### Agenda

• Motivation

• Technical Solution

• Introduction to Theorem Proving and Resolution

• Description Logics

• Logic Programming

• Summary

• References

### Motivation

• Basic results of mathematical logic show:
• We can do logical reasoning with a limited set of simple (computable) rules in restricted formal languages like First-order Logic (FOL)
• Computers can do reasoning

• FOL is interesting for this purpose because:
• It is expressive enough to capture many foundational theorems of mathematics
• Many real-world problems can be formalized in FOL
• It is the most expressive logic that one can adequately approach with automated theorem proving techniques

### Motivation

• Due to its theoretical properties (decidability & complexity) First-Order Logic is not always an ideal solution
• This motivates research towards formalisms with more practically oriented computitional properties and expressivity
• Description Logics
• Syntactic fragments of FOL
• Focus on decidability and optimized algorithms key reasoning tasks (terminological reasoning / schema reasoning)
• Logic Programming
• Provides different expressivity than classical FOL (non-monotonic reasoning with non-classical negation)
• Provides a very intuitive way to model knowledge
• Efficient reasoning procedures for large data-sets

### Introduction – Basic Notions

• A proof system is collection of inference rules of the form:
 P 1 … P n ____________ C

where C is a conclusion sequent, and P i ‘s are premises sequents .

• If an infererence rule does not have any premises that are taken to be true (called an axiom ), its conclusion automatically holds.
• Example:
Modus Ponens: From P, P → Q infer Q,
Universal instantiation: From (∀x)p(x) infer p(A)
• Theorems:
• Expressions that can be derived from the axioms and the rules of inference.

### Resolution - Principle

• Resolution is a refutation system .

• To prove a statement we attempt to refute its negation

• Basic idea: Given a consistent set of axioms KB and goal sentence Q , we want to show that KB Q

• This means that every interpretation I that satisfies KB also satisfies Q

• Any interpretation I only satisfies either Q or ¬ Q, but not both

• Therefore, if in fact KB Q holds , an interpretation that satisfies KB , satisfies Q and does not satisfy ¬ Q

Hence KB { ¬ Q } is unsatisfiable, i.e., it is false under all interpretations

### Resolution - Principle

• Resolution commonly requires sentences to be in clause normal form (also called conjunctive normal form or CNF)
• A clause is a disjunction of literals (implicitly universally quantified)
• E.g. : l1 ∨ … ∨ ln
• A formula in clause normal form conjunction of a set of clauses
• E.g.: C1 ∧ … ∧ Cn = (ln ∨ … ∨ lm) ∧ … ∧ (li ∨ … ∨ lj)
• High level steps in a resolution proof:
• Put the premises or axioms into clause normal form (CNF)
• Add the negation of the to be proven statement, in clause form, to the set of axioms
• Resolve these clauses together, using the resolution rule , producing new clauses that logically follow from them
• Derive a contradiction by generating the empty clause.
• The substitutions used to produce the empty clause are those under which the opposite of the negated goal is true

### Resolution - The Resolution Rule

• The resolution rule is a single inference that produces a new clause implied by two clauses (called the parent clauses) containing complementary literals

• Two literals are said to be complements if one is the negation of the other (in the following, ai is taken to be the complement to bj).
• The resulting clause contains all the literals that do not have complements. Formally:

• Where a1, …, an, b1, …, bm are literals, and ai is the complement to bj
• The clause produced by the resolution rule is called the resolvent of the two input clauses.
• A literal and its negation produce a resolvent only if they unify under some substitution σ.
• σ Is then applied to the resolvent

### Resolution - The Resolution Rule

• When the two initial clauses contain more than one pair of complementary literals, the resolution rule can be applied (independently) for each such pair

• However, only the pair of literals that are resolved upon can be removed: All other pairs of literals remain in the resolvent clause

• Modus ponens (see last lecture) can be seen as a special case of resolution of a one-literal clause and a two-literal clause

• Example: Given clauses

• P(x, f(a)) v P(x, f(y)) v Q(y) and ¬P(z, f(a)) v ¬ Q(z)

• P(x, f(a)) and ¬P(z, f(a)) unify with substitution σ = {x/z}

• Therefore, we derive the resolvent clause (to which σ is applied):

• P(z, f(y)) v Q(y) v ¬ Q(z)

### Resolution – Complete Resolution Algorithm

1. Convert all the propositions of a knowledge base KB to clause normal form (CNF)

2. Negate Q (the to be proven statement) and convert the result to clause form. Add it to the set of clauses obtained in step 1.

