Knowledge From the Logical Point of View

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Knowledge From the Logical Point of View

• Tutorial by
• Marie Duží and Petr Jirku
• 2004

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Motto

• Zeal without knowledge is a runaway horse.
• There is no royal road to learning.

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Overview

• Introduction
• The role of logic in understanding the concept of
  knowledge
• Logical languages and systems for knowledge
  representation
• Theoretical background of programming languages
  suitable to express knowledge

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Why Do We Need Knowledge?

• To understand the external world
• To communicate with each other(explicit
  knowledge)
• To be able to act in an adequate way even in
  critical situations (derivation of implied
  knowledge is necessary)
• Knowledge is
  power!

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What Is Knowledge?

• každému soudu, který odpovídá pravde,
  propujcuji jméno poznatku. .
  Bernard Bolzano
• Knowledge is a true justified belief
• Possible characteristics of knowledge
  (acquisition)(a priori knowledge vs. a
  posteriori knowledge, knowledge by acquiatance
  and knowledge by description, occurrence vs.
  disposition)
• Knowledge owner (agent/s)
• Knowledge (of P) is a justified belief of an
  agent that P is true

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Logic for KnowledgeWhat we can expect from logic?

• Languages and logics for knowledge representation
  (classical propositional logic, first-order
  logic, high-order logic modal, many-valued and
  fuzzy logics)
• Intentional logics (Montague, transparent
  intentional logic)
• Epistemic logics (Kripkean semantics, syntactic
  approaches)

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Labyrinth of Knowledge Representation Tools

• Declarative and Procedural knowledge,
• Implicite and Explicit knowledge,
• Reasoning (forward, backward, abductive,
  monotonic vs. nonmonotonic),
• If-then rules, Cognitive structures,
• Production systems,
• Frames, Objects, Concepts,
• Neural networks, Semantic nets
• Planning, Intelligent search
• Unique vs. Multiple Representations

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Possible Definitions of Knowledge

• Knowledge is truth justifiable belief
• (analytical necessity)
• Knowledge is relation between agent A and a
  meaning of proposition P
• (nomic necessity)
• Knowledge is justifiable belief of agent A that
  proposition P is true
• (epistemic necessity)

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Different Systems

• Intentional Systems
• Hyperintensional Systems
• Extensional Systems

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Epistemic Logics

• Axioms
• All zero-order tautologies
• (K) K f ? K (f ? ? ) ? K ?
• axiom of logical rationality
• (epistemic modality K is closed wrt
  implication)
•  
• Inference rules 
• (MP) Modus ponens From formulas f and f ? ?
  derive ? .
• (NEC) Necesitation From a formula f derive K f.

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Stronger Epistemic Logics

• (T) K f ? f knowledge implies truth
• (D) K f ? ? K ? f logical racionality
• (4) K f ? K K f positive introspection
• (5) ? K f ? K ? K f negative introspection 

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Other Nonclassical Logics

• Conditional Logic
• Deontic Logic
• Dynamic Logic
• Erotetic Logic (Logic of Questions)
• Intuitionistic Logic
• Modal Logic
• Many-valued and Fuzzy Logic
• Paraconsitent Logic
• Partial Logic
• Temporal Logic

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Reasoning

• Hide not your talents, they for use were made.
• Whats a Sun-dial in the Shade?
• (Benjamin
  Franklin)

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Classical Logical Derivability(A. Tarski)

• F set of formulas
• Cn P(F) gt P(F) operation on F such that
• Reflexivity X ? Cn(X)
• Monotonicity If X ? Y then
• 
  Cn(X) ? Cn(Y)
• Transitivity Cn(Cn(X)) Cn(X)
• hold.

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Various types of inferences

• De-duction
• In-duction
• Ab-duction
• -duction

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Deduction

• All the rabbits in the hat are white.
• These rabbits are from the hat.
• Therefore These rabbits are from the hat.
• Rule (premise)
• Fact (premise)
• Fact (conclusion)

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Induction

• These rabbits are white.
• All the rabbits are from the hat.
• These rabbits are from the hat.
• Fact (premise)
• Fact (premise)
• Rule (conclusion)

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Abduction

• All the rabbits in the hat are white.
• These rabbits are white.
• That is so These rabbits are from the hat.
• Rule (premise)
• Fact (premise)
• Fact (conclusion)

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Abduction II

• B Theoretical Background (deductively closed)
• G - Set of goals that should be explaned
• How to find a set H of hypotheses for which
• 1. ( B ? H) ? G, or G ? Cn(B ? H)
•    2. ( B ? H) is not inconsistent
•    3. H ? A a G ? H ?
•   4. not ( B ? G )
•    5. There is no set H ? H, such that
• B ? H ? G

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Abduction III

• Example (minimality of explanation)
• p gt r
• p and q gt r
• p, q is an explanation for r but it is not
  minimal for r while p is minimal
• Basicality of explanation (Feyrabend)
• Expanation is basic if is not explainable in
  terms of other explanations

