1. Default Reasoning By Naval Chopra(07010015) ‏ Pranay Bhatia (07005005) ‏ Pradeep Kumar(07D05020) ‏ Siddharth Chinoy(07D05005) ‏ Vaibhav Chhimpa(07D05011) ‏

2. Motivation The problem of what to do next Implicit assumptions, rules of thumb, inferential shortcuts Default reasoning : "Inference to the first unchallenged alternative" Is this reasoning? Yes, it often leads to useful knowledge. Real people, unlike philosophers, have to make judgments and decisions all the time, not just when they're good and ready

3. Reasoning Thinking in the pattern of an argument Moving from premise to conclusion People are generally capable of good reasoning When people make mistakes, they are due to deviations from the ideal The 'closer' we are to following the logically valid pattern of argument, the better the reasoning Wrong! Courtesy : Kent Bach Look before you leap - A proverb He who hesitates is lost - Another proverb

4. Introduction Default reasoning is a form of Defeasible Reasoning used to express facts like “by default, something is true”. Default Logic is a Non-Monotonic Logic proposed by Raymond Reiter to formalize reasoning with default assumptions.

5. Introduction Standard logic can only express that something is true or that something is false. This is a problem because reasoning often involves facts that are true in the majority of cases but not always. Example : “Birds typically fly” vs “All birds fly” Exceptions – Penguins, Ostriches

6. Defeasible Reasoning Reasoning is defeasible when the corresponding argument is rationally compelling but not deductively valid. The truth of the premises of a good defeasible argument provide support for the conclusion, even though it is possible for the premises to be true and the conclusion false.

7. Defeasible Reasoning In other words, the support provided by the premises to the conclusion is a tentative one, potentially defeated by additional information. Defeasible reasoning has typically been limited to inferences involving exception-permitting generalizations, that is, inferring what has or will happen on the basis of what normally happens.

8. Problems with First Order Logic First Order Logic is a Monotonic Logic in the sense that its consequence relation is monotonic. If a sentence φ can be inferred in FOL from a set Γ of premises, then it can also be inferred from any set Δ of premises containing Γ as a subset.

9. Problems with First Order Logic i.e The consequence relation of FOL has the property that if Γ ⊨ φ and Γ ⊆ Δ then Δ ⊨ φ. This property is known as the Monotonic Property. Intuitively, this implies that learning a new piece of knowledge cannot reduce the set of what is known.

10. Problems with First Order Logic There are striking differences between formal logic and the working of the mind when it comes to dealing with Incomplete Knowledge (such as perception, ambiguity, common sense, causality and prediction) ‏ Classical logic lacks tools for describing how to revise a formal theory to deal with inconsistencies caused by new information.

11. Problems with First Order Logic There are mainly two types of revisions : World Model Reorganisation – Very hard problem of revising complex models. Complexity usually stems from part of the model relying on other parts of the model. Eg. Revision of one’s opinion of a friend after discovering his dishonesty

12. Problems with First Order Logic Routine Revision – Easier Problem. Involves maintaining facts which although expressed as universally true have exceptions. Eg. Stating that all birds usually fly, but then on finding out that penguins don’t fly revising the knowledge base to include that fact.

13. Problems with First Order Logic Classical logic has overlooked the above simple cases by altering the notation in which rules are stated. Eg. “All birds except Penguins, Ostriches,... Fly” Classical logic also cannot handle abductive reasoning (consequences deduced as most likely explanations).

14. Default Reasoning Default Reasoning (and Default Logic) was proposed to handle the problems of non-monotonicity and belief revision. It mainly aims at formalising default inference rules without stating all the exceptions.

15. Logics formalizing default reasoning Logics able to deal with arbitrary default assumptions (default logic, defeasible reasoning, and answer set programming) ‏ Logics that formalize the specific default assumption that facts that are not known to be true can be assumed false by default (closed world assumption and circumscription).

