**From:** Jim Hendler (*jhendler@darpa.mil*)

**Date:** 07/13/01

**Next message:**Benjamin Grosof: "Re: DAML RULES -- logic programs, Webizing, RuleML new release (resend)"**Previous message:**Ian Horrocks: "Re: equivalentTo / sameClassAs / sameIndividualAs puzzle"**In reply to:**Frank van Harmelen: "Re: Joint Committee telecon tomorrow 8 May"**Next in thread:**Dieter Fensel: "Re: Joint Committee telecon tomorrow 8 May"**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ] [ attachment ]**Mail actions:**[ respond to this message ] [ mail a new topic ]

Stefan asked us to provide examples of what we would like to do with a rule language. I've put together the following example -- first, I define an ontology that is useful for the expression of proofs (based on traditional notation schemes from philosophical logic). I then have rules expressed in various documents, and want a system that can put them together into a proof document (i.e. an instantiated proof) that could be checked by some other system. Of course, I also want this embedded on the web (i.e. URIs used where possible, etc.) Richard Fikes is also working on a proof ontology, so I've cc'ed him in this message. Some caveats The current example is almost exclusively in RDF(S) - my plan had been to do it in RDFS and then in DAML to show the added power of DAML - I still hope to do that, but just haven't found time. I have purposefully left all the rules as simply strings, which express propositional logic (variables and the like should be easily recognized). I do NOT propose that this is the right way to do it -- rather, I did this to make it easy to see what sort of rules I would like, and to make it so I could plug in various rule contenders to see if they would do what I need. Here we go (following is in RDF/N3 - I merged a set of files quickly, and so I don't promise there are no syntax violations - but this should be enough to convey the idea). --------- @prefix dc: <http://purl.org/dc/elements/1.1/>. @prefix rdfs: <http://www.w3.org/2000/01/rdf-schema#> . @prefix rdf: <http://www.w3.org/1999/02/22-rdf-syntax#>. @prefix daml: <http://www.daml.org/2001/03/daml+oil#> . <> dc:description """ Idea: There is a standard way that students are taught for the exchange of FOL proofs in philosophical logic courses (like the one I took in 1975 - ouch!) One generates a proof by showing that a particular set of assumptions can lead to a particular conclusion via a set of proof steps permitted by the axioms of FOL. (Propositional cases are easier to show than Predicate, but same thing works for full FOL). The idea is simple - each step of a proof is made up of 1. The set of assumptions on which a step depends 2. A step number (or arbitrary label) 3. A new assertion 4. The logical axiom on which it depends 5. The previous step numbers (labels) that were used by the axiom 6. Comments (real logicians don't use comments) ------------------------------------------------------------------------ Example: Given: A -> B B -> -C C Prove: -A PROOF: 1 (1) A -> B A 1 2 (2) B -> -C A 2 3 (3) C A 3 4 (4) A A 4 ; an assumption is added to make the proof hold 1,4 (5) B MP 1,4 ; modus ponens of 1,4 1,2,4 (6) -C MP 2,5 ; note assumption 4 included since 5 was needed 1,2,3,4 (7) C & -C &-I 3,6 ; and-introduction brings in 3 1,2,3 (8) -A RAA 7 ; we can negate any assumption (we chose 4) by reducio Thus we see -A dependent only on 1,2,3 and are done. ------------------------------------------------------------------------ Bringing this to the web The idea is simply to create an ontology which has proof as a class with a set of statements, each statement containing the stuff above. ------------------------------------------------------------------------ """. @prefix : <proof-ontology#>. @prefix Proof: <proof-ontology#>. :Proof :a rdfs:Class. Proof:Body :a rdf:Property; rdfs:domain Proof:Proof; range [ :collectionOf Proof:Step]. # collectionOf shorthand for Dan's stuff Proof:Step a rdfs:Class. :AssumptionSet a rdf:Property; rdfs:domain Proof:Step; rdfs:range [ :collectionOf :StepID]. :StepID a daml:UniqueProperty; rdfs:domain Proof:Step; rdfs:range :STRING. # how would one do unique property in RDFS? Proof:Assertion a rdfs:Class. Proof:AsBody a rdf:Property; rdfs:domain Proof:Assertion; rdfs:range :STRING. # :StepAssertion a rdf:Property; rdfs:domain Proof:Step; rdfs:range Proof:Assertion. :RuleName a rdfs:Class. # IMPORTANT: RuleName can be any URI! :StepRule :a rdf:Property; rdfs:domain :ProofStep; rdfs:range :RuleName. :StepDependsOn a rdfs:Property; rdfs:domain :ProofStep; rdfs:range [ :collectionOf :StepID]. :ResourceDescription a rdfs:class; :TrustedResource a rdfs:class. :TrustedLogic a rdfs:class. :CompoundAssertion a rdfs:Class. :AssertStep a rdf:property; rdfs:range CompoundAssertion; rdfs:Domain [ :CollectionOf :Assertion]. <> dc:description """ Below is an example in which a proof