symbolic reasoning

10/1/2010

Symbolic Reasoning under uncertainty

Artificial Intelligence

• Story so faro We have described techniques for reasoning with a complete,

consistent and unchanging model of the world. But in many problem domains, it is not possible to create such models. So here we are going to explore techniques for solving problems with incomplete and uncertain models.

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• Introduction to Non-monotonic Reasoning o The murder mystery exampleo Non monotonic reasoning is one in which the axioms and/or the

rules of inference are extended to make it possible to reason with incomplete information.

o These systems preserve, however, the property that , at any given moment, a statement is either believed to be true, believed to be false, or not believed to be either.

o Statistical Reasoning : in which the representation is extended to allow some kind of numeric measure of certainty(rather than true or false) to be associated with each statement.

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• At times we need to maintain many parallel belief spaces, each of which would correspond to the beliefs of one agent.

• Such techniques are complicated by the fact that the belief spaces of various agents, although not identical, are sufficiently similar that it is unacceptably in efficient to represent them as completely separate knowledge bases.

• Conventional reasoning systems, such as FOPL are designed to work with information that has three important properties.o It is complete with respect to domain of interest.o It is consistent.o The only way it can change is that new facts can be added as

they become available. All this leads to monotonicity.

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• If any of these properties is not satisfied, conventional logic based reasoning systems become inadequate. Non monotonic reasoning systems, are designed to be able to solve problems in which all of these properties may be missing

• Issues to be addressedo How can the knowledge base be extended to allow inferences to

be made on the basis of lack of knowledge as well as on the presence of it?

o How can the knowledge base be updated properly when a new fact is added to the system(or when the old one is removed)?

o How can knowledge be used to help resolve conflicts when there are several in consistent non monotonic inferences that could be drawn?

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• Logic for monotonic reasoningo Monotonicity is kind of a definition to FOPL, we have to find some

alternative to support non monotonic reasoning. o No single formalization has all the desired properties.o We want to find a formalism that does all of the following things

Define the set of possible worlds that could exist given the facts that we do have.

Provide a way to say that we prefer to believe in some models rather than others.

Provide the basis for a practical implementation of this kind of reasoning.

Corresponds to our intuitions about how this kind of reasoning works.

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• Default Reasoningo We use non monotonic reasoning to perform, what is commonly

called Default Reasoning.o We want to draw conclusions based on what is most likely to be

true.o Two approaches are

Nonmonotonic Logic Default Logic

o Two common kinds of nonmonotonic reasoning that can be defined in these logics : Abduction Inheritance

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• Non Monotonic Logic(NML)o It is one in which the language of FOPL is augmented with a

modal operator M, which can be read as “is consistent.”o Vx,y : Related(x,y) ^ M GetAlong(x,y) WillDefend(x,y)o For all x and y, if x and y are related and if the fact that x gets

along with y is consistent with everything else that is believed, then conclude that x will defend y.

o If we are allowing statements in this form, one important issue that must be resolved if we want our theory to be even semi decidable, we must decide what “is consistent ” means.

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• Default Logico An alternative logic for performing default-based reasoning is

Reiter’s Default Logic(DL) in which a new class of inference rules is introduced. In this approach, we allow inference rules of this form A : B C The rule should be read as – If A is provable and it is

consistent to assume B, then conclude C

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• Abductiono Standard logic performs deductions. Given 2 axioms

Vx : A(x) B(x) A(C)

• We conclude B(C) using deduction• Vx : Measels Spots(x)• To conclude that if somebody has spots will surely have measels is

incorrect, but it may be the best guess we can make about what is going on. Deriving conclusions in this way is this another form of default reasoning. We call this abductive reasoning.

• To accurately define abductive reasoning we may state that – Given 2 wff’s AB and B, for any expression A & B, if it is consistent to assume A, do so.

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• Inheritanceo A rule for the Baseball Player can be as

Baseball-Player(x) : height(x,6-1) [This is a rule] height(x,6-1)

Adult-Male(x) : height(x,5-10) [This is a rule] height(x,5-10) Adult-Male(x) : --- Baseball-Player(x) ^ height (x,5-10) [ Rule] height(x,5-10) Vx: Adult-Male(x) ^ --AB(x,aspect1) height(x,5-10) and so on

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• Minimalist Reasoningo We describe methods for saying a very specific and highly useful

class of things that are generally true.o These methods are based on some variant of the idea of a

minimal model.o We will define a model to be minimal if there are no other models

in which fewer things are true.o The idea behind using minimal models as a basis for

nonmonotonic reasoning about the world is the following – There are many fewer true statements than false ones. If

something is true and relevant it makes sense to assume that it has been entered into our knowledge base. Therefore, assume that the only true statements are those that necessarily must be true in order to maintain the consistency.

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• The Closed World Assumptiono CWA says that the only objects that satisfy any predicate P are

those that must. o Eg. A company’s employee database.o Airline example

• Although the CWA is both simple & powerful, it can fail to produce an appropriate answer for either of the two reasons.o The assumptions are not always true in the world; some parts of

the world are not realistically “closable”. - unrevealed facts in murder case

o It is a purely syntactic reasoning process. Thus, the result depend on the form of assertions that are provided.

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• Augment a Problem- Solvero How to write a program that solves problems using axioms.o Even uncertain knowledge can be solved using forward &

backward reasoning. o As a result there are 2 approaches to this kind of problem solving

Reason forward from what is known. Treat non monotonically derivable conclusions the same way monotonically derivable ones are handled.

Everything is same as monotonic except that here we have forward chaining rules but with UNLESS clause.

Reason backward to determine whether some expressions P is true?

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• Augment a Problem- Solver Reason backward to determine whether some expressions P

is true?(or perhaps to find a bindings for its variables that make it true) Non-monotonic reasoning systems that support this kind of reasoning may do either or both of the following

Allow default(unless) clauses in backward ruling. Support a kind of debate in which an attempt is made to

construct arguments both in favor of P and opposed to it.

symbolic reasoning

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