16. Syntax of Default Logic A default theory is a pair W is a set of logical formulae, called the background theory, that formalize the facts that are known for sure. D is a set of default rules, each one being of the form: Prerequisite : Justification 1, …,Justification n Conclusion

17. Syntax of Default Logic According to this default, if we believe that Prerequisite is true, and each of Justification i is consistent with our current beliefs, we are led to believe that Conclusion is true. The logical formulae in W and all formulae in a default were originally assumed to be first-order logic formulae, but they can potentially be formulae in an arbitrary formal logic.

18. Examples for Syntax The default for “Birds typically fly” is formalised by the following default : The rule means that if X is a bird and it can be assumed that it flies, then we conclude that it flies A background theory for the above is :

19. Example for Syntax What we can conclude : Flies(Condor) ‏ What we cannot conclude : Flies(Penguin) ‏ Bird(Eagle) ‏

20. This default rule is applicable if we can prove from our beliefs that John is an American and an adult, and believing that there is some car that is owned by John does not lead to an inconsistency. If these two sets of premises are satisfied, then the rule states that we can conclude that John owns a car. Courtesy : http://www.rci.rutgers.edu/~cfs/472_html/Logic_KR/DefaultTheory.html

21. Another Example A common default assumption is that what is not known to be true is believed to be false. This is known as the Closed World Assumption, and is formalized in default logic using a default like the following one for every fact F.

22. Restrictions A default is categorical or prerequisite- free if it has no prerequisite (or it’s prerequisite is tautological). A default is normal if it has a single justification that is equivalent to its conclusion. A default is supernormal if it is both categorical and normal. A default is seminormal if all its justifications entail its conclusion.

23. Semantics of Default Logic A default rule can be applied to a theory if its precondition is entailed by the theory and its justifications are all consistent with the theory. When the theory is such that no other default can be applied, the theory is called an extension of the default theory. The default rules may be applied in different order, and this may lead to different extensions.

24. Nixon Diamond Usually, Quakers are pacifist Usually, Republicans are not pacifist Richard Nixon is both a Quaker and a Republican Courtesy : http://www.cs.cf.ac.uk/Dave/AI2/node80.html

25. Nixon Diamond Since Nixon is a Quaker, one could assume that he is a pacifist; since he is Republican, however, one could also assume he is not a pacifist.

26. Entailment Entailment of a formula from a default theory can be defined in two ways: ◦ Skeptical  A formula is entailed iff it is entailed by all its extensions  Since Nixon can neither be proved to be a pacifist nor the contrary, no conclusion is drawn. ◦ Credulous  A formula is entailed iff it is entailed by at least one of its extensions.  Since Nixon can be proved to be a pacifist in at least one case, he is believed to be a pacifist; however, since he can also be proved not be a pacifist, he is also believed not to be a pacifist.

27. Some examples The following default theory has no extension: Normal default theory (At least one extension present):

28. Alternate Default Inference Rules Justified - differs from the original one in that a default is not applied if thereby the set T becomes inconsistent with a justification of an applied default Concise - a default is applied only if its consequence is not already entailed by T (the exact definition is more complicated than this one; this is only the main idea behind it) ‏

29. Alternate Inference Rules Constrained - a default is applied only if the set composed of the background theory, the justifications of all applied defaults, and the consequences of all applied defaults (including this one) is consistent. Rational - similar to constrained default logic, but the consequence of the default to add is not considered in the consistency check

30. Alternate Default Inference Rules Cautious - defaults that can be applied but are conflicting with each other (like the ones of the Nixon diamond example) are not applied.

31. Variations Priorities on defaults ◦ Relative priority of defaults are explicitly specified. Among the defaults that are applicable to a theory, only one of the most preferred one is applied. ◦ One form of implicit priority : more specific defaults (those that are applicable in fewer cases) are preferred over less specific ones. http://www.rci.rutgers.edu/~cfs/472_html/Logic_KR/KnowledgeRepToc.html

32. Variations Statistical variant ◦ Statistical default is a default with an attached upper bound on its frequency of error; in other words, the default is assumed to be an incorrect inference rule in at most that fraction of times it is applied. ◦ Extensions for statistical default logic are constructed in the usual way, except that the operator ‘terminates’ when inference reaches a specified threshold.

