Transcript Description Logics - clic

INTRODUCTION TO ARTIFICIAL
INTELLIGENCE
Massimo Poesio
LECTURE 4: Semantic Networks and
Description Logics
SEMANTIC NETWORKS
• Around end of the ’70s researchers in Psychology thought
they had found evidence that knowledge was organized
more or less as expected on the basis of ideas about
taxonomies
• This led to the development of formalisms for knowledge
representation in AI called SEMANTIC NETWORKS
• Soon researchers like Schubert (1975) and Hayes (1979)
demonstrated that these formalisms were just notational
variants of logic
• The ‘logical’ approach to semantic networks has however
led to the development of so-called DESCRIPTION LOGICS, a
family of logics which also includes logics with better
computational properties than first order logic
SOME RESULTS FROM COGNITIVE PSYCHOLOGY:
SEMANTIC NETWORKS
• Collins & Quillian, 1969: knowledge appears to be
organized around objects and in a taxonomic way
– A canary is yellow
– A canary has feathers
– A canary eats food
• Haviland & Clark 1974, Sanford & Garrod 1979:
‘associated’ knowledge available when concepts are
mentioned
– I looked around the house.
– The lounge was very spacious.
SEMANTIC NETWORKS
• Hypothesis: commonsense knowledge is
organized in networks whose nodes are types and
instances of types, and whose relations encode
– Taxonomic relations (as in Aristotle)
– Attributes
• The key inference that such theories want to
model: INHERITANCE
– Semantic networks also called INHERITANCE
NETWORKS
AN EXAMPLE OF SEMANTIC
NETWORK
ANIMAL
eats food
can move
can fly
has feathers
BIRD
scales
CANARY
sings
yellow
FISH
swims
AN EXAMPLE OF INHERITANCE
BIRD has FEATHERS
CANARY IS-A BIRD
∴ CANARY has FEATHERS
DESCRIPTION LOGICS
• Brachman & Levesque (1985) proposed a
formal approach to knowledge bases
organized as semantic networks, encoding
inheritance reasoning
SPECIFYING A KNOWLEDGE BASE:
TBOX AND ABOX
• According to Description Logics, a knowledge
base contains two types of knowledge:
– Generic knowledge about concepts, contained in
the TBOX (ie: SEMANTIC MEMORY)
• Bicycles have two wheels
• Parents have children
– Knowledge about the instances of these concepts,
contained in the ABOX (ie: EPISODIC MEMORY)
• Massimo’s bicycle is grey
• Distinct logical languages for each of them
EXAMPLE OF TBOX
KEY TERMS
• Nodes: CONCEPTS
• Subtype relation: ISA
• Properties: ATTRIBUTES or ROLES
CONCEPT DEFINITION SYNTAX
• Intersection of concepts: C ∩ D
– E.g., ANIMAL ∩ FLY
– Interpretation: ANIMAL(x) ∩ CANFLY(x)
• Attributes: ∃R.C
– E.g., ∃hasFeather.FEATHER
• Value restriction: ∀R.C
– E.g., ∀hasWheel.WHEEL
• Number restriction: (≤ n R), (≥ n R)
– E.g., (≤ 2 hasWheel)
• Negation: ¬ C
– E.g., ¬ FEMALE
EXAMPLES OF COMPLEX CONCEPTS
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BIRD ∩ YELLOW ∩ SINGS
ANIMAL ∩ RATIONAL
PERSON ∩ ¬ FEMALE
VEHICLE ∩(≤ 2 hasWheel)
VEHICLE ∩∃hasEngine.ENGINE
TBOX DEFINITIONS
• NECESSARY AND SUFFICIENT
– CANARY ≡ BIRD ∩ YELLOW ∩ SINGS
– HUMAN ≡ ANIMAL ∩ RATIONAL
– WOMAN ≡ PERSON ∩ FEMALE
– MALE ≡ PERSON ∩ ¬ FEMALE
– BICYCLE ≡ VEHICLE ∩(≤ 2 hasWheel) ∩
¬∃hasEngine.ENGINE
• PRIMITIVE
– BEAR ⊂ ANIMAL
SEMANTICS
• TBOX concepts denote SETS
– ∩ denotes INTERSECTION
– ¬ denotes COMPLEMENTATION
– Etc
• The resulting language is a subset of FOL
ABOX DEFINITIONS
• PERSON ∩ FEMALE (Maria)
• hasCHILD(Maria,Gigino)
INFERENCE IN DL
• Description Logics were developed to model
inheritance reasoning
• In fact, they model a more complex form of
reasoning: SUBSUMPTION
• They are intended to be the
COMPUTATIONALLY LEAST EXPENSIVE logics in
which such reasoning is possible
SUBSUMPTION
• Concept D subsumes concept C, written
C⊆D
• If D is MORE GENERAL than C, i.e., if the set
denoted by C is a subset of the set denoted by
D
EXAMPLE OF (TBOX) SUBSUMPTION:
INHERITANCE
BIRD≡ ANIMAL ∩ CANFLY
∩ ∃hasFeather.FEATHER
CANARY ≡ BIRD ∩ YELLOW ∩ CANSING
∴ CANARY ⊆ CANFLY
EXAMPLE OF (TBOX) SUBSUMPTION
ENGINED_OBJECT ≡∃hasEngine.ENGINE
CAR≡ VEHICLE ∩(= 4 hasWheel) ∩
∃hasEngine.ENGINE
∴ CAR ⊆ ENGINED_OBJECT
EXAMPLE OF (ABOX) SUBSUMPTION
WOMAN ≡ PERSON ∩ FEMALE
PERSON ∩ FEMALE (Maria)
∴ WOMAN(Maria)
SUBSUMPTION AND MODERN PSYCHOLOGICAL
THEORIES OF CONCEPTUAL KNOWLEDGE
• As we will see in the next lectures, modern
theories of concepts in cognitive science
(since Rosch) have abandoned the position
that conceptual knowledge is organized
taxonomically in favour of the `clustering’
views from PROTOTYPE THEORY
• Subsumption is a modern approach to
inheritance that does NOT depend on the
existence of special ISA links
UNDECIDABILITY, COMPLEXITY, and
LOGIC
• One would want to have a logic as expressive
as possible
– ideally, as expressive as natural language
• But there is a tight connection between the
expressive power of a logic and the cost of
reasoning with that logic
• It is known from Goedel and Turing that FOL is
undecidable
• Even the propositional calculus is NP-complete
THE COMPLEXITY OF DESCRIPTION
LOGICS
• The simplest form of DL is DECIDABLE and
POLYNOMIAL (i.e., relatively efficient)
• But even minor additions result in exponential
complexity
• DL-FOL is undecidable
READINGS
• Nardi & Brachman, An introduction to
Description Logics, ch. 1 of Handbook of
Description Logics (on the site)