Logical Derivation Research Articles

Curry-Howard for incomplete first-order logic derivations using one-and-a-half level terms

The Curry-Howard correspondence connects derivations in natural deduction with the lambda-calculus. Predicates are types, derivations are terms. This supports reasoning from assumptions to conclusions, but we may want to reason backwards; from the desired conclusion towards the assumptions. At intermediate stages we may have a partial derivation, with holes. This is natural in informal practice but it can be difficult to formalise. The informal act of filling holes in a partial derivation suggests a capturing substitution, since holes may occur in the scope of quantifier introduction rules. As other authors have observed, this is not immediately supported by the lambda-calculus. Also, universal quantification requires a ‘fresh name’ and it is not immediately obvious what formal meaning to assign to this notion if derivations are incomplete. Further issues arise with proof-normalisation; this corresponds with lambda-calculus reduction, which can require alpha-conversion to avoid capture when beta-reducing, and it is not immediately clear how to alpha-convert a name in an incomplete derivation. We apply a one-and-a-half level technique based on nominal terms to construct a Curry-Howard correspondence for first-order logic. This features two levels of variable, but with no lambda-abstraction at the second level. Predicates are types, derivations are terms, proof-normalisation is reduction — and the two levels of variable are, respectively, the assumptions and the holes of an incomplete derivation. We give notions of proof-term, typing, alpha-conversion and beta-reduction for our syntax. We prove confluence, we exhibit several admissible rules including a proof that instantiation of level two variables is type-safe — this corresponds with the act of filling holes in an incomplete derivation, and can be viewed as a form of Cut-rule — and we explore the connection with traditional Curry-Howard in the case that the derivation is in fact complete. Our techniques are not specifically tailored to first-order logic and the same ideas should be applicable without any essential new difficulties to similar logical systems.

A novel active vibration isolator applied for micro-gyro by array of deposited electrodes

Micro-gyroscopes are usually driven into resonance so that the sensitivity and resolution can be enhanced. However, if any substantial vibration of the gyroscope base, over which the seismic proof mass is seated, was present, then the preset resonant frequency would be altered and the performances of the gyroscopes could be much degraded. In this paper, an innovative three degree-of-freedom (DOF) isolation system is proposed to attenuate the undesirable vibrations caused by the ambient environments. The mathematic model of the proposed 3-DOF micro-machined isolation system is established and analyzed such that the transfer function of transmissibility is obtained. The pull-in instability and associated pull-in voltage for the actuators of the isolation system are numerically unveiled so that the interval of the applied voltage to generate the electrostatic control force can be set to ensure the stability of the suspension system. In addition, a fuzzy logic proportion and derivative (PD) controller is synthesized for disturbance rejection. Five sensing electrodes, in cooperation with the isolator, are used to provide the feedback signals of the relative displacements of the proof mass with respect to the base, i.e., pitch, yaw and lateral linear displacement. Ten tuning electrodes are utilized to generate the required electrostatic forces to preserve the seismic proof mass from external disturbance. In comparison with the traditional PD action, the proposed fuzzy logic PD control strategy is verified by intensive simulations to illustrate its superior vibration isolation capability.

Epistemology, social technology, and expert judgement: Olaf Helmer's contribution to futures research

The article discusses Olaf Helmer's contribution to the development of futures studies by focusing on four basic theses defining his approach. (1) Quasi-laws in social sciences and futures studies can in fact be treated in the same way that the natural laws of the physical sciences. (2) In order to make predictions, one need not appeal to a strict logical derivation, as the “covering laws” doctrine of logical-empiricism suggests. (3) Prediction and explanation are not logically symmetrical as positivists believe, thus the conditions needed for explanation are not those required for prediction. (4) Local, tacit, personal and expert knowledge are crucial in developing a foresight methodology. In conjunction, these four theses open the way to a unique theory of social prediction and to variety of “unorthodox items of methodological equipment for the purposes of prediction in the inexact sciences.”

Asymptotic Models for the Topological Sensitivity Versus the Topological Derivative

An approach, based on the refined method of matched asymptotic expansions, is proposed for the construction of asymptotic models for the topological sensitivity of the energy functional with respect to the creation of a small hole in the geometrical domain. It is shown that the asymptotic model provides more information for calculations than the topo- logical derivative.

