November 11, 2004 July 20, 2004 Common Logic (CL) Development Current project Harry Delugach Univ. of Alabama in Huntsville

November 11, 2004 July 20, 2004 OverviewOverview First order logic language for knowledge interchange Provides a core semantic framework for logic Provides the basis for a set of syntactic forms (dialects) all sharing a common semantics

November 11, 2004 July 20, 2004 First Order Semantics Entities - things, states, attributes –Harry, idleness, color, etc. Relations - between entities, attributes –Marriage, eye-color, etc. Quantification - single or multiple instance –Definition, uniqueness, etc. Negation - explicit falsehood –Harry is not President of the United States Iteration - over elements a set –Age of each member of a population NOT INCLUDED: –actors, ontologies, knowledge bases, …

November 11, 2004 July 20, 2004 Goal of Common Logic Two agents, A and B, each have a first-order formalization of some knowledge A and B wish to communicate their knowledge to each other so as to draw some conclusions. –Any inferences which B draws from A's input should also be derivable by A using basic logical principles, and vice versa The goal of Common Logic is to provide a logical framework which can support this kind of use and communication without requiring complex negotiations between the agents.

November 11, 2004 July 20, 2004 Issue - Syntax A and B may have used different surface syntactic forms to express their knowledge. –A well-known problem –Usually solved by defining a standard syntax into which others can be translated, such as KIF –CL provides a common 'interlingua' syntax XCL into which the others can be translated. –XCL uses XML concepts and design principles Provides a clean separation between the description of logical form and the surface syntactic form appropriate to a particular usage. Allows for linking of CL text across documents and conveying CL written in non-XCL syntaxes between applications using XML protocols.

November 11, 2004 July 20, 2004 Issue - Different Axiomatic Styles A and B may have made divergent assumptions about the logical signatures of their formalizations. –A uses relation name where B uses function –A and B use same relation with different argument orderings or different numbers of arguments. –A particular concept, such as marriage, might be represented by A as an individual, but by B as a relation. Can be solved by mappings between the logical forms of such divergent choices –CL removes conventional limitations on first-order signatures For example, a name in CL may serve both as an individual name and as a relation name.

November 11, 2004 July 20, 2004 Example: Rule-based vs. logic Rule-based system: CL: (implies (equal date “Jan 1”) pay_vacation_rate ) Logic system CL: not ( (equal date “Jan 1”) and not ( pay_vacation_rate ) ) if date =“Jan 1” then pay_vacation_rate not ( date = “Jan 1” and not pay_vacation_rate )

November 11, 2004 July 20, 2004 Issue - Multiple Domains A and B may have been writing with different intended universes of discourse in mind –Assertions in a domain might be interpreted to be talking about things that they have not even conceived of –E.g., taxonomic classifications of animals “complement of the set of mammals” may be taken to include fruit, sodium molecules, styles of avant-garde paintings or the names of fictional characters in movies. CL has a 'top-level' syntactic form called a module which automatically gives a name to the universe of discourse of a named ontology –Automatically inserts “namespace” on any contained quantifiers when information is combined.

November 11, 2004 July 20, 2004 Issue - Shareable Semantics Any meaningful exchange of utterances depends upon the prior existence of an agreed set of semantic and syntactic rules -- ISO TR 9007:1987 (“Helsinki principles”) The recipients of the utterances must use only these rules to interpret the received utterances, if it is to mean the same as that which was meant by the utterer -- ISO TR 9007:1987 (“Helsinki principles”) Interpretation of the symbols forms an ontology -- a “namespace” which must be internally self-consistent.

November 11, 2004 July 20, 2004 CL Semantics Based on model theory Assumes a universe of individuals U I and a way of associating individuals with symbols (e.g., “Jack”) An interpretation of vocabulary (V O,V R,V F ) is defined by a set and three mappings: –A nonempty set U I called the universe; –A mapping int I from V O to U I maps individuals to symbols –A mapping rel I from V R to the set Rel I of relational extensions over U I –A mapping fun I from V F to the set Fun I of functional extensions over U I. Once given U I and mappings, every system will have the same interpretation of CL sentences

November 11, 2004 July 20, 2004 Surface Forms/Syntax (  )(Boy(x)  (  )(Girl(y) & Kissed(x,y))) *x] [If: (Boy ?x) [Then: [*y] (Girl ?y) (Kissed ?x ?y) ]] Girl Boy kiss not Boy not Girl

