University College Dublin DEPARTMENT OF COMPUTER SCIENCE COMP 4.19Multi-Agent Systems(MAS)

1. University College Dublin DEPARTMENT OF COMPUTER SCIENCE COMP 4.19Multi-Agent Systems(MAS) Lectures 19&20

2. Why use Modal Logic • There are two primary reasons why Modal Logic would be used rather than • First Order Predicate Logic (FOPL) when reasoning within Agent based systems. • These are:- • Does not adhere to the syntax; • Referential opaqueness;

3. Syntactic Reason Consider the following belief. Gregory believes that Rome is the capital of Italy. In a naïve manner we could represent this as a first order structure. Bel(Gregory, capitalof(Rome, Italy)). The problem with this of course is that the second argument to the Bel operator is not a term but rather a formula. Thus syntactically it does not adhere to necessary conditions.

4. Referential Opaqueness The second problem is more subtle. Consider where Rome is known by another name Roma. Clearly these constants would refer to the same geographic place. We could thus encode this in first order logic as (Rome = Roma) Bel(Gregory(capitalof(Roma,Italy)). This intuitively is not the case. First order logic is unable to cope with this because notions of belief and desire are referentially opaque. In such cases the standard Substitution rules employed in First order logic are invalid.

5. Referential Opaqueness II First order logic is truth Functional. The semantic value of a formula is dependent upon the the denotations or semantic value of its sub-expressions. a AND ~c dependent upon the truth values of a and ~c respectively. In intentional systems where Bel(Gregory(b)). It is not possible to substitute various values for b because the truth value of This sentence is not merely dependent upon the truth value of b.

6. Logical Omniscience Problem • The logical omniscience problem is one that has proven • Problematic for agent reasoners. • There are two aspects of this : • Consistency; • Equivalent propositions are not equivalent beliefs; We will consider each in turn.

7. Consistency As human reasoners we offen holdinconsistent beliefs. For example I could believe Two propisitions a and b where a implies ~b. We are unlikely to be aware of such inconsistencies. The reasoners advocated in possible world models however cannot have such inconsistent beliefs. This is because beliefs are closed under logical consequence. This seems counterintuitive because there are many cases where we are blissfully aware of the logical consequences that can be drawn from our beliefs and knowledge. Wooldridge (2002) cites the example of if our reasoner were to know Peano’s Axioms then they may well be able to deduce Fermat’s last theorm. Something that took centuries to achieve. Inconsistent beliefs under possible world models cannot occur without believing every formula of the logical language. Since consequential closure of a set of inconsistent formulae is in fact the set of all formulae.

8. Logical Consistency is Too Strong Konolige (1986) has argued that logical consistency is too strong a property to try to enforce on resource bounded reasoners. He has argued that a weaker property of non-contradiction is what should be expected. Non-contradiction means that an agent would not simultaneously believe a and ~a Even though they may hold logically inconsistent beliefs.

9. Equivalent Propositions are not Equivalent Beliefs Consider two propositions :- (1) Gregory likes ice-cream. (2) Gregory likes ice-cream and all ice-creams are large. Let’s assume the second conjunct in the second proposition to be valid. Given that agents are ideal reasoners within possible world semanticsthey would believe the two propositions are logically equivalent. Logically equivalent propositions are not equivalent as beliefs.