3. Repeat until either a contradiction is found, no progress can be made, or a predetermined amount of effort has been expended:

a) Select two clauses. Call these the parent clauses.

b) Resolve them together using the resolution rule . The resolvent will be the disjunction of all the literals of both parent clauses with appropriate substitutions.

c) If the resolvent is the empty clause , then a contradiction has been found. If it is not, then add it to the set of clauses available to the procedure.

### Resolution – Algorithm in Pseudocode

 procedure resolution-refutation(KB, Q) ;; KB a set of consistent, true sentences ;; Q is a goal sentence that we want to derive ;; return success if KB ⊢ Q, and failure otherwise KB = union(KB, ¬Q) while false not in KB do pick 2 sentences, S1 and S2, in KB that contain literals that unify if none return "failure“ resolvent = resolution-rule(S1, S2) KB = union(KB, resolvnt) return "success" 

### Resolution - Converting to Clause Normal Form

• Resolution expects input in clause normal form
• Step 1: Eliminate the logical connectives → and ↔
• a b = (a b) (b a)
• a b = ¬ a ∨ b
• Step 2: Reduce the scope of negation
• ¬ ( ¬ a) = a
• ¬ (a b) = ¬ a ¬ b
• ¬ (a b) = ¬ a ¬ b
• ¬ ( X) a(X) = ( X) ¬ a(X)
• ¬ ( X) b(X) = ( X) ¬ b(X)

### Resolution - Converting to Clause Normal Form

• Step 3: Standardize by renaming all variables so that variables bound by different quantifiers have unique names

• ( X) a(X) ( X) b(X) = ( X) a(X) ( Y) b(Y)

• Step 4: Move all quantifiers to the left to obtain a prenex normal form

• A formula is inn prenex normal form if it is written as a string of quantifiers followed by a quantifier-free part

• Step 5: Eliminate existential quantifiers by using skolemization

### Resolution - Converting to Clause Normal Form

• Step 5: Skolemization
• Skolem constant
• (∃X)(dog(X)) may be replaced by dog(fido) where the name fido is picked from the domain of definition of X to represent that individual X.
• Skolem function
• If the predicate has more than one argument and the existentially quantified variable is within the scope of universally quantified variables, the existential variable must be a function of those other variables.
(∀X)(∃Y)(mother(X,Y)) ⇒ (∀X)mother(X,m(X))
(∀X)(∀Y)(∃Z)(∀W)(foo (X,Y,Z,W))
⇒ (∀X)(∀Y)(∀W)(foo(X,Y,f(X,Y),w))

### Resolution - Converting to Clause Normal Form

• Step 6: Drop all universal quantifiers
• Step 7: Convert the expression to the conjunctions of disjuncts
(a ∧ b) ∨ (c ∧ d)
= (a ∨ (c ∧ d)) ∧ (b ∨ (c ∧ d))
= (a ∨ c) ∧ (a ∨ d) ∧ (b ∨ c) ∧ (b ∨ d)
• step 8: Split each conjunction into a separate clauses, which are just a disjunction of negated and nonnegated predicates, called literals
• step 9: Rename so that no variable symbol appears in more than one clause.
(∀X)(a(X) ∧ b(X)) = (∀X)a(X) ∧ (∀ Y)b(Y)

### Resolution - Complete Example

• (Nearly) Classical example: Prove “Fido will die.” from the statements
• “Fido is a dog.”
• “All dogs are animals.”
• “All animals will die.”
• Changing premises to predicates
• (x) (dog(X) animal(X))
• dog(fido)
• Modus Ponens and {fido/X}
• animal(fido)
• (Y) (animal(Y) die(Y))
• Modus Ponens and {fido/Y}
• die(fido)

