Alternative representations: Semantic networks A semantic net is a labeled directed graph, where each node represents an object (a proposition), and each link represents a relationship between two objects. Example: sister-of husband-of wife-of husband-of mother-of father-of mother-of wife-of father-of mother-of father-of Note that only binary relationships can be represented in this model. Ann Carol David Bill Tom Susan John

Semantic nets represent propositional information Semantic nets represent propositional information. Relations between propositions are of primary interest because they provide the basic structure for organizing knowledge. Some important relations are: “IS-A” (is an instance of). Refers to a member of a class, where a class is a group of objects with one or more common attributes (properties). For example, “Tom IS-A bird”. “A-KIND-OF”. Relates one class to another, for example “Birds are A-KIND-OF animals”. “HAS-A”. Relates attributes to objects, for example “Mary HAS-A cat”. “CAUSE”. Expresses a causal relationship, for example “Fire CAUSES smoke”. Note that semantic nets can be easily converted into a set of FOL formulas, and vice versa. Semantic nets, however, have two important advantages: A very simple execution model. Very readable representation, which makes it easy to visualize inference steps.

Inference in semantic networks. The inference procedure in semantic nets is called inheritance, and it allows one node’s characteristics to be duplicated by a descendent node. Example: Consider a class “aircraft”, and assume that “balloons”, “propeller-driven objects” and “jets” are subclasses of it, i.e. “Balloons are A-KIND-OF aircrafts” “Propeller-driven objects are A-KIND-OF aircrafts”, etc. Assume that the following attributes are assigned to aircrafts: “Aircraft IS-A flying object”, “Aircraft HAS-A wings”, “Aircraft HAS-A engines” All properties assigned to the superclass, “aircraft”, will be inherited by its subclasses, unless there is an “exception” link capturing a non-monotonic inference relation.

Multiple inheritance may result in a conflicting inference In some semantic networks, one class can inherit properties of more than one superclass. The “Nixon diamond” example: It is widely accepted that Quakers tend to be pacifists, and Republicans tend not to be. Nixon is known to be both - a Quaker, and a Republican. Not IS-A IS-A IS-A IS-A The resulting conflict can be resolved only if additional information stating a preference to one of the conflicting inferences is provided. Pacifists Republicans Quakers Nixon

Object-attribute-value triplets One problem with semantic nets is that there is no standard definition of link names. To avoid this ambiguity, we can restrict this formalism to a very simple kind of a semantic network, which has only two types of links, “HAS-A” and “IS-A”. Such a formalism is called Object-Attribute-Value (OAV) triplets, and it is a widely used mode of knowledge representation (especially for representing declarative knowledge). It is also the base of the Semantic Web KR model, called RDF. Example: Consider object “airplane”. Some of its attributes are: number of engines; type of engines; type of wing design. Possible values of these attributes are: number of engines: 2, 3, 4. type of engines: jet, propeller-driven. type of wing design: conventional, swept-back.

Object-attribute-value triples (example contd) Object Attribute Value Airplane NumberOfEngines 2 Airplane NumberOfEngines 3 Airplane NumberOfEngines 4 Airplane TypeOfEngines Jet Airplane TypeOfEngines Propeller Airplane TypeOfWings Conventional Airplane TypeOfWings SweptBack Or, as predicates NumberOfEngines (Airplane, 2), … TypeOfEngines (Airline, Jet), … TypeOfWings (Airlines, Propeller), …

Problems with semantic nets and OAV triplets There is no standard definition of link and node names. This makes it difficult to understand the network, and whether or not it is designed in a consistent manner. Inheritance is a combinatorially explosive search, especially if the response to a query is negative. Plus, it is the only inference mechanism built in semantic nets, which may be insufficient for some applications. Initially, semantic nets were proposed as a model of human associative memory (Quinllian, 1968). But, are they an adequate model? It is believed that human brain contains about 10^10 neurons, and 10^15 links. Consider how long it takes for a human to answer “NO” to a query “Are there trees on the moon?” Obviously, humans process information in a very different way, not as suggested by the proponents of semantic networks. Semantic nets are logically and heuristically very weak. Statements such as “Some books are more interesting than others”, “No book is available on this subject”, “If a fiction book is requested, do not consider books on history, health and mathematics” cannot be represented in a semantic network.

