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Semantics of a Propositional Network

Explore the foundations of propositional network semantics, including network structure, basic notions, syntactic hierarchy, and semantic interpretations. Learn about contexts, ontologies, inference, and examples within the framework. Dive deep into the controlled slot-based inference, case frames, and proposition definitions. Understand how entities, relations, and propositions are represented and manipulated in a directed acyclic graph. Discover the application of semantic rules, logical connectives, and relations in building, inferring, and negating propositions.

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Semantics of a Propositional Network

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  1. Semantics of aPropositional Network Stuart C. Shapiro Department of Computer Science & Engineering Center for MultiSource Information Fusion Center for Cognitive Science University at Buffalo, The State University of New York shapiro@cse.buffalo.edu

  2. Setting • Semantics of SNePS 3 • Builds on previous versions of SNePS and predecessor systems • Ideas evolved from pre-1968 to current. S. C. Shapiro

  3. Kind of Graph • Directed Acyclic Graph • With Labeled Edges (arcs) • Cf. Relational graphs • No parallel arcs with same label • Allowed: parallel arcs with different labels • Allowed: multiple outgoing arcs • With same or different labels • To multiple nodes • Each node identified by a URI S. C. Shapiro

  4. Basic Notions 1 • Network representsconceptualized mental entities of a believing and acting agent. • Entities include • Individuals • Classes • Properties • Relations • Propositions • Acts • etc… S. C. Shapiro

  5. Basic Notions 2 • 1-1 relation between entities and nodes. • Every node denotes a mental entity. • Arbitrary and Indefinite Terms replace variables. • Every mental entity denoted by one node. • No two nodes with same arcs to same nodes. • Some, not all, proposition-denoting nodes are asserted. S. C. Shapiro

  6. Contexts • Delimit sub-graph of entire network. • Contain and distinguish hypotheses and derived propositions. • Organized as a rooted DAG for inheritance of asserted propositions. S. C. Shapiro

  7. Top-Level Domain Ontology • Entity • Proposition • Act • Policy • Thing • Use many-sorted logic. S. C. Shapiro

  8. Syntactic Hierarchy • Node • Atomic • Base (Individual constants) • Variable • Arbitrary • Indefinite • Molecular • Generic • … S. C. Shapiro

  9. Semantics ofIndividual Constants • Base node, ni • No outgoing arcs • Denotes some entity ei • niis created because • When ei is conceived ofno other node “obviously” denotes it. S. C. Shapiro

  10. Frame View of Molecular Nodes Multiple arcs with same label forms a set. mi:(R1 (n1 … n11k) … Rj (nj1 … njjk)) mi … R1 R1 Rj Rj nj1 n1 n11k njjk … … S. C. Shapiro

  11. Set/Frame View MotivatesSlot-Based Inference E.g.From(member (Fido Lassie Rover) class (dog pet))To(member (Fido Lassie) class dog) S. C. Shapiro

  12. Slot-Based Inference & Negation From(not (member (Fido Lassie Rover) class (dog pet)))To(not (member (Fido Lassie) class dog))OK. From(not (siblings (Betty John Mary Tom)))To(not (siblings (Betty Tom)))Maybe not. S. C. Shapiro

  13. Relation DefinitionControls Slot-Based Inference • Name • Type (of node(s) pointed to) • Docstring • Positive • Adjust (expand, reduce, or none) • Min • Max • Negative • Adjust (expand, reduce, or none) • Min • Max • Path S. C. Shapiro

  14. Example: member • Name: member • Type: entity • Docstring: “Points to members of some category.” • Positive • Adjust: reduce • Min: 1 • Max: nil • Negative • Adjust: reduce • Min: 1 • Max: nil (member (Fido Lassie Rover) class (dog pet))├ (member (Fido Lassie) class dog) (not (member (Fido Lassie Rover) class (dog pet)))├(not (member (Fido Lassie) class dog)) S. C. Shapiro

  15. Example: siblings • Name: siblings • Type: person • Docstring: “Points to group of people.” • Positive • Adjust: reduce • Min: 2 • Max: nil • Negative • Adjust: expand • Min: 2 • Max: nil (siblings (Betty John Mary Tom)) ├(siblings (Betty Tom)) (not (siblings (Betty John Mary))) ├(not (siblings (Betty John Mary Tom)) S. C. Shapiro

  16. Case Frames • Function “symbols” of the SNePS logic. • Denote nonconceptualized functions in the domain. S. C. Shapiro

  17. Case Frame Definition • Type (of created node) • Docstring • KIF-mapping • Relations S. C. Shapiro

  18. Example Case Frame • Type: proposition • Docstring:“the proposition that [member] is a [class]” • KIF-mapping: (‘Inst member class) • Relations: (member class) S. C. Shapiro

  19. Example Proposition M1!:(member (Fido Lassie Rover) class (dog pet)) (Inst (setof Fido Lassie Rover) (setof dog pet)) The proposition that Rover, Lassie, and Fido is a pet and dog. S. C. Shapiro

  20. Arbitrary Terms mi propositions (any x restrict p1 … pn) No two that are just renamings. any restrict restrict pn p1 nk nj x … … [Shapiro KR’04] S. C. Shapiro

  21. Example: Dogs are furry. M3!:(object (any x restrict (member x class dog)) property furry) S. C. Shapiro

  22. Indefinite Terms, Example There’s some citizen of every country whom Mike believes is a spy. M9!:(agent Mike belief (member (some x ((any y (member y class country))) (relation citizen subject x object y)) class spy)) S. C. Shapiro

  23. Closed Indefinite Terms, Example Mike believes that some citizen of every country is a spy. M15!:(agent Mike belief (close (member (some x ((any y (build member y class country))) (relation citizen subject x object y)) class spy))) S. C. Shapiro

  24. Other Topics • Logical Connectives • (andor (i j) P1 … Pn) • (thresh (i j) P1 … Pn) • (i=> (setof A1 … An) (setof C1 … Cn)) • Supports for ATMS S. C. Shapiro

  25. Summary • Nodes denote mental entities. • Individual constants. • Arbitrary and Indefinite terms. • Functional terms, including: • Atomic Propositions; • Nonatomic Propositions. • Some propositions are asserted. • Labeled arcs indicate argument position. • Case Frames denote nonconceptualized functions. S. C. Shapiro

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