33. Variations Weak extensions ◦ Rather than checking whether the preconditions are valid in the theory composed of the background theory and the consequences of the applied defaults, the preconditions are checked for validity in the extension that will be generated. ◦ In other words, the algorithm for generating extensions starts by guessing a theory and using it in place of the background theory; what results from the process of extension generation is actually an extension only if it is equivalent to the theory guessed at the beginning. ◦ This variant of default logic is related in principle to autoepistemic logic, where a theory has a model in which x is true just because, assuming true, the formula supports the initial assumption. ◦ A logic allowing such a self-support of beliefs is called not strongly grounded to differentiate them from strongly grounded logics, in which self-support is not possible. Strongly grounded variants of autoepistemic logic exist.

34. Circumscription Non-monotonic logic that formalizes the common sense assumption that things are as expected unless otherwise specified. In the Missionaries and Cannibals Problem, the solution “go half a mile south and cross the river on the bridge” is intuitively not valid The problem statement does not mention a bridge. Does not exclude it’s existence either.

35. Circumscription in FOL Given a FOL formula T containing a predicate P, circumscribing this predicate amounts to selecting only the models of T in which P is assigned to true on a minimal set of tuples. The extension of a predicate in a model is a set of tuples that this predicate assigns to true in that model The circumscription of a predicate P in a formula T is obtained by selecting only the models of T with a minimal extension of P.

36. Circumscription in FOL Where p<P = p is a predicate having the same arity as P and x is an n-tuple The above formula means that there exists no predicate p which assigns false to every value that P assigns false and is not P By adding an extra literal ~Abnormal(..) to each fact stating that it holds only in normal conditions. Minimizing the extension allows for reasoning under the implicit assumption that things are as expected and that this assumption is made only if possible

37. Closed World Assumption The presumption that what is not currently known to be true, is false. Negation as failure is related as it amounts to believing false every predicate that cannot be proved to be true. In formal logic it is achieved by adding to the knowledge base the negation of the literals that are not currently entailed by it Resultant system may not be consistent

38. Implementations DeReS, XRay, GADel XRay : An experimental theorem prover for query-answering from incomplete knowledge bases. Implementation relies heavily on Mark Stickel's Prolog Technology Theorem Prover, PTTP. S. Brüning and T. Schaub. A model-based approach to consistency-checking.

39. Implementations ◦ Deres - written in C, run under Unix ◦ Relaxed stratification - Divide and conquer approach to computing extensions ◦ Original default theory partitioned into several sub theories called strata ◦ Extensions of theory are recomputed via the extensions of the strata. ◦ Efficient, when strata are small.

40. The intermediate language - PTTP for formulas in negation normal form. Here's its grammar: An example file : % facts adult. % default rules adult :- student : adult. not_employed :- student : not_employed. not_married :- student : not_married. employed:-adult:not_student. married:-adult:not_student. % initial query query:-married;employed. M. Stickel. A Prolog technology theorem prover. New Generation Computing, 2:371-383, 1984.

41. References 1.M. Stickel. A Prolog technology theorem prover. New Generation Computing, 2:371-383, 1984. 2.S. Brüning and T. Schaub. A model-based approach to consistency-checking. In Z. Ras and M. Michalewicz, editors, Proceedings of the Ninth International Symposium on Methodologies for Intelligent Systems, volume 1079 of Lecture Notes in Artificial Intelligence, pages 315-324. Springer Verlag, 1996. 3.Elaine Rich & Kevin Knight, "Artificial Intelligence", McGraw-Hill Science/Engineering/Math; 2nd edition. 4.Russel S. and Norvig P., "Artificial Intelligence: a Modern Approach", Prentice Hall, 1998. 5.Nilsson, N.J., "Artificial Intelligence, a New Approach", Morgan Kaufmann, 2000. 6.http://www.cs.cf.ac.uk/Dave/AI2/node74.html 7.http://plato.stanford.edu/entries/reasoning-defeasible/ 8.http://plato.stanford.edu/entries/logic-nonmonotonic/ 9.http://online.sfsu.edu/~kbach/defaultreasoning.pdf 10.http://www.cs.cmu.edu/~jgc/publication/Default_Reasoning_Inheritance_Mechanis ms_SIGART_1980.pdf

42. Thank you! Questions ?