Decomposable theories

2007 ◽  
Vol 7 (5) ◽  
pp. 583-632 ◽  
Author(s):  
KHALIL DJELLOUL
Keyword(s):  
General Algorithm ◽  
Boolean Combination ◽  
First Order ◽  
Free Variables ◽  
Infinite Trees ◽  
Decomposable Theory ◽  
Rewriting Rules

AbstractWe present in this paper a general algorithm for solving first-order formulas in particular theories called decomposable theories. First of all, using special quantifiers, we give a formal characterization of decomposable theories and show some of their properties. Then, we present a general algorithm for solving first-order formulas in any decomposable theory T. The algorithm is given in the form of five rewriting rules. It transforms a first-order formula ϕ, which can possibly contain free variables, into a conjunction φ of solved formulas easily transformable into a Boolean combination of existentially quantified conjunctions of atomic formulas. In particular, if ϕ has no free variables then φ is either the formula true or ¬true. The correctness of our algorithm proves the completeness of the decomposable theories. Finally, we show that the theory ${\cal T}$ of finite or infinite trees is a decomposable theory and give some benchmarks realized by an implementation of our algorithm, solving formulas on two-partner games in ${\cal T}$ with more than 160 nested alternated quantifiers.

scholarly journals Theory of finite or infinite trees revisited

2008 ◽  
Vol 8 (04) ◽  
pp. 431-489 ◽  
Author(s):  
KHALIL DJELLOUL ◽  
THI-BICH-HANH DAO ◽  
THOM FRÜHWIRTH
Keyword(s):  
Constraint Satisfaction ◽  
Decision Procedure ◽  
Constraint Satisfaction Problem ◽  
Free Variable ◽  
Constraint Solver ◽  
First Order ◽  
Infinite Trees ◽  
Order Constraint ◽  
Infinite Set ◽  
Rewriting Rules

AbstractWe present in this paper a first-order axiomatization of an extended theoryTof finite or infinite trees, built on a signature containing an infinite set of function symbols and a relationfinite(t), which enables to distinguish between finite and infinite trees. We show thatThas at least one model and prove its completeness by giving not only a decision procedure, but a full first-order constraint solver that gives clear and explicit solutions for any first-order constraint satisfaction problem inT. The solver is given in the form of 16 rewriting rules that transform any first-order constraintinto an equivalent disjunction φ of simple formulas such that φ is either the formulatrueor the formulafalseor a formula having at least one free variable, being equivalent neither totruenor tofalseand where the solutions of the free variables are expressed in a clear and explicit way. The correctness of our rules implies the completeness ofT. We also describe an implementation of our algorithm in CHR (Constraint Handling Rules) and compare the performance with an implementation in C++ and that of a recent decision procedure for decomposable theories.

scholarly journals Characterization of Dissipative Structures for First-Order Symmetric Hyperbolic System with General Relaxation

Mathematics ◽  
2021 ◽  
Vol 9 (7) ◽  
pp. 728
Author(s):  
Yasunori Maekawa ◽  
Yoshihiro Ueda
Keyword(s):  
Hyperbolic System ◽  
Dissipative Structure ◽  
Dissipative Structures ◽  
Algebraic Characterization ◽  
Symmetric Hyperbolic System ◽  
Full Generality ◽  
Order Linear ◽  
First Order

In this paper, we study the dissipative structure of first-order linear symmetric hyperbolic system with general relaxation and provide the algebraic characterization for the uniform dissipativity up to order 1. Our result extends the classical Shizuta–Kawashima condition for the case of symmetric relaxation, with a full generality and optimality.

scholarly journals On Consensus Index of Triplex Star-Like Networks: A Graph Spectra Approach

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1248
Author(s):  
Da Huang ◽  
Jian Zhu ◽  
Zhiyong Yu ◽  
Haijun Jiang
Keyword(s):  
Algebraic Connectivity ◽  
Multi Agent Systems ◽  
Graph Spectra ◽  
Asymptotic Behaviors ◽  
Agent Systems ◽  
First Order ◽  
Laplacian Spectra ◽  
Multi Agent ◽  
Consensus Index

In this article, the consensus-related performances of the triplex multi-agent systems with star-related structures, which can be measured by the algebraic connectivity and network coherence, have been studied by the characterization of Laplacian spectra. Some notions of graph operations are introduced to construct several triplex networks with star substructures. The methods of graph spectra are applied to derive the network coherence, and some asymptotic behaviors of the indices have been derived. It is found that the operations of adhering star topologies will make the first-order coherence increase a constant value under the triplex structures as parameters tend to infinity, and the second-order coherence have some equality relations as the node related parameters tend to infinity. Finally, the consensus related indices of the triplex systems with the same number of nodes but non-isomorphic graph structures have been compared and simulated to verify the results.

scholarly journals Numerical Characterization of Cohesive and Non-Cohesive ‘Sediments’ under Different Consolidation States Using 3D DEM Triaxial Experiments

