Structural Templates In Type Theory

1. Structural Templates In Type Theory Henson Graves June, 2012

2. Air System Air Vehicle Pilot Pilot Air Frame Power Control Avionics Models Are Used To Represent Manufactured Products, Molecular Structure, & Biomedical Systems Recent change in perspective Models are becoming the authoritative source of information about an application Models get validated using empirical data then become ground truth Aircraft Product Model • 22,000 unique part numbers and over 300,000 part counts

3. Results Embedding a Model As Axiom Set Enable Engineering Problems To Become Logic Problems Logic Language • Results • Class model fragment of UML/SysML to DL (Berardi, Calvanese, De Giacomoa, 2005) • Static SysML to type theory (Graves, Bijan, 2012) • Enables • Automated reasoning to answer engineering questions (Graves, 2012) Modeling Language Model Axioms Embedding Inference Engine

4. Many Engineering Questions Can Be Answered By Translation Into Logic Questions Is a design specification is consistent Does a design satisfies its requirements Is the design implementable For structures that conform to a template do symptoms imply disease conditions (fault)

5. A Diagram Is Used To Describe And Reason About Structures Which Conform To Pattern Detailed designs, molecular structure, family relationships, biomedicine • Detailed designs, molecular structure, family relationships, biomedicine • Embedding results do not sufficiently constrain models to conform to diagram (Motik, et al., 2008) The known embeddings do not constrain the models satisfactorily to achieve template property

6. A Structure Diagram Can Be Used To Generate a Template Axiom Set Structure Diagram Template • Structure diagram included in signature of template axiom set • Templates have a minimal model • Structures that satisfy a template are isomorphic • Additional axioms can be added retaining decidability of consistency • Diagrams use functional relations with domain and range • Two kinds of relations, parts and connections resulting in different kinds of axioms • parts provide a tree like structure • classes & properties have orthogonal properties

7. Type Theory Is Good Choice For Embedding Structural Descriptions And Producing Templates • Diagram embedded as part of signature of multi-sorted logic • Nodes map to disjoint subtypes of type Thing, arrows map to subtypes of Product type • Template axioms uses Description Logic constructions & equalizer type construction • Decidability uses connection between internal and external logic to show axioms equivalent to monadic Ackermann class

8. Type Theory Is Good Choice As Embedding Logic For Modeling Languages SysML Model of Heart Full type theory expressiveness needed for applications Using graphical syntax of modeling languages such as SysML/UML greatly simplifies developing complex models Enables full integration of formal methods with engineering

9. Current Results Are Only The Tip of The Iceberg Of What Is Needed From This To This Getting results is a matter of careful study of applications