nature of Alloy specifications, and has been partly solved by the developers of Alloy; however, their proposed solution requires a complicated and unstructured characterization of executions.We propose to overcome the first problem by translating relational logic to the equational calculus of fork algebras . Fork algebras provide a purely relational formalism close to Alloy, which possesses a complete equational deductive calculus. Regarding the second problem, we propose to extend Alloy by adding actions . These actions, unlike Alloy functions, do modify the state. Much the same as programs in dynamic logic, actions can be sequentially composed and iterated, allowing them to state properties of execution traces at an appropriate level of abstraction.Since automatic analysis is one of Alloy's main features, and this article aims to provide a deductive calculus for Alloy, we show that:---the extension hereby proposed does not sacrifice the possibility of using SAT solving techniques for automated analysis,---the complete calculus for the relational logic is straightforwardly extended to a complete calculus for the extension of Alloy.

On modal μ-calculus with explicit interpolants

This paper deals with the extension of Kozen's μ-calculus with the so-called “existential bisimulation quantifier”. By using this quantifier one can express the uniform interpolant of any formula of the μ-calculus. In this work we provide an explicit form for the uniform interpolant of a disjunctive formula and see that it belongs to the same level of the fixpoint alternation hierarchy of the μ-calculus than the original formula. We show that this result cannot be generalized to the whole logic, because the closure of the third level of the hierarchy under the existential bisimulation quantifier is the whole μ-calculus. However, we prove that the first two levels of the hierarchy are closed. We also provide the μ-logic extended with the bisimulation quantifier with a complete calculus.

Percutaneous Antegrade Removal of Impacted Upper-Ureteral Calculi: Still the Treatment of Choice in Developing Countries

To assess the safety and efficacy of antegrade percutaneous nephrolithotomy (PCNL) of large impacted proximal-ureteral calculi. Between July 1998 and October 2003, a total of 66 patients (43 male and 23 female; mean age 37.5 years) underwent PCNL for impacted proximal-ureteral calculi. The inclusion criteria were calculi >15 mm that were densely impacted and located between the ureteropelvic junction and the lower border of the 4th lumbar vertebra. The mean drop in hemoglobin, operating time, analgesic requirement, and hospital stay were assessed. Sixty-five patients (98.5%) had complete calculus clearance in a single session through a single tract. The mean operating time and hospital stay were 47 minutes and 46 hours, respectively. The mean analgesic requirement was 65 mg of pethidine (meperidine). The mean follow-up was 14 months. There were no significant postoperative complications. Antegrade PCNL is a safe and effective option for large, impacted proximal-ureteral calculi.

Ultrastructural study of calculus–enamel and calculus–root interfaces

The attachment of dental calculus to the tooth (enamel or cementum) surface affects the ease or difficulty of its removal. Understanding the ultrastructural features of the calculus-tooth interface will help in the development of efficient strategies for efficient removal of dental calculus. The aim of this study was to determine the ultrastructural characteristics of the calculus-tooth interface in relation to the occurrence of calculus fracture. Investigation of the ultrastructural characteristics of the calculus-tooth interface was made on eight human molars with mature supragingival and subgingival calculus using scanning electron microscopy (SEM), transmission electron microscopy (TEM) and fourier transform infra-red (FT-IR) spectroscopy. Fractures were shown by SEM to consistently occur within the calculus itself, but not at the calculus-tooth interface. Higher magnification revealed that the enamel apatite crystals (in the case of supragingival calculus) or the cementum apatite crystals (in the case of subgingival calculus) appeared intimately connected with the calculus crystals at the calculus-enamel or calculus-cementum interface. TEM micrographs confirmed this intimate direct connection or fusion (epitaxial growth) of calculus crystals with enamel and cementum apatite crystals. FT-IR showed lower concentrations of organic phase attributed to microorganisms and higher concentrations of collagen at the calculus-cementum interface compared to that in the calculus away from the interface. Difficulty in complete calculus removal from tooth surfaces (especially from cementum or dentin) may be due in part to the intimate contact between the calculus and the tooth, due to the chemical bonding between the calculus crystals and the tooth apatite crystals and occasional fusion (i.e., epitaxial growth) of the calculus calcium phosphate crystals with the enamel, dentin or apatite crystals. This cohesive bonding results in fracture planes occurring within the calculus instead of at the calculus-tooth interface.