Processes ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1252
Author(s):  
Hadar Elyashiv ◽  
Revital Bookman ◽  
Lennart Siemann ◽  
Uri ten Brink ◽  
Katrin Huhn
Keyword(s):  
Mechanical Response ◽  
Burial Depth ◽  
Cohesive Sediments ◽  
Third Order ◽  
First Order ◽  
Numerical Characterization ◽  
Response To Stress ◽  
Bond Strengths ◽  
Individual Bond

The Discrete Element Method has been widely used to simulate geo-materials due to time and scale limitations met in the field and laboratories. While cohesionless geo-materials were the focus of many previous studies, the deformation of cohesive geo-materials in 3D remained poorly characterized. Here, we aimed to generate a range of numerical ‘sediments’, assess their mechanical response to stress and compare their response with laboratory tests, focusing on differences between the micro- and macro-material properties. We simulated two endmembers—clay (cohesive) and sand (cohesionless). The materials were tested in a 3D triaxial numerical setup, under different simulated burial stresses and consolidation states. Variations in particle contact or individual bond strengths generate first order influence on the stress–strain response, i.e., a different deformation style of the numerical sand or clay. Increased burial depth generates a second order influence, elevating peak shear strength. Loose and dense consolidation states generate a third order influence of the endmember level. The results replicate a range of sediment compositions, empirical behaviors and conditions. We propose a procedure to characterize sediments numerically. The numerical ‘sediments’ can be applied to simulate processes in sediments exhibiting variations in strength due to post-seismic consolidation, bioturbation or variations in sedimentation rates.

Amphibole and mica 40Ar/39Ar ages from the Kaipokok and Aillik domains, Makkovik Province, Labrador: towards a characterization of back-arc processes in the Paleoproterozoic

2002 ◽  
Vol 39 (5) ◽  
pp. 749-764 ◽  
Author(s):  
Nicholas Culshaw ◽  
Peter Reynolds ◽  
Gavin Sinclair ◽  
Sandra Barr
Keyword(s):  
Thermal Effect ◽  
Shear Zone ◽  
Second Order ◽  
Greenschist Facies ◽  
Metamorphic Rocks ◽  
Arc Magmatism ◽  
First Order ◽  
Back Arc ◽  
Order Pattern

We report amphibole and mica 40Ar/39Ar ages from the Makkovik Province. Amphibole ages from metamorphic rocks decrease towards the interior of the province, indicating a first-order pattern of monotonic cooling with progressive migration of the province into a more distal back-arc location. The amphibole data, in combination with muscovite ages, reveal a second-order pattern consisting of four stages corresponding to changing spatial and temporal configurations of plutonism and deformation. (1) The western Kaipokok domain cooled through muscovite closure by 1810 Ma, long after the cessation of arc magmatism. (2) The Kaipokok Bay shear zone, bounding the Kaipokok and Aillik domains, cooled through amphibole closure during 1805–1780 Ma, synchronous with emplacement of syn-tectonic granitoid plutons. (3) Between 1740 and 1700 Ma, greenschist-facies shearing occurred along the boundary between the Kaipokok domain and Nain Province synchronous with A-type plutonism and localized shearing in the western Kaipokok domain, cooling to muscovite closure temperatures in the Kaipokok Bay shear zone, and A-type plutonism and amphibole closure or resetting in the Aillik domain. (4) In the period 1650–1640 Ma, muscovite ages, an amphibole age from a shear zone, and resetting of plutonic amphibole indicate a thermal effect coinciding in part with Labradorian plutonism in the Aillik domain. Amphibole ages from dioritic sheets in the juvenile Aillik domain suggest emplacement between 1715 and 1685 Ma. Amphibole ages constrain crystallization of small mafic plutons in the Kaipokok domain (reworked Archean foreland) to be no younger than 1670–1660 Ma. These ages are the oldest yet obtained for Labradorian plutonism in the Makkovik Province.

Formalization of metatheory of the Lambda Calculus in constructive type theory using the Barendregt variable convention

2021 ◽  
pp. 1-20
Author(s):  
Ernesto Copello ◽  
Nora Szasz ◽  
Álvaro Tasistro
Keyword(s):  
Type Theory ◽  
Lambda Calculus ◽  
Proof Assistant ◽  
First Order ◽  
Constructive Type Theory ◽  
Fundamental Part ◽  
Constructive Type ◽  
Free Variables ◽  
The Church ◽  
Theoretical Results

Abstarct We formalize in Constructive Type Theory the Lambda Calculus in its classical first-order syntax, employing only one sort of names for both bound and free variables, and with α-conversion based upon name swapping. As a fundamental part of the formalization, we introduce principles of induction and recursion on terms which provide a framework for reproducing the use of the Barendregt Variable Convention as in pen-and-paper proofs within the rigorous formal setting of a proof assistant. The principles in question are all formally derivable from the simple principle of structural induction/recursion on concrete terms. We work out applications to some fundamental meta-theoretical results, such as the Church–Rosser Theorem and Weak Normalization for the Simply Typed Lambda Calculus. The whole development has been machine checked using the system Agda.

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