Computational Logic and Cognitive Science: An Overview

1
Computational Logic and Cognitive Science An
Overview

• Session 1 Logical Foundations
• ICCL Summer School 2008
• Technical University of Dresden
• 25th of August, 2008
• Helmar Gust Kai-Uwe Kühnberger
• University of Osnabrück

Helmar Gust Kai-Uwe Kühnberger Universität
Osnabrück
ICCL Summer School 2008 Technical University of
Dresden, August 25th August 29th, 2008
2
Who we are
Helmar Gust Interests Analogical Reasoning,
Logic Programming, E-Learning Systems,
Neuro-Symbolic Integration
Kai-Uwe Kühnberger Interests Analogical
Reasoning, Ontologies, Neuro-Symbolic Integration
Where we work University of Osnabrück Institute
of Cognitive Science Working Group Artificial
Intelligence
Helmar Gust Kai-Uwe Kühnberger Universität
Osnabrück
ICCL Summer School 2008 Technical University of
Dresden, August 25th August 29th, 2008
3
Cognitive Science in Osnabrück

• Institute of Cognitive Science
• International Study Programs
• Bachelor Program
• Master Program
• Joined degree with Trento/Rovereto
• PhD Program
• Doctorate ProgramCognitive Science
• Graduate SchoolAdaptivity in Hybrid Cognitive
  Systems
• Web www.cogsci.uos.de

Helmar Gust Kai-Uwe Kühnberger Universität
Osnabrück
ICCL Summer School 2008 Technical University of
Dresden, August 25th August 29th, 2008
4
Who are You?

• Prerequisites?
• Logic?
• Propositional logic, FOL, models?
• Calculi, theorem proving?
• Non-classical logics many-valued logic,
  non-monotonicity, modal logic?
• Topics in Cognitive Science?
• Rationality (bounded, unbounded, heuristics),
  human reasoning?
• Cognitive models / architectures (symbolic,
  neural, hybrid)?
• Creativity?

Helmar Gust Kai-Uwe Kühnberger Universität
Osnabrück
ICCL Summer School 2008 Technical University of
Dresden, August 25th August 29th, 2008
5
Overview of the Course

• First Session (Monday)
• Foundations Forms of reasoning, propositional
  and FOL, properties of logical systems, Boolean
  algebras, normal forms
• Second Session (Tuesday)
• Cognitive findings Wason-selection task,
  theories of mind, creativity, causality, types of
  reasoning, analogies
• Third Session (Thursday morning)
• Non-classical types of reasoning many-valued
  logics, fuzzy logics, modal logics, probabilistic
  reasoning
• Fourth Session (Thursday afternoon)
• Non-monotonicity
• Fifth Session (Friday)
• Analogies, neuro-symbolic approaches
• Wrap-up

Helmar Gust Kai-Uwe Kühnberger Universität
Osnabrück
ICCL Summer School 2008 Technical University of
Dresden, August 25th August 29th, 2008
6
Forms of Reasoning Deduction, Abduction,
Induction

• Theorem Proving,
• Sherlock Holmes,
• and All Swans are White...

Helmar Gust Kai-Uwe Kühnberger Universität
Osnabrück
ICCL Summer School 2008 Technical University of
Dresden, August 25th August 29th, 2008
7
Basic Types of Inferences Deduction

• Deduction Derive a conclusion from given axioms
  (knowledge) and facts (observations).
• Example
• All humans are mortal. (axiom)
• Socrates is a human. (fact/ premise)
• Therefore, it follows that Socrates is mortal.
  (conclusion)
• The conclusion can be derived by applying the
  modus ponens inference rule (Aristotelian logic).
• Theorem proving is based on deductive reasoning
  techniques.

Helmar Gust Kai-Uwe Kühnberger Universität
Osnabrück
ICCL Summer School 2008 Technical University of
Dresden, August 25th August 29th, 2008
8
Basic Types of Inferences Induction

• Induction Derive a general rule (axiom) from
  background knowledge and observations.
• Example
• Socrates is a human (background knowledge)
• Socrates is mortal (observation/ example)
• Therefore, I hypothesize that all humans are
  mortal (generalization)
• Remarks
• Induction means to infer generalized knowledge
  from example observations Induction is the
  inference mechanism for (machine) learning.