Frames (Minsky, 1975) Semantic nets represent shallow knowledge, because all of the information must be represented in terms of nodes and links which are propositions. What if the objects in the domain are, in turn, complex structures? For example, consider object “animal”. We may want to incorporate as part of the object’s description all of the important properties of this object. Example: AIMA (first edition), page 318, Figure 10.7 Note that “frames” comprising the nodes of this frame-based network represent “typical examples” or “stereotypes” of described objects. Such typical example are called concepts, and they are like data structures where data, in turn, contain data. The underlying assumption of the frame theory is that when one encounters a new situation (or a substantial change in one’s view of the situation occurs), one selects from his/her memory the frame representing a given concept and changes it to reflect the new reality. Note: Case-based reasoning systems exclusively utilize frames as a knowledge representation formalism.

Problems with Frames Frames cannot represent exceptions. By definition, frames represent typical objects. But, consider Clyde, an elephant with 3 legs. Is he NOT an elephant? Frames lack a well-defined semantics. Consider Sam, a bat. Is he a typical mammal? It is difficult to incorporate heuristic information into the frame. For example, consider a medical ES which uses frames to represent a typical patient or a typical disease. It is difficult to represent how specific symptoms relate to each other and how they must be used in the diagnostic process. In general, frames are good for representing class hierarchies to model domains where objects are well defined and the classification is clear cut.

Description logics Description logics are decidable fragments of FOL intended to represent categories (classes/concepts), the relations between classes (roles) and individuals. Example: Consider category “Student”. One possible definition of this category is Student  And(takes-classes, does-homeworks, is-responsible) Note that this definition can be translated into the following FOL sentence  x (Student(x) <=> Takes-classes(x) & Does-homeworks(x) & & Is-responsible(x)) Example: Consider category of men with at least three sons who are all unemployed and married to doctors, and at most two daughters who are all professors in physics or chemistry departments. Man3SS2SD  And(Man, At-Least(3, Son), At-Most(2, Daughter), All(Son, And(Unemployed, Married, All(Spouse, Doctor))), All(Daughter, And(Professor, Fills(Department, Physics, Chemistry)))). DLs is a family of logics which defer by their expressivity depending on the constructors employed to build complex descriptions from simple ones.

General Architecture of Description Logics DL knowledge base consists of 3 parts: The TBox, which contains terminological knowledge (i.e. knowledge about concepts in a domain). Examples: Student  Person ⊓  takesCourse . Course Student ⊑ Person The ABox, which contains assertion knowledge (i.e. knowledge about individuals). Example: Student (Bob), takesCourse (Bob, ComputerScience) The RBox, R, which contains role-centric knowledge (i.e. knowledge about interdependences between roles/properties) Example: teachesGradCourse ⊑ teachesAtUniv RBox is not required; only more expressive DLs have constructors to handle RBoxes. The smallest deductively complete DL is called ALC. It contains ⊓, ⊔, and  constructors, and quantifiers restricting the domain and the range of roles, i.e.  R . C and  R . C It also contains class inclusion axiom (ex. Student ⊑ Person) and class equivalence axiom (ex. PhDThesis  Dissertation) ALC -- Attributive Language with Complements

ALC (Attributive Language with Complements): Syntax ALC Atomic types: Concept names: A, B, C, … Top and bottom concepts, ᴛ and  Role names: R, S, … ALC Constructors: ⊓, ⊔, ,  R . C and  R . C Class inclusion and equivalence axioms: ⊑,  Complex class relations are built from atomic types and ⊓, ⊔,  Example: Instructor ⊑ (Staff ⊓ Professor) ⊔ Lecturer In FOL: x Instructor(x)  (Staff(x)  Professor(x))  Lecturer(x) Quantifiers on Roles: Strict binding of the range of a role to a class. Example: A thesis must be authored by a student, i.e. Thesis ⊑  author . Student In FOL: x (Thesis(x)  y(author(x, y)  Student(y))) Open binding of the range of a role to a class. Example: Every student has at least one advisor, i.e. Student ⊑  advisor . Professor In FOL: x (Student(x)  y (advisor(x, y)  Professor(y)))

ALC: Formal Syntax Production rules for creating classes in ALC : C, D ::= A | T | | C | C ⊓ D | C ⊔ D |  R . C |  R . C where A is an atomic class, C and D are complex classes, and R is a role. An ALC TBox contains assertions of the forms C ⊑ D and C  D An ALC ABox contains assertions of the form C(a) and R(a, b), where a and b are individuals. An ALC Knowledge Base = {TBox, ABox}.