### Resolution – Complete Example

• Equivalent proof by Resolution
• Convert predicates to clause normal form
 Predicate form Clause form 1. ∀ (x) (dog(X) → animal(X)) ¬ dog(X) ∨ animal(X) 2. dog(fido) dog(fido) 3. ∀ (Y) (animal(Y) → die(Y)) ¬ animal(Y) ∨ die(Y)
• Negate the conclusion
4. ¬ die(fido)                      ¬ die(fido)

### Resolution – Further Considerations

• Resolution can be used to establish that a given sentence is entailed by a set of axioms
• However, it cannot , in general, be used to generate all logical consequences of a set axioms
• Also, the resolution cannot be used to prove that a given sentence is not entailed by a set of axioms

• Resolution defines a search space
• The decision which clauses will be resolved against which others define the operators in the space
• A search method is required

• Worst case: Resolution is exponential in the number of clauses to resolve

### Resolution – Strategies

• Order of clause combination is important

• N clauses N2 ways of combinations or checking to see whether they can be combined

• Search heuristics are very important in resolution proof procedures

• Strategies

• Set of Support Strategy

• Unit Preference Strategy

• Linear Input Form Strategy

### Concrete System: Prover9

• First-Order theorem prover
• Homepage: http://www.cs.unm.edu/~mccune/prover9/
• Successor of the well known “Otter” theorem prover
• Under active development, available for several platforms and released under the GPL
• Graphical user-interface and extensive documentation make Prover 9 comparatively user friendly
• Core of system builds on resolution / paramodulation as inference method
• Prover9 works in parallel with external component „Mace4“
• Mace4 searches for finite models and counterexamples
• This helps to avoid wasting time searching for a proof by first finding a counterexample

### Limitations of First-OrderTheorem Proving

• Entailment: KB ⊧ Q
• Entailment is a relation that is concerned with the semantics of statements
• Q is entailed by KB (a set of premises or assumptions) if and only if there is no logically possible world in which Q is false while all the premises in KB are true

• Entailment for FOL is only semi-decidable : If a conclusion follows from premises, then a complete proof system (like resolution) will find a proof.
• If there’s a proof, we’ll halt with it (eventually)
• However, If there is no proof (i.e. a statement does not follow from a set of premises), the attempt to prove it may never halt

### Limitations of First-OrderTheorem Proving

• From a practical point of view this is problematic
• We cannot distinguish between the non-existence of a proof or the failure of an implementation to simply find a proof in reasonable time.
• Theoretical completeness of an inference procedure does not make a difference in this cases
• Does a proof simply take too long or will the computation never halt anyway?

• Due to its complexity and remaining limitations FOL is often not suitable for practical applications

• Often less expressive (but decideable) formalisms or formalisms with different expressivity are more suitable:
• Description Logics
• Logic Programming

### Description Logic

• Most Description Logics are based on a 2-variable fragment of First Order Logic
• Classes (concepts) correspond to unary predicates
• Properties correspond to binary predicates
• Restrictions in general:
• Quantifiers range over no more than 2 variables
• Transitive properties are an exception to this rule
• No function symbols (decidability!)
• Most DLs are decidable and usually have decision procedures for key reasoning tasks
• DLs have more efficient decision problems than First Order Logic
• We later show the very basic DL ALC as example
• More complex DLs work in the same basic way but have different expressivity

### Description Logic Syntax

• Concepts/classes (unary predicates/formulae with one free variable)

• E.g. Person, Female

• Roles (binary predicates/formulae with two free variables)

• E.g. hasChild

• Individuals (constants)

• E.g. Mary, John

• Constructors allow to form more complex concepts/roles

• Union ⊔: Man ⊔ Woman

• Intersection ⊓ : Doctor ⊓ Mother

• Existential restriction ∃: ∃hasChild.Doctor (some child is a doctor)

• Value(universal) restriction ∀: ∀hasChild.Doctor (all children are doctors)

• Complement /negation¬: Man ⊑ ¬Mother

• Number restriction ≥n, ≤n

• Axioms

• Subsumption ⊑ : Mother ⊑ Parent

### Description Logic Syntax - Concepts

• Classes/concepts are actually a set of individuals

• We can distinguish different types of concepts:

• Atomic concepts: Cannot be further decomposed (i.e. Person)

• Incomplete concepts (defined by ⊑)

• Complete concepts (defined by ≡)

• Example incomplete concept defintion:

• Man ⊑ Person ⊓ Male

• Intended meaning: If an individual is a man, we can conclude that it is a person and male.