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Abduction IV

• Example
• r gt g
• w gt g
• g gt s
• Interpretation
• s shoes are wet, r - reined last night,
• w watering-can was on, g grass is wet

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Strengh of abductive closures

• How good is hypothesis H independently on
  alternatives
• How decisively H overcome alternatives
• How complete was searching in space of
  alternatives

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Logic and Programming

• Languages for Logically-Oriented Knowledge Bases
  consist of
• K - language of formulas describing a kb (facts
  and/or rules)
• Q - language of questions
• A - language of answers
• QA System answ K x Q ? A

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Most Typical Examples

• First Order Theory
• Relational Data Bases
• Simple Deductive Data Bases
• Disjunctive Deductive Data Bases
• General Logic Programs

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First Order Theory

• K Q A, In classical logical consequence
  operation Cn (or relation ?, which is monotonic
  relation on the set of wffs).

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Relational Data Bases

• K set of ground atomic formulas (positive
  facts) represented by tables or relations
• Q SQL
• A yes, no
• It is non-monotonic since it is represented as
  set difference in relational algebra

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Simple Deductive Data Bases

• K set of positive facts and rules of the form A
  - B1, , BN.,
• where A is an atomic first-order formula Bi
  are literals and negation is treated as failure.
• Q set of atomic formulas
• A yes, no
• In linear resolution with selected element

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Disjunctive Data Bases

• K set of disjunctions of literals and rules as
  in simple deductive db
• Q set of literals
• A yes, no with substitution
• In linear resolution

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General Logic Programs

• They are equivalent to closed first order
  theories.
• K set of general clauses
• Q set of general clauses
• In classical logical consequence operation (or
  logical derivability relation)

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Hierarchy of DifferentMonotonic Derivabilities

• Theory is persistent if true/false formulas
  remain true/false after adding new formulas.
• Theory is reliable if truth/falsity of a formula
  in partial models entail its truth/falsity in
  every information completion.
• Theory is determined if each formula is
  determined, i.e. its truth/falsity is uniquely
  determined in the complete model.
• If the system is both determined and persistent,
  then it is reliable.

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Dynamics of Knowledge

• Expansion (T, f)
• Contraction (T, f)
• Revision (T, f)
• Postulates of rationality
• Peter
  Gardenfors

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Hirarchy of Different Nonmonotonic Derivabilities
I

• A formula f is arguable when there is fixpoint f
  of nm-Cn operation that includes it.
• A formula f is conceivable when its negation is
  not derivable from T.
• Formula is doubtless, if its negation is not
  arguable.

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Hierarchy of Different Nonmonotonic
Derivabilities II

• Safe
• Forseable
• Plausible
• Uncontroversial
• Realizable
• Undeniable

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Major Nonmonotonic Logics

• Fixpoint logics (default logics, modal
  nonmonotonic logics, epistemic logics, TMS, RMS)
• Model preference logics (close-world assumptions,
  circumscription, conditional logic)
• Systems for abductive reasoning (dependency
  networks, assumption-based TMS, RMS)

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Default Logic

• R. Reiter, 1970
• Default rules are rules of the form
• If a and if also ß can be consistently
• assumed
  then ?.
• (a ß / ?)

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Default Theories

• T F, D
• F set of first-order formulas
• D finite set of (closed) defaults
• Extension(s) of default theories
• fixpoints of nonmonotonic consequence
  operation which involves all facts and it is
  closed to both logical rules and defaults

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Examples I

• Theory with two extensions
• F b gt a and c
• D ( a / a), ( b / b), ( c / c)
• E1 Cn( F b)
• E2 Cn( F a, c)

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Examples II

• Theory with just one extension
• F
• D ( a / b) ( b / c), ( c / d)
• E Cn(b, d)

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Examples III

• Theories without extension
• F
• D ( true a / a
• F a
• D (a b and c / c), (c b / b)

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Various kinds of defaults

• General defaults
• (a ß / ?)
• Seminormal defaults
• (true ß and ? / ?)
• Normal defaults
• (true ? / ?) guarrantee existence of extension
  

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Lambda kalkul I

• Teory of algorithms, recursion, metalanguage
  for Lisp
• ?-calcul Alonzo Church 30ties in last century.
• Axioms and rules (Scott, Plotkin)
•                      Axioms for ?-abstraction
• (? x . M) (?y . M x / y ) if y is not free
  in M ?-rule
• ((? x.M) N) M x / N ?-rule

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Lambda kalkul II

•                Rules for equalities of terms
• Reflexivity
• Symetry
• Transitivity

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Lambda kalkul III

•                  Inference rules
• From MM derive NM NM
• From MM odvod MN MN
• From MM infer
• M (? x . M) (? x .
  M)

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Lambda kalkul IV

• When we understand binary relations of
  reduction (?) and equality () as primitive (not
  defined) terms, we can ?-calculus equivalently
  describe by the following axioms and rules (after
  Barendreght)
• (? x . M) N ? Mx / N) (ß reduction)
• (? x . Mx) ? M (?
  reduction)