Wittgenstein on the Infinity of Primes

It is controversial whether Wittgenstein's philosophy of mathematics is of critical importance for mathematical proofs, or is only concerned with the adequate philosophical interpretation of mathematics. Wittgenstein's remarks on the infinity of prime numbers provide a helpful example which will be used to clarify this question. His antiplatonistic view of mathematics contradicts the widespread understanding of proofs as logical derivations from a set of axioms or assumptions. Wittgenstein's critique of traditional proofs of the infinity of prime numbers, specifically those of Euler and Euclid, not only offers philosophical insight but also suggests substantive improvements. A careful examination of his comments leads to a deeper understanding of what proves the infinity of primes.

Reasoning about taxonomies in first-order logic

Experts often disagree about the organization of biological taxa. The shifting definitions of taxonomic names complicate otherwise simple queries concerning these taxa. For example, a query such as “how many occurrences of specimens in genus G are recorded in database D” will return different answers depending on whose definition of genus G is used. In our proposed framework, taxonomic classifications of multiple experts are captured using first-order logic (FOL). Specifically, taxonomies, and relationships between them, are viewed as sets of first-order formulas, constraining the possible interpretations of names and concepts in the taxonomies. The formalization of taxonomies and the relationships between them via our FOL language L tax allows us to clarify (a) what it means for a taxonomy to be consistent, (b) to be inconsistent, (c) whether a new relationship between two taxa (e.g., a congruence A ≡ B) is implied, thus “filling logic gaps”, and (d) whether two taxonomies from different authorities, together with a taxonomy mapping (articulation) from a third authority, are mutually consistent. We illustrate our logic-based formalization and the resulting opportunities for automated reasoning support for biological taxonomies using examples involving the classification of a genus of plants. We elaborate on (a–d) above and give some example derivations in logic. We also show that while reasoning in L tax is decidable, it might still be computationally hard (e.g., NP-complete) and thus infeasible over large taxonomies and articulations. By employing results from the spatial algebra RCC-5, we identify an important class of efficient taxonomy articulations, i.e., whose satisfiability can be checked in polynomial time.

Theory and Practice in Law

Lawis characterised by a fundamental gap between its social embeddedness and theapparently formal automatism it operates, which gap is basically bridged by thelaw's ultimate practicality under the guise of its mere logicity. This seemingcontradiction is resolved by judicial decisions as responsible and responsivepractical actions which are to result from the necessary conceptualtransformation(s) of the law's wording in the course of its officialapplication, which does involve a necessary jump in logical derivation. This isto say that on final analysis and inpractical terms, law is what gets actualised through the actual uses of it. Black-box effect such as this is helped by thevariety-and owing to the magical transforming effects-of legal techniques.Eventually, it is legal culture that provides a medium in which legaltechniques can at all be selected and used. On a conceptual plane, one of thefilters is offered by legal dogmatics. This very complex includes dialectics aswell, for there is no motion without counter-motion, therefore, it is notrealistic to pursue any human ambition without some safety valves inserted. Or,regarding, e.g., law, no homo­genisationis feasible without some re-heterogenisation at the same time. Paradoxicallyspeaking, while modern formal legal development went in the direction tomechanise the judge, the realisation was also made that law had ever been tooserious an undertaking to be just left alone to the logification by someimpersonally formalistic apparatus. Therefore, simultaneously with the veryfirst act of formalisation, law has ever built in its scheme the possibility ofde-formalisation as well.

A Quick Primer on Logic and Rationality

Logic does not permit contradictions. Ordinary language sometimes uses contradictions meaningfully. Examples are provided of logical derivations. Yet one might ask how one might prove that logic itself is rational. Gottlob Frege answered that logic constitutes rationality.

Las redes interinstitucionales en la construcción de un sistema regional de innovación. El caso de Aguascalientes

Este artículo es una versión condensada qu ofrece los resultados finales del proyecto de investigación en el marco del Doctorado Interinstitucional en Ciencias Sociales de la UAA-UAM/Iztapalapa. El trabajo se ubica en el relativamente reciente campo de estudio de los Sistemas Nacionales de Innovación (SNI), del que los sistemas regionales (SRI) constituyen una derivación lógica aunque, hasta ahora, insuficientemente trabajada por los especialistas.