ALC: Semantics An interpretation, I = (I, .I), is defined by: A domain of individuals, I An interpretation function .I which maps Individual names, a, to domain elements aI  I Class names, C, to a set of domain elements CI  I Role names, R, to a set of pairs of domain elements RI  I  I An interpretation, I, for axioms: C(a) holds iff aI  CI R(a, b) holds iff (aI, bI)  RI C ⊑ D holds iff CI  DI C  D holds iff CI = DI An interpretation, I, for complex classes is defined as follows: TI = I and I =  (C ⊓ D)I = CI  DI , (C ⊔ D)I = CI  DI , ( C)I = I \ CI  R . C = {a  I |(  b  I ) ((a, b)  RI  b  CI )}  R . C = {a  I | (b  I ) ((a, b)  RI  b  CI )} The Model-theoretic semantics for ALC is defined via interpretaions

More DL constructors that are beyond ALC Number restrictions for roles. Example:  20 hasStudent Qualified number restrictions for roles. Example: 6 hasStudent . Graduate Nominals (definition by enumeration): {CS151, CS152, CS253} Concrete domains (datatypes): hasStudent . ( 20) Inverse roles: hasChild-  hasParent Transitive roles: has Ancestor ⊑+ has Ancestor Role composition: hasParent . hasBrother (uncle) Constructors allowed define the specific DL. Examples S: ALC + Transitive roles; R: Role constructors; O: Nominals; I: Inverse roles; Q: Qualified number restrictions; (D): datatypes. SROIQ(D) is the DL behind the latest version of OWL 2.

Questions DLs inference must address Does the knowledge base make sense? Are there empty classes? Are two classes equivalent? Does one class subsume another class? Are two classes disjoined? Is a given individual a member of a specified class? Find all individuals of a given class. The so-called tableaux algorithm (initially developed for FOL) is adapted to DLs to guarantee that these questions are answered in finite time. Tableaux algorithm proves unsatisfiability of a KB using the refutation method and transformation rules similar to Wang’s algorithm (called here tableaux extension rules)

Tableaux algorithm: PL example Given an expression in disjunctive normal form, construct the tableaux / decision tree where each node is a logical formula, each path from the root note to a leaf note is a conjunction of formulas, and each branch of a path is a disjunction of formulas. To build the tree, we apply the following rules: -- -Rules: (A & B) => A, B Example: (A  B) => A, B (A  B) => A, B (A & B & C)  (D & A)  (B & D) -- -Rules: A -rule A  B B (A & B & C) (D & A)  (B & D) A -rule -rule (A & B) B A (D & A) (B & D) A A  B B D B B -- Double negation rule: A => A C A D -- True => False, False => True

Tableaux Algorithm for ALC Step 1: Translate the DL KB to a negation normal form (NNF) using the following transformations: Substitute C  D by C ⊑ D and D ⊑ C Substitute C ⊑ D by C ⊔ D Apply all possible NNF transformations to complex classes: NNF ( C)  NNF (C) NNF (C ⊔ D)  NNF (C) ⊔ NNF (D) NNF (C ⊓ D)  NNF (C) ⊓ NNF (D) NNF ((C ⊔ D))  NNF (C) ⊓ NNF (D) NNF ((C ⊓ D))  NNF (C) ⊔ NNF (D) NNF ( R . C)  ( R) . NNF (C) NNF ( R . C)  ( R) . NNF (C) NNF ( R . C)  ( R) . NNF (C) NNF ( R . C)  ( R) . NNF (C)

