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Semantic WebSemantic WebDL Relationships with other Formalisms

Habib Rostami

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Outline• Approaches to knowledge representation• Deductive/logical methods

– Forward-chaining production rule systems– Semantic networks– Frame-based systems– Description logics

• Abductive/uncertain methods– What’s abduction?– Why do we need uncertainty?– Bayesian reasoning– Other methods: Default reasoning, rule-based methods, Dempster-

Shafer theory, fuzzy reasoning

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Introduction to some Introduction to some formalismsformalisms

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Introduction• Related to Description Logics for historical reasons

– Semantic networks

– Frame systems

• Had or have a strong influence on Description Logics– Modal logics

• Have similar expressive power– Semantic data models

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Semantic Networks (historical relationship)

• A semantic network is a simple representation scheme that uses a graph of labeled nodes and labeled, directed arcs to encode knowledge.– Usually used to represent static, taxonomic, concept dictionaries

• Semantic networks are typically used with a special set of accessing procedures that perform “reasoning”– e.g., inheritance of values and relationships

• Semantic networks were very popular in the ‘60s and ‘70s but are less frequently used today.– Often much less expressive than other KR formalisms

• The graphical depiction associated with a semantic network is a significant reason for their popularity.

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Nodes and Arcs

• Arcs define binary relationships that hold between objects denoted by the nodes.

john 5Sue

age

mother

mother(john,sue)age(john,5)wife(sue,max)age(max,34)...

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age

father

Max

wifehusband

age

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Semantic Networks• The ISA (is-a) or AKO (a-

kind-of) relation is often used to link instances to classes, classes to superclasses

• Some links (e.g. hasPart) are inherited along ISA paths.

• The semantics of a semantic net can be relatively informal or very formal– often defined at the

implementation level

isa

isa

isaisa

Robin

Bird

Animal

RedRusty

hasPart

Wing

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Reification

• Non-binary relationships can be represented by “turning the relationship into an object”

• This is an example of what logicians call “reification”– reify v : consider an abstract concept to be real

• We might want to represent the generic give event as a relation involving three things: a giver, a recipient and an object, give(john,mary,book32)

give

mary book32

john

recipient

giver

object

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Individuals and Classes

• Many semantic networks distinguish–nodes representing individuals and those representing classes

–the “subclass” relation from the “instance-of” relation

subclass

subclass

instanceinstance

Robin

Bird

Animal

RedRusty

hasPart

Wing

instance

Genus

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Inference by Inheritance

• One of the main kinds of reasoning done in a semantic net is the inheritance of values along the subclass and instance links.

• Semantic networks differ in how they handle the case of inheriting multiple different values.– All possible values are inherited, or

– Only the “lowest” value or values are inherited

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Conflicting inherited values

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Multiple inheritance

• A node can have any number of superclasses that contain it, enabling a node to inherit properties from multiple “parent” nodes and their ancestors in the network.

• These rules are often used to determine inheritance in such “tangled” networks where multiple inheritance is allowed:– If X<A<B and both A and B have property P, then X inherits A’s

property.

– If X<A and X<B but neither A<B nor B<Z, and A and B have property P with different and inconsistent values, then X does not inherit property P at all.

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From Semantic Nets to Frames

• Semantic networks morphed into Frame Representation Languages in the ‘70s and ‘80s.

• A frame is a lot like the notion of an object in OOP, but has more meta-data.

• A frame has a set of slots.

• A slot represents a relation to another frame (or value).

• A slot has one or more facets.

• A facet represents some aspect of the relation.

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Facets• A slot in a frame holds more than a value.

• Other facets might include:– current fillers (e.g., values)

– default fillers

– minimum and maximum number of fillers

– type restriction on fillers (usually expressed as another frame object)

– attached procedures (if-needed, if-added, if-removed)

– salience measure

– attached constraints or axioms

• In some systems, the slots themselves are instances of frames.