An Embedded Microprocessor for Intelligent Control

The conventional approach for the implementation of the knowledge base of a planning agent, on an intelligent embedded system, is solely of software nature. It requires the existence of a compiler that transforms the initial declarative logic program, specifying the knowledge base, to its equivalent procedural one, to be programmed to the embedded system's microprocessor. This practice increases the complexity of the final implementation (the declarative to sequential transformation adds a great amount of software code for simulating the declarative execution) and reduces the overall system's performance (logic derivations require the use of a stack and a great number of jump instructions for their evaluation). The design of specialized hardware implementations, which are only capable of supporting logic programs, in an effort to resolve the aforementioned problems, introduces limitations in their use in applications where logic programs need to be intertwined with traditional procedural ones in a desired application. In this paper, we exploit HW/SW codesign methods to present a microprocessor, capable of supporting hybrid applications using both programming approaches. We take advantage of the close relationship between attribute grammar (AG) evaluation and knowledge engineering methods to present a programmable hardware parser that performs logic derivations and combine it with an extension of a conventional RISC microprocessor that performs the unification process to report the success or failure of logic derivations. The extended RISC microprocessor is still capable of executing conventional procedural programs, thus hybrid applications can be implemented. The presented implementation increases the performance of logic derivations for the control inference process (experimental analysis yields an approximate 1000% -- 10 times increase in performance) and reduces the complexity of the final implemented code through the introduction of an extended C language called C-AG that simplifies the programming of hybrid procedural-declarative applications.

Extracting meaning from temporal nouns and temporal prepositions

This article provides a compositional semantics for temporal nouns and temporal prepositions that are annotated as temporal prepositional phrases or noun phrases by an automatic tagging system (e.g., last Monday, on Dec. 1 st , for three weeks or before Christmas ). Current temporal tagging systems rely on an ad-hoc-representation for temporal date and time expressions, but the more demanding tasks of temporal question-answering and automatic text summarization require a sound logical derivation and representation of temporal expressions. Our proposal draws from two formal accounts of temporal prepositional phrases by Pratt and Francez [2001] and von Stechow [2002b], and is realized within an automatic temporal tagging system for German newspaper articles.

Optimum design of multiple intersecting spheres deep-submerged pressure hull

The multiple intersecting spheres (MIS) pressure hull is a logical derivative of the single unstiffened sphere, which is frequently used for deep operating, small submersibles because of its attractive low buoyancy factor. This paper investigates the optimum design of an MIS deep-submerged pressure hull subjected to hydrostatic pressure, using a powerful optimization procedure combined the extended interior penalty function method (EIPF) with the Davidon–Fletcher–Powell (DFP) method. In this study, the thickness of the shell, the width of the rib-ring, the inner radius of the rib-ring and the angle of intersection of the spherical shell are selected as design variables, and structural failure and human requirements are considered to minimize the buoyancy factor. Additionally, a sensitivity analysis is performed to study the influence of the design variables on the optimal structural strength design. The results reveal that the shell thickness is most important to lobar buckling strength, and that rib-ring width, rib-ring inner radius and spherical shell intersection angle are most important to rib-ring hoop strength. Optimization results may provide a valuable reference for designers.

Belief revision via Lamarckian evolution

We present a system for performing belief revision in a multi-agent environment. The system is called GBR (Genetic Belief Revisor) and it is based on a genetic algorithm. In this setting, different individuals are exposed to different experiences. This may happen because the world surrounding an agent changes over time or because we allow agents exploring different parts of the world. The algorithm permits the exchange of chromosomes from different agents and combines two different evolution strategies, one based on Darwin's and the other on Lamarck's evolutionary theory. The algorithm therefore includes also a Lamarckian operator that changes the memes of an agent in order to improve their fitness. The operator is implemented by means of a belief revision procedure that, by tracing logical derivations, identifies the memes leading to contradiction. Moreover, the algorithm comprises a special crossover mechanism for memes in which a meme can be acquired from another agent only if the other agent has "accessed" the meme, i.e. if an application of the Lamarckian operator has read or modified the meme.Experiments have been performed on the n -queen problem and on a problem of digital circuit diagnosis. In the case of the n-queen problem, the addition of the Lamarckian operator in the single agent case improves the fitness of the best solution. In both cases the experiments show that the distribution of constraints, even if it may lead to a reduction of the fitness of the best solution, does not produce a significant reduction.