Tableaux Algorithm for ALC (contd.) Step 2: Apply the following tableaux expansion rules to derive new ABox facts until no more rules can be applied (in which case we say that the tableaux is fully expanded). ⊓ -Rule if (C ⊓ D) (a)  ABox and {C(a), D(a)}  ABox, then ABox’ = Abox  {C(a), D(a)} ⊔ -Rule if (C ⊔ D) (a)  ABox and {C(a), D(a)}  ABox = , then ABox’ = ABox  {C(a)} and ABox” = ABox  {D(a)} -Rule if  R . C(a)  ABox and there is no b such that C(b)  ABox and R(a, b)  ABox then Abox ‘ = Abox  {C(z), R(a, z)} for a new individual z  ABox -Rule if R . C(a)  ABox and R(a, b)  Abox, but C(b) ABox then Abox ‘ = Abox  {C(b)} The algorithm returns TRUE if there is a clash-free tableaux and FALSE if the tableaux is closed. The tableaux is closed if all its paths are closed, where a path is closed if a formula and its negation occur along that path, or if false occurs. To prove X, we must obtain a closed tableaux for X.

Example Let TBox = {Professor ⊑ (Person ⊓ Staff) ⊔ (Person ⊓ Student)} Prove: Professor ⊑ Person Step 1: Negate the conclusion you want to prove.  (Professor ⊑ Person) Step 2: Convert all formulas in the NNF: TBox = { Professor ⊔ (Person ⊓ Staff) ⊔ (Person ⊓ Student)} NNF ( (Professor ⊑ Person)) = NNF ( (Professor ⊔ Person)) = = NNF (Professor) ⊓ NNF ( Person) = Professor ⊓  Person Step 3: Apply the tableaux extension rules to derive new ABox facts from the TBox extended with the negated theorem.

Example: step 3 (contd.) Professor(a) ⊓  Person(a) (Consider a new individual a for whom you are disproving the theorem) (2) Apply ⊓-rule to (1): Professor(a) closed path Apply ⊓-rule to (1):  Person(a) closed path Professor(a) ⊔ (Person(a) ⊓ Staff(a)) ⊔ (Person(a) ⊓ Student(a)) Apply ⊔-rule to (4): Professor(a) (6) Apply ⊔-rule to (4): (Person(a) ⊓ Staff(a)) ⊔ (Person(a) ⊓ Student(a)) (7) Apply ⊔-rule to (6): (Person(a) ⊓ Staff(a)) (8) Apply ⊔-rule to (6): (Person(a) ⊓ Student(a)) closed path (9) Apply ⊓-rule to (7): Person(a) (10) Apply ⊓-rule to (7): Staff(a) (11) Apply ⊔-rule to (8): Person(a) (12) Apply ⊔-rule to (8): Student(a) No more rules can be applied – the tableaux is fully expanded and closed. Therefore, the original theorem is proved.

Advantages and disadvantages of description logics The two inference tasks in DLs, subsumption (i.e. checking if one category is a subset of another based on their descriptions) and classification (i.e. checking if an object belongs to a category) have polynomial complexity. Some DLs may include also consistency checking, i.e. whether a category definition is satisfiable. DLs have clear semantics, which makes them a good KR formalism for applications with large declarative component. The Semantic Web language OWL is build upon DLs. Disadvantages: Polynomial inference is assured by the definitions of categories. If they are small and correctly defined, then the inference tasks are easy to carry out. But, if categories are hard to define in a concise manner, then their descriptions may become (exponentially) large. Weak inference capabilities (to preserve decidability), which cannot handle imprecisely specified concepts and not fully enforced relations between them.

Subsumption example Given the following KB: Woman  Person ⊓ Female Man  Person ⊓ Woman Mother  Woman ⊓ hasChild . Person Father  Man ⊓ hasChild . Person Parent  Mother ⊔ Father Grandmother  Woman ⊓ hasChild . Parent Prove: Grandmother ⊑ Person. 1. Negate the goal and add it to the KB, i.e. add (Grandmother ⊓ Person). 2. Translate the KB into the negation normal form i.) substitute A  B with A ⊑ B and B ⊑ A Woman ⊑ Person ⊓ Female Person ⊓ Female ⊑ Woman Man ⊑ Person ⊓ Woman Person ⊓ Woman ⊑ Man Mother ⊑ Woman ⊓ hasChild . Person Woman ⊓ hasChild . Person ⊑ Mother Father ⊑ Man ⊓ hasChild . Person Man ⊓ hasChild . Person ⊑ Father Parent ⊑ Mother ⊔ Father Mother ⊔ Father ⊑ Parent Grandmother ⊑ Woman ⊓ hasChild.Parent Woman ⊓ hasChild.Parent ⊑ Grandmother