• Man(x) ⇒ Person(x) ∧ Male(x)

• Example complete concept definition:

• Man ≡ Person ⊓ Male

• Intended meaning: Every individual which is a male person is a man, and every man is a male person.

• Man(x) ⇔ Person(x) ∧ Male(x)

### Description Logic Syntax- Roles

• Roles relate to individuals to each other
• I.e. directedBy(Pool Sharks, Edwin Middleton), hasChild(Jonny, Sue)

• Roles have a domain and a range

• Example:
• Domain(directedBy, Movie)
• Range(directedBy, Person)
• Given the above definitions we can conclude that Pool Sharks is a move and that Edwin Middleton is (was) a person.

• Functional Roles
• Roles which have exactly one value
• Usually used with primitive datavalues
• A special case of (unqualified) number restriction ≤1 R

### Description Logic Syntax- Roles

• Transitive Roles
• Example: hasAncestor
Simple in a rule language: hasAncestor(X,Z) :- hasAncestor(X,Y), hasAncestor(Y,Z).
• Requires more than one variable!
• Transitivity can be captured in DLs by role hierarchies and transitive roles:

• Symmetric Roles
• Roles which hold in both directions
• I.e. hasSpouse, hasSibling
• Inverse Roles
• Roles are directed, but each role can have an inverse
• I.e. hasParent ≡ hasChild
hasChild(X,Y) hasChild(Y,X)

### Description Logic Knowledge Bases

• Typically a DL knowledge base (KB) consists of two components
• Tbox (terminology): A set of inclusion/equivalence axioms denoting the conceptual schema/vocabulary of a domain
• Bear ⊑ Animal ⊓ Large
• transitive(hasAncestor)
• hasChild ≡ hasParent

• Abox (assertions): Axioms, which describe concrete instance data and holds assertions about individuals
• hasAncestor(Susan, Granny)
• Bear(Winni Puh)

• From a theoretical point of view this division is arbitrary but it is a useful simplification

### A basic Description Logic - ALC

• Smallest propositionally closed DL is ALC
• Only atomic roles
• Concept constructors: ⊔ , ⊓, ¬
• Restricted use of quantifiers: ∃, ∀

• “Propositionally closed” Logic in general:
• Provides (implicitly or explicitly) conjunction, union and negation of class descriptions

• Example:
• Person ⊓ ∀hasChild.(Doctor ⊔ ∃hasChild.Doctor)

### A basic Description Logic - ALC

• What can we express in ALC?
• ALC concept descriptions can be constructed as following:

### A basic Description Logic - ALC

• Individual assertions :
• a ∈ C
• Mary is a Woman.

• Role assertions:
• 〈a, b〉 ∈ R
• E.g. Marry loves Peter.

• Axioms:
• C ⊑ D
• C ≡ D, because C ≡ D ⇔ C ⊑ D and D ⊑ C
• E.g.: A Dog is an animal. A man is a male Person.

### The Description Logic Family

• Description Logics are actually a family of related logics
• Difference in expressivity and features, as well as complexity of inference
• Description Logics follow a naming schema according to their features
• ALC = Attributive Language with Complements
• S often used for ALC extended with transitive roles
• Additional letters indicate other extensions, e.g.:
• H for role hierarchy
• O for nominals, singleton classes
• I for inverse roles (e.g., isChildOf ≡ hasChild–)
• N for number restrictions
• Q for qualified number restrictions
• F for functional properties
• R for limited complex role inclusion axioms, role disjointness
• (D) for datatype support

### Description Logic - Semantics

• Semantics follow standard FOL model theory

• Description Logics are a fragment of FOL

• The vocabulary is the set of names (concepts and roles) used

• I.e. Mother, Father, Person, knows, isRelatedTo, hasChild, …

• An interpretation I is a tuple (∆I, ⋅I)

• I is the domain (a set)

• I is a mapping that maps:

• Names of objects (individuals) to elements of the domain

• Names of unary predicates (classes/concepts) to subsets of the domain

• Names of binary predicates (properties/roles) to subsets of ∆I x ∆I

### Description Logic - Semantics

• The semantics of DL are based on standard First Order Model theory

• A translation is usually very straightforward, according to the following correspondences (for ALC):

• A description is translated to a first-order formula with one free variable

• An individual assertion is translated to a ground atomic formula

• An axiom is translated to an implication, closed under universal implication

• More complex DLs can be handled in a similar way

### Description Logic - Semantics

• Mapping ALC to First Order Logic:

### Description Logic - Reasoning

• Main reasoning tasks for DL systems:

• Satisfiability : Check if the assertions in a KB have a model

• Instance checking : Check if an instance belongs to a certain concept

• Concept satisfiability : Check if the definition of a concept can be satisfied

• Subsumption : Check if a concept A subsumes a concept B (if every individual of a concept B is also of concept A)

• Equivalence : A ≡ B ⇔ B ⊑ A and A ⊑ B

• Retrieval : Retrieve a set of instances that belong to a certain concept

### Description Logic - Reasoning

• Reasoning Task are typically reduced to KB satisfiability sat(A) w.r.t. to a knowledge base A

• Instance checking : instance(a,C, A) ⇔¬sat(A ⋃ {a: ¬ C})

• Concept satisfiability : csat(C) ⇔ sat(A ⋃ {a: ¬ C})

• Concept subsumption : B ⊑ A ⇔ A ⋃ {¬B ⊓ C} is not satisfiable ⇔ ¬sat(A ⋃ {¬B ⊓ C})

• Retrieval : Instance checking for each instance in the Abox

• Note: Reduction of reasoning tasks to one another in polynomial time only in propositionally closed logics

• DL reasoners typically employ tableaux algorithms to check satisfiability of a knowledge base

### Concrete System: Pellet

• Description Logic reasoner
• Homepage: http://clarkparsia.com/pellet
• Written in Java and available from Dual-licensed AGPL license for open-source applications
• Proprietary license available for commercial applications
• Sound and complete reasoner aimed at OWL-DL inference based on tableaux procedure
• Covers all constructs in OWL-DL
• Supporting major part of OWL 2 specification
• Integrates with popular toolkits and editors
• E.g. Jena, Protege, TopBraid Composer
• Comprehensive hands-on tutorial available:
• http://clarkparsia.com/pellet/tutorial/iswc09

### Concrete System: Pellet

• Pellet supports expected standard DL reasoning tasks

• Consistency, classification, realization

• Conjunctive query answering

• Concept satisfiability

• Additionally Pellet supports:

• SPARQL-DL queries

• Datatype reasoning

• User-defined datatypes

• N-ary datatype predicates

• Rules support (DL-safe SWRL rules)

• Explanation and debugging features

• Axiom pinpointing service

### Concrete System: Pellet

• Explanation & Debugging support

• Motivation: It is hard to understand large and/or complex ontologies

• Examples:

• Why is a specific subclass relation inferred?

• Why is an ontology inconsistent?

• Pellet provides axiom pinpointing service:

• For any inference, returns the (minimal set of) source axioms that cause the inference

• Applications can track additional provenance information

• Axiom pinpointing is the first step to explanations

• Precise justifications (pinpoint parts of axioms) are ongoing work in Pellet‘s developement

### Concrete System: Pellet

• Pellet is integrated in OwlSight and performs classification of loaded ontologies
• Lightweight , web-based ontology browser: http://pellet.owldl.com/owlsight/

### Concrete System: Pellet

• Classification Results in the Protege Editor

### Logic Programming

• What is Logic Programming?
• Various different perspectives and definitions possible:
• Computations as deduction
• Use formal logic to express data and programs
• Theorem Proving
• Logic programs evaluated by a theorem prover
• Derivation of answer from a set of initial axioms
• High level (non-precedural) programming language
• Logic programs do not specifcy control flow
• Instead of specifying how something should be computed, one states what should be computed
• Procedural interpretation of a declarative specification of a problem
• A LP systems procedurally interprets (in some way) a general declarative statement which only defines truth conditions that should hold

