proof ontology in real N3 [was: KC...]

From: Dan Connolly <connolly@w3.org>
Date: Thu, 24 May 2001 15:16:39 -0500
Message-ID: <3B0D6C27.135D8AE0@w3.org>
To: Jim Hendler <jhendler@darpa.mil>, timbl@w3.org, www-archive@w3.org
```Jim,

Attached find your proof ontology cleaned up to
parse as N3. I'm sure there are still bugs in it,
but this is a start.

I'm copying www-archive, which makes the attatchment
available somewhere in
http://www.w3.org/Archives/Public/www-archive/

--
Dan Connolly, W3C http://www.w3.org/People/Connolly/
@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

------------------------------------------------------------------------

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. In very rough
N3/DAML (details at some later date - namespace issues ignored)
------------------------------------------------------------------------
""".

@prefix : <proof-ontology#>.
@prefix Proof: <proof-ontology#>.

:Proof :a rdfs:Class.

Proof:Body :a rdf:Property;
rdfs:domain Proof:Proof;
range [ :collectionOf Proof:Step]. #  [Dan - I will need your help for how to do collection, etc.]

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. #              Better to use URIs?

Proof:Assertion a rdfs:Class.

Proof:AsBody a rdf:Property;
rdfs:domain Proof:Assertion;
rdfs:range :STRING.     #          [being lazy here - shouldn't be a string, but member of some sort
#                                of enumerated class]

: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].

<> dc:description """
------------------------------------------------------------------------

Thus:

the example above becomes:
""".

@prefix Proof: <proof-ontology#>.
@prefix : <#>.

:ExampleProof a Proof:Proof;
Proof:Body
(PF1 PF2 PF3 PF4 PF5 PF6 PF7 PF8).

:Assumption1 a  Proof:Assertion;
:PfAsBody "A -> B" .                  #  daml:rule preferred over strings

:Assumption2 a  Proof:Assertion;
:PfAsBody "B -> -C" .

:Assumption3 a  Proof:Assertion;
:PfAsBody "C" .

@prefix PropLogic: <PropLogic#>.

:PF1 a Proof:Step;
Proof:AssumptionSet (PF1);
Proof:StepID "PF1";          #         set notation would be a good N3 addition
Proof:StepAssertion :Assumption1;
Proof:StepRule PropLogic:Assumption;  #    (assumptions defined in PropLogic)
Proof:StepDependsOn (:PF1).

:PF2 a Proof:Step;
:AssumptionSet (:PF2);
Proof:StepID "PF2";
Proof:StepAssertion :Assumption2;
Proof:StepRule PropLogic:Assumption;
Proof:StepDependsOn (:PF2).

:PF3 a Proof:Step;
Proof:AssumptionSet (:PF3);
Proof:StepID "PF3";
Proof:StepAssertion :Assumption3;
Proof:StepRule PropLogic:Assumption;
Proof:StepDependsOn (:PF3).

:Assumption4 a  Proof:Assertion;
Proof:AsBody "A" ;
rdfs:label "I added this one to negate later".

:PF4 a Proof:Step;
Proof:AssumptionSet (:PF4);
Proof:StepID "PF4";
Proof:StepAssertion :Assumption4;
Proof:StepRule PropLogic:Assumption;
Proof:StepDependsOn (:PF4).

:PF5 a :ProofStep;
Proof:AssumptionSet (:PF1 PF4);
Proof:StepID "PF5";
Proof:StepAssertion Conclusion1;
Proof:StepRule PropLogic:ModusPonens;  #     (ModusPonens a standard inference in Logic)
Proof:StepDependsOn (:PF1 :PF4).

:Conclusion1 a Proof:Assertion;
Proof:AsBody "B".

@prefix HendlerOldProof: <HendlerOldProof#>.
@prefix cwmLogic: <cwmLogic#>.

:PF6 a Proof:Step;
Proof:AssumptionSet (:PF1 :PF2 :PF4);
Proof:StepID "PF6";
Proof:StepAssertion HendlerOldProof:Conclusion29; # Had a "-C" string hanging around
Proof:StepRule cwmLogic:ModusPonens; #    Rules by URI is web power at its best
Proof:StepDependsOn (:PF2 :PF5).

@prefix DanCwm20010604: <DanCwm20010604#>.
@prefix LogicAllowedByAll: <LogicAllowedByAll#>.

:PF7 a Proof:Step;
Proof:AssumptionSet (:PF1 :PF2 :PF3 :PF4);
Proof:StepID "PF7";
Proof:StepRule LogicAllowedByAll:AndIntroduction;
Proof:StepDependsOn (:PF3 :PF6).

@prefix LogicAllowedBySome: <LogicAllowedBySome#>.

:PF8 a Proof:Step;
Proof:AssumptionSet (:PF1 :PF2 :PF3);
Proof:StepID "PF8";
Proof:StepAssertion :ProvenAssertion;
Proof:StepDependsOn (:PF7);
rdfs:comment "QED.".

:ProvenAssertion a Proof:Assertion;
Proof:AsBody "-A" .

<> dc:description """
Lots of stuff left to do - but basic idea works.

------------------------------------------------------------------------

features: I think we don't even need ordering - that is implied solely by
the StepDependsOn relations. Proof Checker must agree to all rules specified
as StepRules, check that the proof is sound (i.e. all StepIDs defined, all
inferences legitimized), and it can agree w/conclusion. No need for HOL,
works for any FOL system, could be made to work for any logic system
definable by rules in step (includes modal and conditional). Predicate
Logics include use of "existential introduction, universal elimination" and
the like (see basic logic book).
""".
```
Received on Thursday, 24 May 2001 16:16:45 UTC

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