Theorem proving with built-in hybrid theories

A growing number of applications of automated reasoning exhibits the necessity of flexible deduction systems. A deduction system should be able to execute inference rules which are appropriate to the given problem. One way to achieve this behavior is the integration of different calculi. This led to so called hybrid reasoning [22, 1, 10, 20] which means the integration of a general purpose foreground reasoner with a specialized background reasoner. A typical task of a background reasoner is to perform special purpose inference rules according to a built-in theory. The aim of this paper is to go a step further, i.e. to treat the background reasoner as a hybrid system itself. The paper formulates sufficient criteria for the construction of complete calculi which enable reasoning under hybrid theories combined from sub-theories. For this purpose we use a generic approach described in [20]. This more detailed view on built-in theories is not covered by the known general approaches [1, 3, 6, 20] for building in theories into theorem provers. The approach is demonstrated by its application to the target calculi of the algebraic translation [9] of multi-modal and extended multi-modal [7] logic to first-order logic.

Root surface debridement and endotoxin removal.

This study assessed associations between the number of standardized scaling strokes and the reduction of endotoxin on the root surface. Therapy of periodontally involved teeth attempts removal of accretions by scaling and root planing. The amount of mechanical therapy required to free the root surface from etiologic factors remains unknown. Twenty-four extracted human caries-free single rooted teeth with at least 5 mm attachment loss were used. A region of interest (ROI) which contained subgingival calculus was defined on the root surface of each tooth. Standardized force instrumentation was applied using a force-measuring curet. Fifty working strokes were applied to every ROI. Forces applied were recorded. The force recordings were converted from Millivolts into Newtons (N). After every unit of 5 strokes, presence of calculus was evaluated and scaling debris was collected. Endotoxin concentration was determined in the debris samples. The endotoxin values for strokes 1-5 were statistically significantly greater than the values from all other stroke intervals. Complete calculus removal occurred after a mean of 9.3 strokes. The endotoxin concentration reached a minimal level with concentrations of 0.03-0.306 EU/ml after calculus removal was complete. These findings suggest that completion of calculus removal coincides with endotoxin levels associated with clinically healthy teeth.

The possibility of detecting subgingival calculus by laser-fluorescence in vitro.

The aim of the present study was to investigate laser-fluorescence effects on the root surface. The possibility of detecting subgingival calculus should be evaluated to control complete calculus removal after scaling and root planing. Twenty freshly extracted human teeth, partially covered with calculus on the root surface, were fixed on a translation stage. The specimens were irradiated by light from a laser source (diode laser, <1 mW) with a wavelength of 655 nm using the Diagnodent trade mark -system. To simulate the conditions in a periodontal pocket during the measurement procedure, the teeth were covered by a layer of physiological saline solution or blood in order to determine the influence of different inflammatory fluids. The root surface was scanned on a line of about 1 cm in steps of 0.5 mm. The start and end was marked by a notch. The results of the laser-fluorescence detection were compared to clinical and histological findings. Clinically apparent calculus on the root surface was always accompanied with a statistically significant increase of laser-fluorescence values ( p < 0.05). These values were not influenced by the different fluids ( p < 0.05). The laser-fluorescence values were reproducible ( p > 0.05). After scaling, when no debris remained at the root surface, laser-fluorescence values decreased significantly ( p < 0.05). The histological findings supported the clinical results. In conclusion, the present study indicates that laser-fluorescence values on the root surface are strongly correlated to the presence of calculus. This fluorescence effect can basically be used to develop a new detecting system for subgingival calculus in periodontal pockets.

Lie Brackets on Affine Bundles

Natural affine analogs of Lie brackets on affine bundles are studied.In particular, a close relation to Lie algebroids and a duality withcertain affine analog of Poisson structure is established as well asaffine versions of complete lifts and Cartan exterior calculi.

Cancellative Abelian Monoids and Related Structures in Refutational Theorem Proving (Part II)

Cancellative superposition is a refutationally complete calculus for first-order equational theorem proving in the presence of the axioms of cancellative Abelian monoids, and, optionally, the torsion-freeness axioms. Thanks to strengthened ordering restrictions, cancellative superposition avoids some of the inefficiencies of classical AC-superposition calculi. We show how the efficiency of cancellative superposition can be further improved by using variable elimination techniques, leading to a significant reduction of the number of variable overlaps. In particular, we demonstrate that in divisible torsion-free Abelian groups, variable overlaps, AC-unification and AC-orderings can be avoided completely.