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Realization (languages for knowledge
representation)

• Languages for AI
• (FRL, KRL, KL-One, )
• Logic Programming
• Algorithmic Programming

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Two programming languages supported by well
understandable mathematical theories

• (Horn) fragnent of predicate logic
  (implementation Prolog with various extensions)
• Recursion theory and/or lambda calculus
  (implementation Lisp and its clones)

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Lisp

• List processing language
• S-expressions atoms, lists. The empty list
  Manipulating lists (car, cdr, cons)
• M-expressions
• Recursive definitions
• Conditional expressions
• Lambda expressions

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S-expressions

• Symbolic expressions atoms (natural numbers,
  words, lists of atoms or sublists)
• Examples
• (a b c) (1 3 5) (2 2 2) (2 2 2 )
• ( ) nil
• ((Monday Tuesday) (a 1 b 2))
• Extra blanks are ignored, not extra
  parentheses. They can completely change the
  meaning of expression!

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Manipulating lists (car, cdr, cons)

• Car, cdr are for splitting and/or constructing
  lists (car returns head i.e. the first element of
  a list, cdr returns its tail, cons joins a head
  to a tail) Examples (in M-notation)
• car (a b c) gt a
• cdr (a b c) gt (b c)
• cdr cdr (a b c) gt (c)
• cons a nil gt a
• cons (a) (b c) gt (a b c)

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Conditional expressions

• Three-argument function
• (if predicate then-value else-value)

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Lambda expressions

• Lambda expressions are used to define functions
• (lambda
• (list-of-parameter-names) function-body)
• Examples
• ( lambda (x y) (cons y (cons x nil)))
• (lambda (x y) cons y cons x nil A B) gt (B A)
• 
  (f A B) gt (B A)

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How to bind a function symbol with the function
definition?

• Define (f x y) cons y cons x nil
• (lambda (f ) (f A B)
• lambda (x y) cons y cons x nil)

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Prolog

• Programming in logic
• 1972 A. Colmerauer, P. Roussel
• (Université Marseille-Luminy)
• 1977 Warren (University of Edinburgh)
• Logické symboly ?? ? ??? ? ? ?x

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Definite Clause Programs

• Logic Programming as Mechanized Deduction
• Resolution and Unification
• SLD Resolution and Procedural Semantics
• Herbrand Models
• Declarative and Fixpoint Semantics

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Some useful Logical Equivalences

• ?x P equiv ?x P
• ?x ( P if Q(x)) equiv P if ?x Q(x)
• A if (B and C) equiv (A if B) if C
• A if (B and C) equiv (A if C) and B
• (A and B) equiv A or B
• A or B equiv A if B
• A equiv false if A

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Logic Programming is Mechanized Reasoning

• Program given assumptions A
• Output desired conseuence C
• Computation deduction of C from A
• Meaning of program
• all consequences of
  A

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(Standard) Logic Programming

• Uses FOL to Describe Knowledge
• Uses Inferences to Proces Knowledge
• Uses Clausal Form
• Uses SLD Resolution as an Inference Method
• Definite Clause

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Herbrand Models

• P logic program
• Domain H(P) ground terms of the language in
  question
• Herbrand Base B(P) set of all atomic formulas
  constructed from H(P) and predicate symbol of the
  language
• Logic program then compute with terms from H(P)
  and ground instances in B(P)

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The not Predicate

• not(P) - call(P), fail.
• not(P).
• Negation as failure in derivation. It is the
  source of non-monotonicity.

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Example

• x is a member of a list y
• member(X, X Y).
• member(X, _ Y) - member(X, Y).

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Description logic

• Concept and role descriptions
• Restrictions on role interpretations
• Concept constructors
• Role constructors
• Axioms

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References I

• Franz Baader Diego Calvanese Deborah
  McGuinness Daniele Nardi Peter
  Patel-Schneider (eds.) The Description Logic
  Handbook Theory, Implementation and Applications.
  Cambridge Univ. Press 2003.
• Gregory J. Chaitin The Unknowable. Springer
  1999.
• Melvin Fitting Eva Orlowska (eds.) Beyond Two
  Theory and Applications of Multiple-Valued Logic.
  Physica Verlag 2003.
• Peter Gärdenfors Knowledge in Flux. Modelling
  the Dynamics of Epistemic States. The MIT Press
  1988.

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References II

• Jaakko Hintikka Knowledge and Belief. An
  Introduction to the Logic of the two notions.
  Cornell Univ. Press 1962.
• Raymond Turner Truth and Modality for Knowledge
  Representation. Pitman 1990.
• Reasoning about Knowledge. The MIT Press
• Russell, Bertrand Logic and Knowledge. (Essays
  1901-1950), edited by Robert Charles Marsh,
  Routledge, London and New York 1956.

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Eventually End

• This tutorial has been prepared
  nonmonotonically by Mary and Peter distinctly
  from monotonical Mary and Paul.