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Proof System

Proof System. HY-566. Proof layer. N ext layer of SW is logic and proof layers. allow the user to state any logical principles, computer can to infer new knowledge by applying these principles on the existing data.

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Proof System

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  1. Proof System HY-566

  2. Proof layer • Next layer of SW is logic and proof layers. • allow the user to state any logical principles, • computer can to infer new knowledge by applying these principles on theexisting data. • This project implements a defeasible reasoning system which presents explanations to users for the answersto their queries.

  3. Defeasible Logic • A defeasible theory is a triple (F, R, >) • F is a set of literals (called facts) • R a finite set of rules • > a superiority relation on R • Two kind of rules • Strict rules (A→P) can’t be defeated • Defeasible rules (A=>P) can be defeated

  4. Defeasible Logic Metaprogram • Simulates the proof theory ofdefeasible logic contains clauses : • denite provability • defeasible provability • When rule is blocked/unblocked • When rule is defeated/undefeateds

  5. Explanation Example • An e-shop tell to Bob’s that he owns 30$ • Explanation (facts and rules): • purchase(Bob,DVD) - fact • price(DVD,30) - fact • delivered(DVD,Bob) - fact • purchase(Bob,DVD), price(DVD,30), delivered(DVD,Bob) → owes(Bob,30) - rule

  6. Extension of RULML • RuleML is an XML based language that supports rule representation for theSemantic Web. • A new XMLschema, extension of RuleML, is proposed for explanation representation in defeasible logic

  7. Ext. RULML - Atom, Fact • Atom: operator, var or const, optionaly NOT <Atom> <Not> <Op> rich </Op> <Ind> Bob</Ind> </Not> </Atom> • AFact is consisted by an Atom that comprise a certainknowledge

  8. Ext. RULML - Rules • Strict -Defeasible Rules. • Head:is an Atom • Body: number of Atoms <Defeasible_rule Label="r1"> <Head> <Atom> <Op> rich </Op> <Ind> Bob </Ind> </Atom> </Head> <Body> <Atom> <Op> wins_lotto </Op> <Ind> Bob </Ind> </Atom> </Body> </Defeasible_rule>

  9. Ext. RULML - Explanations • Definitely Provable Explanations • Denote the Atom • Definite Proof • Definite Proof • Fact for that Atom • Strict Rule with Head the Atom and Body (multiple) atoms that must be proved definitely.

  10. <Definitely_provable> <Atom> <Op> rich </Op> <Ind> Bob </Ind> </Atom> <Definite_Proof> <Strict_rule Label="r1"> <Head> <Atom> <Op> rich </Op> <Ind> Bob </Ind> </Atom> </Head> <Body> <Atom> <Op> wins_lotto </Op> <Ind> Bob </Ind> </Atom> </Body> </Strict_rule> <Definitely_provable> <Definite_Proof> <Fact> <Atom> <Op> wins_lotto </Op> <Ind> Bob </Ind> </Atom> </Fact> </Definite_Proof> </Definitely_provable> </Definite_Proof> </Definitely_provable> Explanations Example

  11. Proof tree construction (1/3) • The foundation of the proof system • Prolog metaprogram implementsrules of Defeasible Logic • The trace of theXSB implementation of prolog • XSB: logic programming engine used torun the metaprogram. • To communicate with the XSB the invocation of the XSB executable was used (Javas exec method) • Send commands to the XSB interpreter • Receive the output that was produced as an effect. • Load the metaprogram and the defeasible theory

  12. Proof tree construction (2/3) • Load the metaprogram and the defeasible theory • At the evaluation of a query XSB will print a message each time a predicate is: • Initially entered (Call) • Successfully returned from (Exit), • Failed back into (Redo), and • Completely failed out of (Fail).

  13. Proof tree construction (3/3) • Atree whose nodes are traced predicates is constructedby the Java XSB invoker when trace is parsed. • Each node has information • A string for the predicates name • The predicates arguments • Whether it was found to be true (Exit) or false (Fail) • Whether it was failed back into (Redo) • Booleanattribute tells if predicate is negated.

  14. Why the tree needs pruning? • XSB trace contains data not needed for proof • A metaprogram to translate the DL into logic programming is used. Additional clauses are needed which add informationto trace • Prolog shows successful and unsuccessful paths • The tree produced by theXSB trace is built according to the metaprogram structure but the final tree needs to be compliant with the XML schema

  15. Pruning Rules • Heuristic rules are used in order to prune the proof tree • According to the truth value and the type of the root nodewe may want to maintain • only successful paths • only failed paths • combinations ofthem.

  16. Pruning Motivation Example1 • Suppose we have the following defeasibly theorytranslated in logic programmingas: • fact(a). • fact(e). • defeasible(r1,b,a). • defeasible(r2,b,e). • defeasible(r3,~(b),d).

  17. Example1: Defeasible provability of b Pruned

  18. Example1: Defeasible provability of b • We are interested in successful (True) paths • The pruning algorithmremoves • the subtree with the false goal to prove that b is denitely provable • the false predicate to find a strict supportive rule for b • the metaprogramadditional negation clause.

  19. Pruning Motivation Example2 • Suppose we have the following defeasibly theorytranslated in logic programmingas: • fact(a). • defeasible(r1,b,a). • defeasible(r2,~(b),a).

  20. Example2: Defeasible provability of b We are interested in unsuccessful paths and the pruning algorithmkeeps the initial proof tree.

  21. Pruning resume • The proof tree after using the heuristic techniques is similar to an explanation derivedby the use of pure DL • Drawback: heuristics arefully dependent on the metaprogram. • Changes at metaprogram => changes at pruning implementation.

  22. Pruning Example The marked rule is pruned

  23. Pruning Example 2 The marked rule is pruned

  24. Agent Interface to the Proof System • The system makes use of two kinds of agents • 'Agent' which issues queries • 'Main Agent' which is responsible to answer the queries. • Both agents are based onJADE (Java Agent DEvelopment Framework) • a software framework to develop agent-based applications

  25. XML Writter 8) Pruned result 9) XML proof 3) Predicate 2) Predicate 1)Question XSB Invoker Main Agent Agent 4) result trace 5) result tree 10) answer or proof 6) result tree 7) Pruned result Pruner Process to answer a query (1/3)

  26. Process to answer a query (2/3) • An agent issues a query to the Main Agent. • predicate::(proof|answer) • Main Agent sends Predicate to the Invoker • Invoker is responsible to communicate with XSB • Invoker executes the Predicate. • XSB returns the full result trace. • Invoker returns result tree to Main Agent. • Main Agent sends result tree to the Pruner

  27. Process to answer a query (3/3) • Pruner returns pruned result to MainAgent. • Main Agent sends the pruned result to the XML writer (only if proof requested) • XML writer returns the XML Proof. • Main Agent returns Answer or Proof • ANSWER(true | false) • PROOF:(proof string) • ERROR:(error message)

  28. Visual Agent

  29. Command Line Agent • Readsin a random way the questions from a configuration file • Sends the question to the Main Agent with the order read • The format of the questions is of the form • predicate::(proof|answer) • The answers and proofs are stored at a files

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