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 BIRD FISH CANARY can fly has feathers yellow sings swims scales

AN EXAMPLE OF SEMANTIC NETWORK

AN EXAMPLE OF INHERITANCE CANARY IS-A BIRD ∴ CANARY has FEATHERS BIRD has FEATHERS

AN EXAMPLE OF INHERITANCE BEAR IS-A MAMMAL ∴ BEAR has VERTEBRA MAMMAL has VERTEBRA

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 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 CANARY ≡ BIRD ∩ YELLOW ∩ CANSING ∴ CANARY ⊆ CANFLY BIRD≡ ANIMAL ∩ CANFLY ∩ ∃ hasFeather.FEATHER

EXAMPLE OF (TBOX) SUBSUMPTION CAR≡ VEHICLE ∩(= 4 hasWheel) ∩ ∃ hasEngine.ENGINE ∴ CAR ⊆ ENGINED_OBJECT ENGINED_OBJECT ≡ ∃ hasEngine.ENGINE

EXAMPLE OF (ABOX) SUBSUMPTION PERSON ∩ FEMALE (Maria) ∴ WOMAN(Maria) WOMAN ≡ PERSON ∩ FEMALE

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)