Complex fuzzy logic

A novel framework for logical reasoning, termed complex fuzzy logic, is presented in this paper. Complex fuzzy logic is a generalization of traditional fuzzy logic, based on complex fuzzy sets. In complex fuzzy logic, inference rules are constructed and "fired" in a manner that closely parallels traditional fuzzy logic. The novelty of complex fuzzy logic is that the sets used in the reasoning process are complex fuzzy sets, characterized by complex-valued membership functions. The range of these membership functions is extended from the traditional fuzzy range of [0,1] to the unit circle in the complex plane, thus providing a method for describing membership in a set in terms of a complex number. Several mathematical properties of complex fuzzy sets, which serve as a basis for the derivation of complex fuzzy logic, are reviewed in this paper. These properties include basic set theoretic operations on complex fuzzy sets - namely complex fuzzy union and intersection, complex fuzzy relations and their composition, and a novel form of set aggregation - vector aggregation. Complex fuzzy logic is designed to maintain the advantages of traditional fuzzy logic, while benefiting from the properties of complex numbers and complex fuzzy sets. The introduction of complex-valued grades of membership to the realm of fuzzy logic generates a framework with unique mathematical properties, and considerable potential for further research and application.

On categorical equivalence of Gentzen-style derivations in IMLL

A new deciding algorithm for categorical equivalence of derivations in Intuitionistic Multiplicative Linear Logic (IMLL) is proposed. The algorithm is based uniquely on manipulations with Gentzen-style derivations. The algorithm has low polynomial complexity. The paper also contains results concerning permutability of rules and its connection with categorical equivalence.

Minimum segments in sequent derivations

In a system of sequents for intuitionistic predicate logic derivations without a special kind of cuts (maximum cuts) will be considered. The following be shown: in a derivation without maximum cuts there are paths of the same form as paths in a normal derivation of natural deduction, i.e., these paths have the E-part, the I-part, and one minimum part which corresponds to a minimum segment in a normal derivation.

Konovalov correlation for molar heats of combustion of organic compounds: its adoption for coal

Konovalov's correlation for molar heats of combustion of organic compounds has been suitably adopted for coal. Studies of a number of coal models, supported by experimental data on actual coals of varying rank, have revealed that ‘ x’ the structure-related factor remains virtually constant at about 0.43 MJ kg −1 for all high volatile coals (C≤81%) beyond which it decreases almost linearly with increasing rank. A logical derivative of the correlation is also presented to give calorific values of coal on dry basis, from lignite to anthracite. Accuracy of the new correlation has been found to be as good as those of previous correlations, SD being about ±0.16, whereas its potential accuracy has been found to be as low as 0.05 MJ kg −1. Such high accuracy has not been hitherto reported for any other correlation. Significance of this vis-à-vis the values of ‘ x’ and hence the structural pattern of coal is also discussed.

Machine loading problem of FMS: A fuzzy-based heuristic approach

A fuzzy-based solution methodology has been formulated to address the machine-loading problem in a flexible manufacturing system. The objectives considered are minimization of system unbalance and maximization of throughput, whereas the systems technological constraints are posed by availability of machining time and tool slots. The job ordering/job sequence determination before loading is carried out by evaluating the membership contribution of each job to its characteristics such as batch size, essential operation processing time and optional operation processing time. The operation?machine allocation decisions are made based on the evaluation of membership contribution of operation machine allocation vector. The formulation of membership function is based on logical derivations and enjoys reasonable analytical support. The proposed heuristic is tested on 10 problems adopted from literatures and the results reveal substantial improvement in solution quality over, some of existing heuristic-based approaches.

Foundations for Applications of Gibbs Derivatives in Logic Design and VLSI

New technologies and increased requirements for performances of digital systems require new mathematical theories and tools as a basis for future VLSI CAD systems. New or alternative mathematical approaches and concepts must be suitable to solve some concrete problems in VLSI and efficient algorithms for their efficient application should be provided. This paper is an attempt in this direction and relates with the recently renewed interest in arithmetic expressions for switching functions, instead representations in Boolean structures, and spectral techniques and differential operators in switching theory and applications.Logic derivatives are efficiently used in solving different tasks in logic design, as for example, fault detection, functional decomposition, detection of symmetries and co‐symmetries of logic functions, etc. Their application is based on the property that by differential operators, we can measure the rate of change of a logic function. However, by logic derivatives, we can hardly distinguish the direction of the change of the function, since they are defined in finite algebraic structures.Gibbs derivatives are a class of differential operators on groups, which applied to logic functions, permit to overcome this disadvantage of logic derivatives. Therefore, they may be useful in logic design in the same areas where the logic derivatives have been already using. For such applications, it is important to provide fast algorithms for calculation of Gibbs derivatives on finite groups efficiently in terms of space and time.In this paper, we discuss the methods for efficient calculation of Gibbs derivatives. These methods should represent a basis for further applications of these and related operators in VLSI CAD systems.

Logical derivation of computer programs

An introduction to mathematical proof programming and program derivation program derivation examples - computing predicates, sets and relations program derivation examples - computing natural numbers and lists.