### Logic Programming Basics

• Logic Programming is based on a subset of First Order Logic called Horn Logic

• Horn Logic can serve as a simple KR formalism and allows to express

• IF <condition> THEN <result> rules

• Under certain restrictions reasoning over knowledge bases based on such rules is decideable (in contrast to general ATP within First Order Logic)

### Logic Programming Basics – Horn Logic

• Syntactically a LP rule is a First Order Logic Horn Clause
• The semantics of LP are different form the standard Tarski style FOL semantics (details later)
• Herbrand semantics differ from FOL smeantics in the structures it considers to be models (details later)
→ Minimal model semantics
• A FOL Horn clause is a disjunction of literals with one positive literal, with all variables universally quantified:
• (∀) ¬C1 ∨ … ∨ ¬Cn ∨ H
• This can be rewritten to closer correspond to a rule-like form:
• (∀) C1 ∧ … ∧ Cn → H
• In LP systems usually the following (non First Order Logic) syntax is used:
• H :- C1,...,Cn
• Such rules can be evaluated very efficiently , since the resolution of two Horn clauses is a again a Horn clause

### Logic Programming – Syntax

• The LP vocabulary consists of:
• Constants: b, cow, “somestring”
• Predicates: p, loves
• Function symbols: f, fatherOf
• Variables: x, y
• Terms can be:
• Constants
• Variables
• Constructed terms (i.e. function symbol with arguments)
• Examples:
• cow, b, Jonny,
• loves(John)

### Logic Programming – Syntax

• From terms and predicates we can build atoms:
• For n-ary predicate symbol p and terms t1, ..., tn, p(t1, ..., tn) is an atom
• A ground atom is an atom without variables
• Examples:
• p(x)
• loves(Jonny, Mary), worksAt(Jonny, SomeCompany)
• worksAt(loves(Mary), SomeCompany)
• Literals
• A literal is a an atom or its negation
• A positive literal is an atom
• A negative literal is a negated atom
• A ground literals is a literal without variables

### Logic Programming – Syntax

• Rules
• Given a rule of the form H :- B1,...,Bn we call
• H the head of the rule (its consequent)
• B1 …. Bn the body of the rule (the antecedent or conditions)
• The head of the rule consists of one positive literal H
• The body of the rule consists of a number of literals B1, ..., Bn
• B1, …, Bn are also called subgoals
• Examples:
• parent(x) :- hasChild(x,y)
• father(x) :- parent(x), male(x)
• hasAunt(z,y) :- hasSister(x,y), hasChild(x,z)

### Logic Programming – Syntax

• Facts denote assertions about the world:
• A rule without a body (no conditions)
• A ground atom
• Examples:
• hasChild(Jonny, Sue)
• Male(Jonny)).
• Queries allow to ask questions about the knowledge base:
• Denoted as a rule without a head:
• ?- B1,...,Bn.
• Examples:
• ? - hasSister(Jonny,y), hasChild(Jonny , z) gives all the sisters and children of Jonny
• ? - hasAunt(Mary,y) gives all the aunts of Mary
• ?- father(Jonny) ansers if Jonny is a father

### Logic Programming – Semantics

• There are two main approaches to define the semantics of LP
1. Model theoretic semantics
2. Computional semanitcs
• Model-theoretic semantics
• Defines the meaning of a model in terms of its minimal Herbrand model
• Computational semantics (proof theoretic semantics)
• Define the semantics in terms of an evaluation strategy which describes how to compute a model
• These two semantics are different in style, but agree on the minimal model
• LP semantics is only equivalent to standard FOL semantics
• Concerning ground entailment
• As long as LP is not extended with negation

### Logic Programming – Semantics

• Semantics of LP vs Semantics of FOL
• Semantics LP defined in terms of minimal Herbrand model
• Only one minimal model
• Semantics FOL defined in terms of First-Order models
• Typically, infinitely many First-Order models
• The minimal Herbrand model is a First-Order model
• Every Herbrand model is a First-Order model
• There exist First-Order models which are not Herbrand models
• Example:
• p(a), p(x) → q(x)
• The Minimal Herbrand model: {p(a), q(a)}
• This is actually the only Herbrand model!
• First-Order models: {p(a), q(a)}, {p(a), q(a), p(b)}, etc.