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Description Logics

• Description logics provide a family of frame-like KR systems with a formal semantics.– E.g., KL-ONE, LOOM, Classic, …

• An additional kind of inference done by these systems is automatic classification– finding the right place in a hierarchy of objects for a new

description

• Current systems take care to keep the languages simple, so that all inference can be done in polynomial time (in the number of objects)– ensuring tractability of inference

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Conceptual graphs (CGs)

• Besides Description Logics, conceptual graphs can be viewed as descendants of frame systems and semantic networks.

• They are given a formal semantics, e.g., by translating them into (first-order) formulae

• Since CGs can express all of first-order predicate logic, these reasoning problems are undecidable for general CGs

• The most prominent decidable fragment of CGs is the class of simple conceptual graphs (SGs)

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Comparison of DLs and CGs• (i) CGs are translated into closed first-order formulae, whereas

Description Logic concepts are translated into formulae in one free variable;

• (ii) since Description Logics use a variable-free syntax, certain identifications of variables expressed by cycles in SGs and by co-reference links in CGs cannot be expressed in Description Logics;

• (iii) in contrast to CGs, most Description Logics considered in the literature only allow for unary and binary relations but not for relations of arity greater than 2;

• (iv) SGs are interpreted by existential sentences, whereas almost all Description Logics considered in the literature allow for universal quantification.

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Simple Conceptual Graphs

• They are defined with respect to a so-called support.

• Roughly spoken, the support is a partially ordered signature that can be used to fix the a primitive ontology of a given application domain.

• It introduces a set of concept types (unary predicates), a set of relation types (n-ary predicates), and a set of individual markers (constants).

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cont

• x0x1x2:(Female(x0) ^ Human(x1) ^ Human(x2) ^ hasChild(x1; x0) ^ hasChild(x0; x2) ^ likes(x2; x1));

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Subsumption

• Subsumption with respect to a support S for two SGs g, h is defined by a so-called projection from h to g : g is subsumed by h iff there exists a mapping from h to g that (1) maps concept nodes marker, because the two resulting generic concept nodes would be interpreted by different variables.

(2) preserves adjacency.

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Logical formalisms

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Modal logics

• The semantics of modal formulae is given by M =(S,Pi,K)

• S is a set of so-called states or worlds (corresponds to individuals)

• Pi is a mapping from the set of propositional letters into sets of states

• K is a binary relation on the states S, the so-called accessibility relation (which can be seen as the interpretation of a single role)

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cont

• Propositional Dynamic Logics

• The -Calculus

• Temporal Logics

• Hybrid Logics

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Guarded fragments

• Guarded fragments is a natural generalizations of modal logics to relations of arbitrary arity.

• Guarded fragments are obtained from first order logic by allowing the use of quantified variables only if these variables are guarded by appropriate atoms before they are used in the body of a formula

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Database models

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Semantic data models (Entity-Relationship)

• The basic elements of the ER model are entities, relationships, and attributes

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Correspondence with Description Logics

• The ER model allows for relations of arbitrary arity, while in traditional Description Logics only unary and binary relations are considered.

• The assumption of acyclicity is unrealistic in an ER shema, while it is common in Description Logics knowledge bases.

• Database states are considered to be finite structures, while no assumption on finiteness is usually made on the interpretation domain of a Description Logic knowledge base

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Object-oriented data models

• Object-oriented data models have been proposed recently with the goal of devising database formalisms that could be integrated with object-oriented programming systems

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cont

• An object-oriented schema is a finite set of class declarations, which impose constraints on the instances of the classes that are used to model the application domain.

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Semistructured data models and XML

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XML

• The structure of the data is usually not rigid, as in conventional databases, and thus it is difficult to describe it using traditional data models

• The Extensible Markup Language (XML) , which has been introduced as a mechanism for representing structured documents on the web, can in fact also be considered a model for semistructured data.

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Relationship between XML and Description Logics

• The encoding of DTDs into Description Logics can be exploited to verify different kinds of properties on DTDs, namely inclusion, equivalence, and disjointness between the sets of documents conforming respectively to two DTDs.