November 11, 2004 July 20, 2004 Mapping to Common Syntax (forall (?x)(implies (and (P ?x) (R ?x)) (PR ?x)))) *x] [If: [P(?x) R(?x)] [Then: PR(?x)]] ( ∀ x)(P(x)&R(x) → PR(x)) (x)not(P(x) R(x) not PR(x))

November 11, 2004 July 20, 2004 Different Axiomatic Styles All can be represented by CL core syntax –(married Jack Jill) –(married (roleset: (husband Jack) (wife Jill))) –(exists (x)(and (married x) (husband Jack x) (wife Jill x))) –(= (when (married Jack Jill)) (hour 3 (pm (thursday (week 12 (year 1997))))) ) –(= (wife (married 32456)) Jill) –(ConjugalStatus married Jack) –((ConjugalStatus Jack) Jill)

November 11, 2004 July 20, 2004 Example: Interchange  CGIF concrete syntax: [Jack: *a] [Jill: *b] (married ?a ?b)  Map  to CL abstract syntax: (married Jack Jill)  Map  to KIF concrete syntax (married (Jack) (Jill) ) Jack Jill married

November 11, 2004 July 20, 2004 Semantic Consistency System A: (married Jack Jill) System B: (married (roleset:(husband Jack)(wife Jill))) How does System B “understand” System A? ( forall (x y) (implies (married x y) (married (roleset:(husband x) (wife y))) ) )

November 11, 2004 July 20, 2004 CL Overview: structure body::header name = Header specifies lexical categories, special assumptions, name classes Body contains asserted sentences (‘axioms’) Both expressed as sets of SCL sentences Ontology name is URI

November 11, 2004 July 20, 2004 (( (?x)not(scl:Rel(?x) scl:Ind(?x)) ;;FOL syntax scl:Rel(Married Kissed Boy Girl) ;;relation names scl:Ind(Jack Jill) ;; individual names scl:Arity(2 Kissed) ;; Kissed can take 2 arguments 2 =scl:Arity(Married);;Married must take 2 arguments 1 =scl:Arity(Boy) ;;Boy must take 1 argument )::( Boy(Jack);;beginning of body (?x)not(Boy(?x) (?y)not (Kissed(?x ?y) Girl(?y) ) ;;every boy kissed a girl Married(Jack Jill) )) CL Overview: headers Header defines syntactic conditions on sentences in body Body contains asserted sentences expressing ontology content

November 11, 2004 July 20, 2004 CL semantics One universe of discourse; all quantifiers range over it. Relations may be in the universe, or may not. Functions are treated as a special class of relations. To allow a fully general syntax, we distinguish relations from their extensions. This also allows intensional theories of relations, providing extra flexibility. (Similar to treatment in RDF.)

November 11, 2004 July 20, 2004 Conformance Issues What if a system/notation is a superset of CL? –E.g., Conceptual Graphs have procedural nodes called actors that serve as functional relations Only that subset of the system/notation that is first-order will be preserved via interchange –E.g., actor will be exchanged as simple relation

November 11, 2004 July 20, 2004 ISO WD Main body Scope, normative references, terms, symbols Common Logic Core –Abstract syntax –Abstract Semantics Conformance Four specific surface syntaxes are conformant KIF, CGIF, XCL, CLCE Provide a mapping from your language to one of those Prove that semantics of your language are preserved for every mapping into CL abstract syntax

November 11, 2004 July 20, 2004 ISO WD Annexes KIF –Normative - Concrete syntax - KIF – EBNF grammar - –Informative - Show KIF to CL abstract syntax CGIF –Normative - CGIF – EBNF grammar –Informative - Show CGIF to CL abstract syntax XCL –Normative - XML-based markup – EBNF grammar –Informative - Show XCL to CL abstract syntax Common Logic Controlled English (CLCE) –Normative - “English-like” grammar –Informative - Show CLCE to CL abstract syntax

November 11, 2004 July 20, 2004 ISO WD Annexes (cont.) Informative - Relationship to other standards and practices –Prolog, Z, OCL (non-ISO-std), OWL (non-ISO- std) –Distinguish CL from Horn clause, description logics, others Especially: meaning of negation, reasoning over sets – see well founded semantics