Subsumption example (contd.) ii.) substitute A ⊑ B with A ⊔ B and then apply all possible transformations to complex classes to bring negation down to atomic classes (see Lecture 7): Woman ⊔ (Person ⊓ Female) (Person ⊓ Female) ⊔ Woman  Person ⊔ Female ⊔ Woman Man ⊔ (Person ⊓ Woman)  (Person ⊓ Woman) ⊔ Man  Person ⊔ Female ⊔ Man Mother ⊔ (Woman ⊓ hasChild . Person)  (Woman ⊓ hasChild . Person) ⊔ Mother  Woman ⊔ hasChild.Parent ⊔ Mother Father ⊔ (Man ⊓ hasChild . Person)  (Man ⊓ hasChild . Person) ⊔ Father  Man ⊔ hasChild.Parent ⊔ Father Parent ⊔ (Mother ⊔ Father)  Parent ⊔ Mother ⊔ Father  (Mother ⊔ Father) ⊔ Parent  (Mother ⊓ Father) ⊔ Parent Grandmother ⊔ (Woman ⊓ hasChild.Parent) (Woman ⊓ hasChild.Parent) ⊔ Grandmother  Woman ⊔ hasChild.Parent ⊔ Grandmother

The (beginning of the) proof Grandmother(a) ⊓ Person(a) # An individual a for whom we are disproving the # theorem. (2) Apply ⊓-rule to (1): Grandmother(a) (3) Apply ⊓-rule to (1): Person(a) closed path (4) Grandmother ⊔ (Woman ⊓ hasChild.Parent) (5) Apply ⊔-rule to (4) : Grandmother(a) (6) Apply ⊔-rule to (4) : Woman ⊓ hasChild.Parent (7) Apply ⊓-rule to (6): Woman(a) (8) Apply ⊓-rule to (6): hasChild.Parent (9) Woman ⊔ (Person ⊓ Female) closed path (10) Apply ⊔-rule to (9): Woman(a) (11) Apply ⊔-rule to (9): Person ⊓ Female closed path (12) Apply ⊓-rule to (11): Person(a) (12) Apply ⊓-rule to (11): Female(a) (13) Person ⊔ Female ⊔ Woman all three paths (14) Apply ⊔-rule to (13): Person(a) are closed (14) Apply ⊔-rule to (13): Female(a) (14) Apply ⊔-rule to (13): Woman(a)

Other reasoning tasks Note: the tableau above is not fully expanded. There are more formulas that can be transformed by means of ⊓-rule, ⊔-rule, -rule and -rule. To prove the original goal, we must show that all paths of the fully expanded tableau are closed. Although fully mechanical, this process is very tedious for a human – automated theorem provers like Pellet and Fact++ (both employed in Protégé) do a much better job. The classification task can be reduced to inconsistency checking as well. To prove that individual x is a member of class A, we can prove that A(x) is inconsistent instead. To find all members of a class (task called instance generation/retrieval), we must show A(x) for all x. To prove that a class A is inconsistent (i.e. has no members), we can add A(x) to the KB and show that the resulting set is inconsistent (i.e. assume that there is a member x  A and proof a contradiction). To prove A ⊔ B (class disjointness), we can prove that adding A ⊓ B to the KB makes it inconsistent.

Inconsistency example Consider the following KB: Bear  Animal ⊓ Omnivore Omnivore  eat . Animal Panda  Bear ⊓ Vegetarian Vegetarian  eat . Animal It is easy to see that this KB is inconsistent. Assume that there exists an individual x which is a member of the class Panda. Then: Panda(x)  Bear(x) AND Vegetarian(x) Vegetarian(x)  x does not eat animals Bear(x)  Animal(x) AND Omnivore(x) Contradiction, Omnivore(x)  x eats animals therefore class Panda must be empty. Note: reasoning over defined classes ONLY!