Proof System HY-566

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 the existing data. This project implements a defeasible reasoning system which presents explanations to users for the answers to their queries.

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

Defeasible Logic Metaprogram Simulates the proof theory of defeasible logic contains clauses : – denite provability – defeasible provability – When rule is blocked/unblocked – When rule is defeated/undefeateds

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

Extension of RULML RuleML is an XML based language that supports rule representation for the Semantic Web. A new XML schema, extension of RuleML, is proposed for explanation representation in defeasible logic

Ext. RULML - Atom, Fact Atom: operator, var or const, optionaly NOT rich Bob A Fact is consisted by an Atom that comprise a certain knowledge

Ext. RULML - Rules Strict - Defeasible Rules. – Head: is an Atom – Body: number of Atoms rich Bob wins_lotto Bob

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.

Explanations Example rich Bob rich Bob wins_lotto Bob wins_lotto Bob

Other Explanations Defeasibly Provable Explanations – arises from the defeasible logic specification. Not Definitely Provable Explanations – consists of all the strict rules with head equal to the negation of the non provable Atom Not Defeasibly Provable Explanations – arises from the defeasible logic specification

Proof tree construction (1/3) The foundation of the proof system – Prolog metaprogram implements rules of Defeasible Logic – The trace of the XSB implementation of prolog XSB: logic programming engine used to run 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

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).

Proof tree construction (3/3) A tree whose nodes are traced predicates is constructed by 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) – Boolean attribute tells if predicate is negated.

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 information to trace – Prolog shows successful and unsuccessful paths The tree produced by the XSB trace is built according to the metaprogram structure but the final tree needs to be compliant with the XML schema

Pruning Rules Heuristic rules are used in order to prune the proof tree According to the truth value and the type of the root node we may want to maintain – only successful paths – only failed paths – combinations of them.

Pruning Motivation Example1 Suppose we have the following defeasibly theory translated in logic programming as: – fact(a). – fact(e). – defeasible(r1,b,a). – defeasible(r2,b,e). – defeasible(r3,~(b),d).

Example1: Defeasible provability of b Pruned

Example1: Defeasible provability of b We are interested in successful (True) paths The pruning algorithm removes – 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 metaprogram additional negation clause.

Pruning Motivation Example2 Suppose we have the following defeasibly theory translated in logic programming as: – fact(a). – defeasible(r1,b,a). – defeasible(r2,~(b),a).

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

Pruning resume The proof tree after using the heuristic techniques is similar to an explanation derived by the use of pure DL Drawback: heuristics are fully dependent on the metaprogram. – Changes at metaprogram => changes at pruning implementation.

Pruning Example The marked rule is pruned

Pruning Example 2 The marked rule is pruned

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 on JADE (Java Agent DEvelopment Framework) – a software framework to develop agent-based applications

Process to answer a query (1/3) Main Agent XSB XML Writter Invoker Pruner Agent 2) Predicate1)Question3) Predicate 4) result trace 5) result tree 7) Pruned result6) result tree 8) Pruned result9) XML proof 10) answer or proof

Process to answer a query (2/3) 1. An agent issues a query to the Main Agent. - predicate::(proof|answer) 2. Main Agent sends Predicate to the Invoker – Invoker is responsible to communicate with XSB 3. Invoker executes the Predicate. 4. XSB returns the full result trace. 5. Invoker returns result tree to Main Agent. 6. Main Agent sends result tree to the Pruner

Process to answer a query (3/3) 7. Pruner returns pruned result to Main Agent. 8. Main Agent sends the pruned result to the XML writer (only if proof requested) 9. XML writer returns the XML Proof. 10. Main Agent returns Answer or Proof - ANSWER(true | false) - PROOF:(proof string) - ERROR:(error message)

Visual Agent

Command Line Agent Reads in 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