is created from distributed sites. Note that the actual rules are simply in strings - this is not to imply this is how to do it, but as examples of what the rule language should allow. At the Site Granting Access (SITE1): we say a resource is accessible by X if X is MyMember, or one of my members says X is authorized. """. @prefix Proof: <proof-ontology#>. @prefix : <#>. :Assert1 a Proof:Assertion; Proof:AsBody "MyMember(URI1)". :Rule1 a Proof:Assertion; Proof:AsBody "MyMember(x) AND AuthBy (user,x) -> Accessible(<>,user)" <URI1> a Proof:TrustedResource. <PropLogic> a Proof:TrustedLogic. <> dc:description """ At URI1 : My employees are authorized and so are recognized substitutes, John is an employee. """. @prefix Proof: <proof-ontology#>. @prefix : <#>. :Assert1 a Proof:Assertion; Proof:AsBody "Employee(JOHN)". :Rule1 a Proof:Assertion; Proof:AsBody "Employee(x) -> AuthBy(x,URI1)" :Rule2 a Proof:Assertion; Proof:AsBody "Substitute(x,y) AND AuthBy(x,UTI1) -> AuthBy(y,URI1)" <URI2> a Proof:TrustedResource. <> dc:description """ At URI2: Mary is allowed to substitute for John """. :Assertion1 a Proof:Assertion; Proof:AsBody "Substitute(JOHN,MARY)"; <> dc:description """ At PropLogic: """. @prefix Proof: <proof-ontology#>. @prefix : <#>. :ModusPonens a Proof:RuleName. :AndIntro a Proof:RuleName. :RuleCitation a Proof:RuleName. :AssumptionCitation a Proof:Rulename. <> dc:description """ ------------------------------------------------------------------------ The following proof could then be validated by a proof checker at SITE1: """. @prefix Proof: <proof-ontology#>. @prefix : <#>. @prefix PropLogic: <PropLogic#>. @prefix U1: <URI1#>. @prefix U2: <URI2#>. :ExampleProof a Proof:Proof; Proof:Body (PF1 PF2 PF3 PF4 PF5 PF6 PF7 PF8 PF9 PF10 PF11). :PF1 a Proof:Step; Proof:AssumptionSet (:PF1); Proof:StepID "PF1"; Proof:StepAssertion SITE1:Assert1; Proof:StepRule PropLogic:AssumptionCitation; Proof:StepDependsOn (:PF1). :PF2 a Proof:Step; :AssumptionSet (:PF2); Proof:StepID "PF2"; Proof:StepAssertion U1:Assert1; Proof:StepRule PropLogic:AssumptionCitation; Proof:StepDependsOn (:PF2). :PF3 a Proof:Step; Proof:AssumptionSet (:PF3); Proof:StepID "PF3"; Proof:StepAssertion U2:Assertion1; Proof:StepRule PropLogic:AssertionCItation; Proof:StepDependsOn (:PF3). :PF4 a Proof:Step; Proof:AssumptionSet (:PF4); Proof:StepID "PF4"; Proof:StepAssertion U1:Rule1; Proof:StepRule PropLogic:RuleCItation; Proof:StepDependsOn (:PF4). :Conclusion1 a Proof:Assertion; Proof:AsBody "Auth(John,URI1)"; :PF5 a :ProofStep; Proof:AssumptionSet (:PF2 PF4); Proof:StepID "PF5"; Proof:StepAssertion :Conclusion1; Proof:StepRule PropLogic:ModusPonens; Proof:StepDependsOn (:PF2 :PF4). :Compound1 a Proof:AndAssertion; Proof:AssertSet (U2:assertion1 :Conclusion1). :PF6 a Proof:Step; Proof:AssumptionSet (:PF2 :PF3 :PF4); Proof:StepID "PF6"; Proof:StepAssertion :Compound1; Proof:StepRule PropLogic:AndIntro; # Rules by URI is web power at its best Proof:StepDependsOn (:PF3 :PF5). :PF7 a Proof:Step; Proof:AssumptionSet (PF7) ; Proof:StepID "PF7"; Proof:StepAssertion U1:Rule2; Proof:StepRule PropLogic:RuleCitation; Proof:StepDependsOn (:PF7). :Conclusion2 a Proof:Assertion; Proof:AsBody "AuthBy(Mary,URI1)". :PF8 a Proof:Step; Proof:AssumptionSet (:PF2 :PF3 :PF4 :PF7); Proof:StepID "PF8"; Proof:StepAssertion :Conclusion2; Proof:StepRule PropLogicModusPonens; Proof:StepDependsOn (:PF6 :PF7). :Compound2 a Proof:AndAssertion; Proof:AssertStep (:Conclusion2 SITE1:Assert1). :PF9 a Proof:Step; Proof:AssumptionSet (:PF1 :PF2 :PF4 :PF6 :PF7) ; Proof:StepID "PF9"; Proof:StepAssertion :Compound2; Proof:StepRule PropLogic:AndIntro; Proof:StepDependsOn (:PF1 :PF8). :PF10 a Proof:Step; Proof:AssumptionSet (:PF10) ; Proof:StepID "PF10"; Proof:StepAssertion Site1:Rule1 ; Proof:StepRule PropLogic:RuleCitation ; Proof:StepDependsOn (:PF10). :PF11 a Proof:Step; Proof:AssumptionSet (:PF1 :PF2 :PF3 :PF4 :PF7 :PF10); Proof:StepID "PF11"; Proof:StepAssertion :ProvenAssertion; Proof:StepRule PropLogic:ModusPonens; Proof:StepDependsOn (:PF9 :PF10). :ProvenAssertion a Proof:Assertion; Proof:AsBody "Accessible(<>, Mary)". <> dc:description """ QED. Lots of detail stuff left to do - but it actually works! """. -- Prof. James Hendler Program Manager DARPA/ISO 703-696-2238 (phone) 3701 N. Fairfax Dr. 703-696-2201 (Fax) Arlington, VA 22203 jhendler@darpa.mil

**Next message:**Benjamin Grosof: "Re: DAML RULES -- logic programs, Webizing, RuleML new release (resend)"**Previous message:**Ian Horrocks: "Re: equivalentTo / sameClassAs / sameIndividualAs puzzle"**In reply to:**Frank van Harmelen: "Re: Joint Committee telecon tomorrow 8 May"**Next in thread:**Dieter Fensel: "Re: Joint Committee telecon tomorrow 8 May"**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ] [ attachment ]**Mail actions:**[ respond to this message ] [ mail a new topic ]

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