scholarly journals Spider Diagrams

2005 ◽  
Vol 8 ◽  
pp. 145-194 ◽  
Author(s):  
John Howse ◽  
Gem Stapleton ◽  
John Taylor
Keyword(s):  
Diagrammatic Reasoning ◽  
Abstract Syntax ◽  
Topological Properties ◽  
Formal Reasoning ◽  
First Order ◽  
Transformation Rules ◽  
Reasoning System ◽  
Concrete Syntax ◽  
Monadic Logic ◽  
Spider Diagrams

AbstractThe use of diagrams in mathematics has traditionally been restricted to guiding intuition and communication. With rare exceptions such as Peirce's alpha and beta systems, purely diagrammatic formal reasoning has not been in the mathematician's or logician's toolkit. This paper develops a purely diagrammatic reasoning system of “spider diagrams” that builds on Euler, Venn and Peirce diagrams. The system is known to be expressively equivalent to first-order monadic logic with equality. Two levels of diagrammatic syntax have been developed: an ‘abstract’ syntax that captures the structure of diagrams, and a ‘concrete’ syntax that captures topological properties of drawn diagrams. A number of simple diagrammatic transformation rules are given, and the resulting reasoning system is shown to be sound and complete.

scholarly journals Spider Diagrams: A Diagrammatic Reasoning System

2001 ◽  
Vol 12 (3) ◽  
pp. 299-324 ◽  
Author(s):  
JOHN HOWSE ◽  
FERNANDO MOLINA ◽  
JOHN TAYLOR ◽  
STUART KENT ◽  
JOSEPH YOSSI GIL
Keyword(s):  
Diagrammatic Reasoning ◽  
Reasoning System ◽  
Spider Diagrams

scholarly journals A case study in programming coinductive proofs: Howe’s method

2018 ◽  
Vol 29 (8) ◽  
pp. 1309-1343 ◽  
Author(s):  
ALBERTO MOMIGLIANO ◽  
BRIGITTE PIENTKA ◽  
DAVID THIBODEAU
Keyword(s):  
Programming Languages ◽  
Operational Semantics ◽  
Lambda Calculus ◽  
Abstract Syntax ◽  
Closure Operation ◽  
Formal Reasoning ◽  
Reasoning System ◽  
Contextual Equivalence ◽  
High Level

Bisimulation proofs play a central role in programming languages in establishing rich properties such as contextual equivalence. They are also challenging to mechanize, since they require a combination of inductive and coinductive reasoning on open terms. In this paper, we describe mechanizing the property that similarity in the call-by-name lambda calculus is a pre-congruence using Howe’s method in the Beluga formal reasoning system. The development relies on three key ingredients: (1) we give a higher order abstract syntax (HOAS) encoding of lambda terms together with their operational semantics as intrinsically typed terms, thereby avoiding not only the need to deal with binders, renaming and substitutions, but keeping all typing invariants implicit; (2) we take advantage of Beluga’s support for representing open terms using built-in contexts and simultaneous substitutions: this allows us to directly state central definitions such as open simulation without resorting to the usual inductive closure operation and to encode very elegantly notoriously painful proofs such as the substitutivity of the Howe relation; (3) we exploit the possibility of reasoning by coinduction in Beluga’s reasoning logic. The end result is succinct and elegant, thanks to the high-level abstractions and primitives Beluga provides. We believe that this mechanization is a significant example that illustrates Beluga’s strength at mechanizing challenging (co)inductive proofs using HOAS encodings.

Dual systems for all: Higher-order, role-based relational reasoning as a uniquely derived feature of human cognition

2019 ◽  
Vol 42 ◽  
Author(s):  
Daniel J. Povinelli ◽  
Gabrielle C. Glorioso ◽  
Shannon L. Kuznar ◽  
Mateja Pavlic
Keyword(s):  
Temporal Reasoning ◽  
Higher Order ◽  
Important Case ◽  
Human Cognition ◽  
Relational Reasoning ◽  
Dual Systems ◽  
First Order ◽  
Reasoning System ◽  
Role Based

Abstract Hoerl and McCormack demonstrate that although animals possess a sophisticated temporal updating system, there is no evidence that they also possess a temporal reasoning system. This important case study is directly related to the broader claim that although animals are manifestly capable of first-order (perceptually-based) relational reasoning, they lack the capacity for higher-order, role-based relational reasoning. We argue this distinction applies to all domains of cognition.

scholarly journals A characterization of first-order topological properties of planar spatial data

2006 ◽  
Vol 53 (2) ◽  
pp. 273-305 ◽  
Author(s):  
Michael Benedikt ◽  
Bart Kuijpers ◽  
Christof Löding ◽  
Jan Van den Bussche ◽  
Thomas Wilke
Keyword(s):  
Spatial Data ◽  
Topological Properties ◽  
First Order

Adimen-SUMO

2012 ◽  
Vol 8 (4) ◽  
pp. 80-116 ◽  
Author(s):  
Javier Àlvez ◽  
Paqui Lucio ◽  
German Rigau
Keyword(s):  
Language Processing ◽  
Intelligent Systems ◽  
Knowledge Engineering ◽  
Formal Reasoning ◽  
Design Decisions ◽  
First Order ◽  
Theorem Provers ◽  
Inference Engines ◽  
Web Infrastructure ◽  
Important Design

In this paper, the authors present Adimen-SUMO, an operational ontology to be used by first-order theorem provers in intelligent systems that require sophisticated reasoning capabilities (e.g. Natural Language Processing, Knowledge Engineering, Semantic Web infrastructure, etc.). Adimen-SUMO has been obtained by automatically translating around 88% of the original axioms of SUMO (Suggested Upper Merged Ontology). Their main interest is to present in a practical way the advantages of using first-order theorem provers during the design and development of first-order ontologies. First-order theorem provers are applied as inference engines for reengineering a large and complex ontology in order to allow for formal reasoning. In particular, the authors’ study focuses on providing first-order reasoning support to SUMO. During the process, they detect, explain and repair several important design flaws and problems of the SUMO axiomatization. As a by-product, they also provide general design decisions and good practices for creating operational first-order ontologies of any kind.

scholarly journals The Hahn Sequence Space Defined by the Cesáro Mean

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Murat Kirişci
Keyword(s):  
Banach Space ◽  
Sequence Space ◽  
Matrix Transformations ◽  
Space Theory ◽  
Topological Properties ◽  
Cesàro Mean ◽  
First Order ◽  
Inclusion Relations ◽  
The Matrix ◽  
Cesaro Mean

The -space of all sequences is given as such that converges and is a null sequence which is called the Hahn sequence space and is denoted by . Hahn (1922) defined the space and gave some general properties. G. Goes and S. Goes (1970) studied the functional analytic properties of this space. The study of Hahn sequence space was initiated by Chandrasekhara Rao (1990) with certain specific purpose in the Banach space theory. In this paper, the matrix domain of the Hahn sequence space determined by the Cesáro mean first order, denoted by , is obtained, and some inclusion relations and some topological properties of this space are investigated. Also dual spaces of this space are computed and, matrix transformations are characterized.

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