74.419 Artificial Intelligence 2004 Non-Classical Logics

Non-Classical Logics• Specific Language Constructs added to classic FOPL• Different Types of Logics• Modal Logics most popular ones, e.g.

• Deontic Logic (allowed and forbidden; ethics; law)

• Epistemic Logic (Knowledge) and Doxastic (Belief) Logic

• Possible World Semantics

Non-Classical Logics 1• (many-) sorted logic

•individuals pre-arranged in sets = sorts• many-valued logic

•more than two truth values (e.g. Lukasiewicz “intermediate truth” = I; "don't know" status)

• fuzzy logic•degree of truth between 0 and 1 (predicate corresponds to fuzzy set; membership in set to a certain degree, not just yes or no)

• non-monotonic logicbelief modeling; defaults; set of true formulae can change (non-monotonicity); TMS

Non-Classical Logics 2

• higher-order logicquantification over predicates (as variables), like P: ..., or statements about statements, e.g. “This sentence is false.”

• modal logics (see later slides)describe “subjunctive” statements in addition to assertional statements, using Modal Operators, i.e. "possible P" and "necessary P"

Non-Classical Logics 3• time logics

• time as temporal modality• time logic based on time points and relations

between them (like “t1 earlier than t2”)• Allen’s model of time intervals

• situational logic; situation calculus (McCarthy)• situation as additional parameter to predicate

expressions, for describing change due to events• additional axioms describe transformations of

ssituations due to actions• used for reasoning with time and planning

Modal Logics 1Uses additional operators and axioms to describe logic. Includes FOPL assertions, and in addition statements using Modal Operators. Different Modalities express different types of statements, e.g.• alethic modality

“necessary” and “possible” as additional operators• temporal modality

with necessary “always” and possible “sometimes”

• deontic modality“permissible” (allowed) and “obligatory” (must)

• epistemic modality“knows” and “beliefs” as operators

Athletic ModalityAlethic modalitySomething is necessarily true, or possibly true.Operators:

“necessary” ٱ and “possible” ◊Axioms: e.g. A1) necessary(P) possible(P)

A2) possible(P) P

“If P is necessarily true, then P is also possible.”

“If P is not possible, then P cannot be true.”

Temporal ModalityTemporal modality Something is always or sometimes true. Operators:

“always” “necessary” “sometimes” “possible”

Axioms: A1) always(P) sometimes(P)

“If P is always true, then P is sometimes true.”A2) always( P) sometimes(P) “If not P is always true, then P is not sometimes true.”

Also for tenses like “past”, “past perfect”, “future”, ...

Deontic Modality

Deontic modality (ethics) Something is permitted or obligatory. Operators:

“permissible” and “obligatory” Axioms:

e.g. obligatory(P) permissible(P)“If P is obligatory, then P is also

permitted.”

Epistemic ModalityEpistemic modality Reasoning about knowledge (and beliefs)Operators:

“Knows” and “Believes”Axioms: e.g. KnowsA(P) P

“If agent A knows P, then P must be true.”KnowsA(P) BelievesA(P) “If agent A knows P, then agent A also believes

P.”KnowsA(P) KnowsA(P Q) KnowsA(Q)

Epistemic Modality - AxiomsMost Common Axioms (Nilsson): 1. Modus Ponens Knowledge

[KnowsA(P) KnowsA(P Q) ] KnowsA(Q)

2. Distribution AxiomKnowsA(P Q) [KnowsA(P) KnowsA(P Q) ]

3. Knowledge AxiomKnowsA(P) P

4. Positive-Introspection AxiomKnowsA(P) (KnowsA(KnowsA(P))

5. Negative Introspection Axiom¬ KnowsA(P) (KnowsA(¬ KnowsA(P))

Epistemic Modality - Inference

Inferential Properties of Agents:Epistemic Necessitation:

from |– αinfer KnowsA(α)

Logical Omniscience:from α |– β and KnowsA(α)infer KnowsA(β)

or:from |– (α β) infer KnowsA(α) KnowsA(β)

Epistemic Modality - Problems 1Problem: "Referential Opaqueness"Different statements refering to the same extension, cannot necessarily be substituted. Agent A knows John's phone number. John's phone number is the same as Jane's phone number. You cannot conclude that A also knows Jane's phone number. Another approach (than ML) is to use Strings instead of plain formulae to model referential opaqueness (cf. Norvig):

e.g. KnowsA(P) KnowsA("P=Q") KnowsA(Q)

Epistemic Modality - Problem 2

Problem: "Non-Compositional Semantics"You cannot determine the truth status of a complex expression through composition as in standard FOPL.From A and α you cannot always determine the truth status of KnowsA(α ).

e.g. From KnowsA(P) and (P Q) not conclude KnowsA(Q)

Modal Logic uses a "Possible Worlds Semantics"

Possible World SemanticsFor modal and temporal logics, semantics is often based on considerations about which “worlds” (set of formulae) are compatible with or possible to reach from a certain given “world” → possible world semanticsRelations between “worlds”:

• accessible• necessary

A world is accessible from a certain world, if it is one possible follow state of that world. A world is a necessary follow state of a certain world, if the formulae in that world must be true, is a necessary conclusion.

Possible World Semantics - Example

Possible World Semantics for Epistemic LogicIf Agent A knows P, then P must be true in all worlds accessible from the current world. That means these worlds are not only accessible but necessary worlds (respective to the agent and its knowledge).If Agent A believes P, then P can be true in some accessible worlds, and false in others.

Possible World Semantics

For modal and temporal logics, semantics is often based on considerations about which “worlds” (described by set of formulae) can be reached or have to be reached given a certain “world” → possible world semanticsRelations between “worlds”:

• accessible• necessary

A World is accessible from a certain world, i.e. if it is a possible follow state of that world. A world is a necessary follow state of a certain world, .

Representing Time

• Time as temporal modality in modal logic• Time in FOPL

add time points and time relations as predicatese.g. "earlier-than" (et) for two time points Axioms:e.g. x,y,z: (et (x,y) et (y,z)) et (x,z)

x,y: et (x,y) et (y,x)

Time Interval Representation (Allen)Allen’s Time Interval LogicTime represented based on Intervals. Relations between time intervals are central :e.g. meet(i,j) for Intervals i and j

Interval i

Interval j

Time points representable as functions on intervals, e.g. start(i) and end(i) specify time points.Axioms:e.g. meet(i,j) time(end(i))=time(start(j))

Additional References

R. A. Frost: Introduction to Knowledge-Based Systems. Collins, London, 1986.

Nils J. Nilsson: Artificial Intelligence – A New Synthesis. Morgan Kaufmann, San Francisco, 1998.