Implementing Euler/Venn Reasoning Systems

This paper proposes an implementation of a Euler/Venn reasoning system using directed acyclic graphs and shows that this implementation is correct with respect to a modified Shin/Hammer mathematical model of Euler/Venn Reasoning. In proving its correctness it will also be shown that the proposed implementation preserves or inherits the soundness and completeness properties of the mathematical model of the Euler/Venn system. Introduction In the following study, we will look at an implementation of a Euler/Venn mechanical reasoning system and show that this implementation captures the essential properties 1 of a system similar to the Shin/Hammer mathematical Euler/Venn system as given in (Shin 1996; Hammer & Danner 1996; Hammer 1995). To do this, we will first look at a modified Shin/Hammer formal mathematical system that is associated with Euler/Venn diagrams. Then a second diagrammatic system representing Euler/Venn reasoning, one lending itself naturally to implementation, will be proposed using DAG’s ~, and the relations between this system and the formal mathematical system associated with Euler/Venn diagrams will be explored. It will be argued that this second representation is in fact true to the formal mathematical model of Euler/Venn reasoning and thereby preserves the properties of being sound and complete. Formal Specification of Mathematical System The mathematical formalization of the diagrammatic language of Euler/Venn, EVE, is defined to be the three-tuple (F, A, ~]), with F as the set of grammatical or well-formed formulae, A the deductive system, and ~ the semantics of the system. EVE is defined 1One system captures the essential properties of another system if there is a translation or mapping between them that preserves deductive and semantic relations. 2A DAG is a Directed Acyclic Graph. to be a traditional Venn system with Euler like extensions (see below.) While this treatment was inspired by and is quite similar to that found in (Hammer 1995), there are a number of important differences that should be noted, the most important of which include that the grammar presented here adds more well-formed diagrams, and that the system’s semantics have been changed to accommodate these new diagrams. By having a modified semantics and more well-formed diagrams, two new inference rules are introduced to maintain the completeness of the system. The Vocabulary 1. Rectangles Each rectangle denotes the domain of discourse to be represented by the diagram. 2. Closed Curves A countably infinite set C1, C2, C3 ... of closed curves. Each closed curve must not intersect itself. These curves denote sets. 3. Shading The shading of any region denotes that the set represented by that region is empty. 4. ® A countably infinite set ®1,®2,®3, ... of individual constants. 5. Lines Lines are used to connect individual constants ®n of the same n, in different regions to illustrate the uncertainty of which set contains that constant. F The Mathematical Grammar Formation rules Formation Rules for well-formed diagrams VEVF of EVF: 1. Any diagram containing only a Rectangle is a member of VEVF. 2. If V E VEVF then: (a) V with the addition of any closed curve C with unique label N completely within the rectangle of V so that the regions intersected by C are split into at most two new regions, is a member of vEv~ .3 3This grammatical stipulation while more general than that used in (Hammer 1995) is still not as general as one 69 From: AAAI Technical Report FS-97-03. Compilation copyright © 1997, AAAI (www.aaai.org). All rights reserved.