1 DCP 1172 Introduction to Artificial Intelligence Chang-Sheng Chen Topics Covered: Introduction to Nonmonotonic Logic

DCP 1172, Ch. 6 2 Monotonic logic Standard logic is monotonic: once you prove something is true, it is true forever Monotonic Logic is not a good fit to reality If the wallet is in the purse, and the purse in is the car, we can conclude that the wallet is in the car But what if we take the purse out of the car?

DCP 1172, Ch. 6 3 Monotonic Logic Given a collection of facts D that entail some sentence s (s is a logical conclusion of D): for any collection of facts D ’ such that D  D ’, D ’ also entails s. in other words: s is also a logical conclusion of any superset of D.

DCP 1172, Ch. 6 4 Nonmonotonic Logic In a nonmonotonic system: the addition of new facts can reduce the set of logical conclusions. S is a conclusion of D, but is not necessarily a conclusion of D+newfact. Humans use nonmonotonic reasoning constantly!

DCP 1172, Ch. 6 5 What is “ Non-monotonic Logic ” ? To understand what nonmonotonic logic means simple consider a standard example: "all birds fly", "Tweety is a bird", "Does Tweet fly?". The obvious answer is yes, however what if later you learned that Tweety had a broken wing, then the answer becomes no, then what if you learned that tweet was an airplane pilot, or had a jet pack, the answer can change again. The important point is that as new information is added the answers change

DCP 1172, Ch. 6 6 Nonmonotonic logic Facts and rules can be changed at any time such facts and rules are said to be dynamic Prolog uses nonmonotonic logic assert(...) adds a fact or rule retract(...) removes a fact or rule assert and retract are said to be extralogical predicates

DCP 1172, Ch. 6 7 Intelligent Reasoning One of the characteristics associated with intelligent systems is adaptability - the ability to deal with a changing environment. Adaptation requires that a system be capable of adding and retracting beliefs as new information is available. This requires nonmonotonic reasoning.

DCP 1172, Ch. 6 8 Uncertainty Another characteristic of intelligent systems is the ability to reason under conditions of uncertainty. Another way of saying this: the ability to reason with an incomplete set of facts.

DCP 1172, Ch. 6 9 Can we implement inheritance using predicate logic? Pat is a Bat. Bats are Mammals. Bats can fly. Bats have 2 legs. Mammals cannot fly. Mammals have 4 legs. How many legs does Pat have?

DCP 1172, Ch Inheritance Reasoning about inheritance of properties from one class to another: Bird(x)  Flies(x) Clearly this is not a good rule, since we know there are exceptions. Bird(x)  Normal(x)  Flies(x) This provides for exceptions, although we must define the conditions that imply Normal(x).

DCP 1172, Ch Normal(x) Assuming we know that: Ostrich(x)  Bird(x)  ~Flies(x) we can derive: Ostrich(x)  ~Normal(x) So an ostrich is not a normal bird. But what about all the the other things that are birds?

DCP 1172, Ch Assumptions and Defaults If there is no reason to believe otherwise, assume that Normal(x) is TRUE. The default is that everything is normal. Now we only need to supply additional information for exceptions.

DCP 1172, Ch How to specify defaults A number of formal systems have been developed to handle defaults. Nonmonotonic logics formalize unsound but reasonable patterns of reasoning with uncertain, incomplete and inconsistent information Default Logic: New rule of inference Abduction: New interpretation of implication.

DCP 1172, Ch And More Logics To Think About! Modal logic is useful for modeling reasoning about knowledge, actions, time or obligations. Epistemic logics apply the techniques of modal logic to reasoning about knowledge. Both individual and group knowledge is studied. The study of epistemic logic is relevant to communication protocols and cooperation. Deontic logic formalizes normative modalities. Deontic logic can be applied to representation of normative (e.g. legal) knowledge.

DCP 1172, Ch Default Reasoning with Nonmonotonic Logic Predicate logic with an extension: a modal operator M which means is “ consistent with everything we know ”. Example:  x,y: Related(x,y)  M GetAlong(x,y)  WillDefend (x,y)

DCP 1172, Ch Default Logic New rule of inference: A : B C If A is true and it is consistent to assume B, then C is true. Same idea, but now used as a rule of inference. The new rule extends the knowledge base to a set of plausible extensions, any new statement that is true in all extensions is added.

DCP 1172, Ch Inheritance with Default Logic Support for inheritance using Default Logic: Mammal(x) : Legs(x,4) Legs(x,4) In the absence of contradictory information, we can assume anything that is a mammal has 4 legs. (also need a rule stating that nothing can have 2 different numbers of legs!)

DCP 1172, Ch Abduction Deduction ( 演繹 ): Given A(x)  B(x) and A(x), we assume that B(x) is true.  Similar to forward reasoning [Cf.] reasoning from the general to the particular (or from cause to effect) Abduction: Given A(x)  B(x) and B(x), we assume that A(x) is true.  Similar to backward reasoning

DCP 1172, Ch Inheritance Diagrams The book shows how we can also express default reasoning using diagrams. Flying Things BirdsOstriches FredTweety Normal Facts Default  Default

DCP 1172, Ch A Problem with NML  x: Republican(x)  M ~ Pacifist(x)  ~ Pacifist(x)  x: Quaker(x)  M Pacifist(x)  Pacifist(x) Republican(Dick) Quaker(Dick)

DCP 1172, Ch Not quite this easy Assuming we have some mechanism for representing defaults, there can still be problems: Is Nixon a pacifist? Nixon QuakersRepublicans Pacifists

DCP 1172, Ch Nixon Dilemma In general we must be prepared to deal with multiple, possibly conflicting consequences of a set of facts. One simple idea - rank all the assumptions and use rank to determine which to believe.

DCP 1172, Ch Other approaches to handling conflicting assumptions. Minimalist Reasoning Assume that there are fewer true statements that false statements in the world. Find the smallest interpretation that satisfies all the statements we know to be true. Closed World Assumption: the only objects that satisfy a predicate are those that must. forces positive assertions to take priority over negative assertions

DCP 1172, Ch Closed world assumption If we are told nothing about Tweety, other than Tweety is a bird, we assume that Tweety's feet are not in concrete, or Tweety's wings are not broken, this is the closed world assumption. Humans regularly make assumptions and when new evidence appears those assumptions can be changed, causing a different answer, thus behaving nonmonotonicaly.

DCP 1172, Ch Using Probabilities Probabilities can also be used determine which defaults apply when contradictions arise. Label each fact with a probability of being true. There is a big split in the A.I. community over whether symbolic methods or numeric methods are best for handling these types of issues.

DCP 1172, Ch Summary - Nonmonotonic logic vs. Probabilty Nonmonotonic logic systems may miss the importance of probability. Probabilistic reasoners can also represent uncertainty, and in a different (probabilistic) way. These systems exhibit a different set of properties, with which nonmonotonic logic can not effectively deal with.

DCP 1172, Ch Nonmonotonic vs. Classical logic Nonmonotonic logic does not have many essential properties of classical first order logic, specifically semi- decidability.  In classical logic, it is possible for a system to halt (be stuck in an infinite loop) trying to prove the negation of something for which there is insufficient information.  In nonmonotonic default logic, rather than return with no answer, the process returns with a wrong (default answer).

DCP 1172, Ch Summary - Nonmonotonic vs. Classical logic First order logic although descriptively universal, is not effective at handling large classes of problems.  If computers are going to handle common sense we need to be able to have some form of default reasoning. Nonmonotonic logic can be used in many domains where classical logic falls short:  such as in the areas of default diagnosis, diagnosis, action, and temporal logic.