### Logic Programming – Semantics

• Recall:
• Terms not containing any variables are ground terms
• Atoms not containing any variables are ground atoms
• The Herbrand Universe U is the set of all ground terms which can be formed from
• Constancts in a program
• Function symbols in a program
• Example: a, b, c, f(a)
• The Herbrand Base B is the set of all ground atoms which can be built from
• Predicate symbols in a program
• Ground terms from U
• Example: p(a), q(b), q(f(a))

### Logic Programming – Semantics

• A Herbrand Interpretation I is a subset of the Herbrand Base B for a program

• The domain of a Herbrand interpretation is the Herbrand Universe U

• Constants are assigned to themselves

• Every function symbol is interpreted as the function that applies it

• If f is an n-ary function symbol (n>0) then the mapping from Un to U defined by (t1, …, tn) → f(t1, …, tn) is assigned to f

### Logic Programming – Semantics

• A Herbrand Model M is a Herbrand Interpretation which makes every formula true, so:
• Every fact from the program is in M
• For every rule in the program: If every positive literal in the body is in M, then the literal in the head is also in M
• The model of a Logic Program P is the minimal Herbrand Model
• This least Herbrand Model is the intersection of all Herbrand Models
• Every program has a Herbrand Model. Thus every model also has a minimal Herbrand Model.
• This model is uniquely defined only for programs without negation
⇒ A very intuitive and easy way to capture the semantics of LP
• As soon as negation is allowed a unique minimal model is not guaranteed anymore

### Logic Programming - Negation

• How do we handle negation in Logic Programs?
• Horn Logic only permits negation in limited form
• Consider (∀) ¬C1 ∨ … ∨ ¬Cn ∨ H
• Special solution: Negation-as-failure (NAF):
• Whenever a fact is not entailed by the knowledge base, its negation is entailed
• This is a form of “Default reasoning”
• This introduces non-monotonic behavior (previous conclusions might need to be revised during the inference process)
• NAF is not classical negation and pushes LP beyond classical First Order Logic
• This allows a form of negation in rules:
• (∀) C1 ∧ … ∧ Ci ∧ not Cn → H
• H :- B1, … Bi, not Bn

### Logic Programming - Negation

• Logic Programs can also contain recursion

• Example:
• ancestor (x,y) :- hasParent(x, y)
• ancestor(x,z) :- ancestor(x,y), ancestor(y,z).

• This is a problem as soon as negation is allowed since the minimal model is not uniquely defined anymore

• It is useful to consider this using a dependency graph
• A predicate is a node in the graph
• There is a directed edge between predicates q and p if they occur in a rule where q occurs in the head and p in the body.
• If the dependency graph contains a cycle then the program is recursive

### Logic Programming - Negation

• As soon as negation is allowed, cycles in a dependency graph become problematic.
• Example: What is the meaning of win(x) :- not win(x) ?

• A Solution: Stratification
• Mark edges with negation in the dependency graph
• Separate predicates which are connected through a positive edge in a individual stratum
• Strata can be (partially) ordered
• If each predicate occurs only in one stratum, then the program is called stratifiable
• Each stratum can be evaluated as usual and independently from other strata

→ This guarantees a unique interpretation of a Logic Program using negation

### Logic Programming - Subsets

• Classical Logic Programming
• Allows function symbols
• Does not allow negation
• Is turing complete

• Full Logic Programming is not decideable
• Prolog programs are not guaranteed to terminate

• Several ways to guarantee the evaluation of a Logic Program
• One is to enforce syntactical restrictions
• This results in subsets of full logic programming
• Datalog is such a subset

### Logic Programming - Datalog

• Datalog is a syntactic subset of Prolog
• Originally a rule and query language for deductive databases

• Considers knowledge bases to have two parts
• Extensional Database (EDB) consists of facts

• Intentional Database(IDB) consists of non-ground rules

• Restrictions:

1. Datalog disallows function symbols

2. Imposes stratification restrictions on the use of recursion + negation

3. Allows only range restricted variables ( safe variables )

• Safe Variables:
• Only allows range restricted variables, i.e. each variable in the conclusion of a rule must also appear in a not negated clause in the premise of this rule.
• This limits evaluation of variables to finitely many possible bindings

### Logic Programming - Reasoning Tasks

• The typical reasoning task for LP systems is query answering
• Ground queries, i.e. ?- loves(Mary, Joe)
• Non-ground query, i.e. ?- loves(Mary, x)

• Non-ground queries can be reduced to a series of ground queries
• ?- loves(Mary, x)
• Replace x by every possible value

• In Logic Programming ground queries are equivalent to entailment of facts
• Answering ?- loves(Mary, Joe) w.r.t. a knowledge base A is equivalent to checking

A ⊧ loves(Mary, Joe)

### Concrete Logic Programming System: IRIS

• Java based Datalog reasoner
• Developed at STI Innsbruck
• Freely available open source project
• Homepage: http://www.iris-reasoner.org/

• Extensions:
• Stratified / Well-founded default negation
• XML Schema data types
• Various built-in predicates (Equality, inequality, assignment, unification, comparison, type checking, arithmetic, regular expressions,... )

• Highly modular and includes different reasoning strategies
• Bottom-up evaluation with Magic Sets optimizations (forward-chaining)
• Top-down evaluation by SLDNF resolution (backward-chaining)

### Concrete Logic Programming System: IRIS

 An example of a concrete combination of components within IRIS: Program Optimization Rewriting techniques from deductive DB research (e.g. Magic sets rewriting) Safety Processing & Stratification Ensure specific syntactic restrictions Rule Re-ordering Minimize evaluation effort based on dependencies between expressions Rule Optimizations Join condition optimization Literal re-ordering Rule compilation Pre-indexing Creation of „views“ on required parts of information

### Concrete Logic Programming System: IRIS

• Example program (also see online demo at http://www.iris-reasoner.org/demo):
man('homer').
woman('marge').
hasSon('homer','bart').
isMale(?x) :- man(?x).
isFemale(?x) :- woman(?x).
isMale(?y) :- hasSon(?x,?y).
• Query:
?-isMale(?x).
• Output:
Init time: 14ms
----------------------------------
Query: ?- isMale(?x). ==>> 2 rows in 1ms
Variables: ?x
('bart')
('homer')

### Concrete Logic Programming System: IRIS

• IRIS performing computations using semi-naiv evaluation in combination with stratification and complex datatypes

### Theorem Proving Summary

• Resolution is a sound and complete inference procedure for First-Order Logic

• Due to its complexity and remaining limitations FOL is often not suitable for practical applications

• Often formalisms expressivity and complexity results are more appropriate:

• Description Logics

• Logic Programming

### Description Logic Summary

• Description Logics are a syntactic fragment of First Order Logic, based on basic building blocks
• Concepts
• Roles
• Individuals

• Limited constructs for building complex concepts, roles, etc.
• Many different Description Logics exist, depending on choice of constructs

• Inference in Description Logics focuses on consistency checking and classification
• Main reasoning task: Subsumption
• Usually reasoning tasks in DLs can all be reduced to satisfiablity checking

• Efficient Tbox (schema) reasoning
• ABox reasoning (query answering) do not scale so well

### Logic Programming Summary

• Logic Programming (without negation) is syntactically equivalent to Horn subset of First Order Logic
• The semantics of a Logic Program are however based on its minimal Herbrand Model

• Logic Programming comes in various variants for different applications (as programming language, for knowledge representation)
• Full Logic Programming including function symbols is not decidable
• Datalog is a syntactic restriction of LP, with desirable computational properties
• Negation-as-failure introduced non-monotonic behavior and pushes LP beyond the expressivity of First Order Logic

• A typical inference task for LP engines is conjunctive query answering

### References and Further Information

• Schöning, U., Logic for Computer Scientists (2nd edition), 2008, Birkhäuser
• Chapter 1 & 2:Normal Forms, Resolution
• Chapter 3: Horn Logic, Logic Programming
• Baader, F. et al., The Description Logic Handbook, 2007, Cambridge University Press
• Chapter 2: Basic Description Logics