Helmar Gust Kai-Uwe Kühnberger Universität
Osnabrück
ICCL Summer School 2008 Technical University of
Dresden, August 25th August 29th, 2008
9
Basic Types of Inferences Abduction

• Abduction From a known axiom (theory) and some
  observation, derive a premise.
• Example
• All humans are mortal (theory)
• Socrates is mortal (observation)
• Therefore, Socrates must have been a human
  (diagnosis)
• Remarks
• Abduction is typical for diagnostic and expert
  systems.
• If one has the flue, one has moderate fewer.
• Patient X has moderate fewer.
• Therefore, he has the flue.
• Strong relation to causation

Helmar Gust Kai-Uwe Kühnberger Universität
Osnabrück
ICCL Summer School 2008 Technical University of
Dresden, August 25th August 29th, 2008
10
Deduction

• Deductive inferences are also called theorem
  proving or logical inference.
• Deduction is truth preserving If the premises
  (axioms and facts) are true, then the conclusion
  (theorem) is true.
• To perform deductive inferences on a machine, a
  calculus is needed
• A calculus is a set of syntactical rewriting
  rules defined for some (formal) language. These
  rules must be sound and should be complete.
• We will focus on first-order logic (FOL).
• ? Syntax of FOL.
• ? Semantics of FOL.

Helmar Gust Kai-Uwe Kühnberger Universität
Osnabrück
ICCL Summer School 2008 Technical University of
Dresden, August 25th August 29th, 2008
11
Propositional Logic and First-Order Logic

• Some rather Abstract Stuff

Helmar Gust Kai-Uwe Kühnberger Universität
Osnabrück
ICCL Summer School 2008 Technical University of
Dresden, August 25th August 29th, 2008
12
Propositional Logic

• Formulas
• Given is a countable set of atomic propositions
  AtProp p,q,r,.... The set of well-formed
  formulas Form of propositional logic is the
  smallest class such that it holds
• ?p ? AtProp p ? Form
• ??, ? ? Form ? ? ? ? Form
• ??, ? ? Form ? ? ? ? Form
• ?? ? Form ?? ? Form
• Semantics
• A formula ? is valid if ? is true for all
  possible assignments of the atomic propositions
  occurring in ?
• A formula ? is satisfiable if ? is true for some
  assignment of the atomic propositions occurring
  in ?
• Models of propositional logic are specified by
  Boolean algebras(A model is a distribution of
  truth-values over AtProp making ? true)

Helmar Gust Kai-Uwe Kühnberger Universität
Osnabrück
ICCL Summer School 2008 Technical University of
Dresden, August 25th August 29th, 2008
13
Propositional Logic

• Hilbert-style calculus
• Axioms
• ? p ? (q ? p)
• ? p ? (q ? r) ? (p ? q) ? (p ? r)
• ? (?p ? ?q) ? (q ? p)
• ? p ? q ? p and ? (p ? q) ? q
• ? (r ? p) ? ((r ? q) ? (r ? p ? q))
• ? p ? (p ? q) and ? q ? (p ? q)
• ? (p ? r) ? ((q ? r) ? (p ? q ? r))
• Rules
• Modus Ponens If expressions ? and ? ? ? are
  provable then ? is also provable.
• Remark There are other possible axiomatizations
  of propositional logic.

Helmar Gust Kai-Uwe Kühnberger Universität
Osnabrück
ICCL Summer School 2008 Technical University of
Dresden, August 25th August 29th, 2008
14
Propositional Logic

• Other calculi
• Gentzen-type calculushttp//en.wikipedia.org/wiki
  /Sequent_calculus
• Tableaux-calculushttp//en.wikipedia.org/wiki/Met
  hod_of_analytic_tableaux
• Propositional logic is relatively weak no
  temporal or modal statements, no rules can be
  expressed
• Therefore a stronger system is needed

Helmar Gust Kai-Uwe Kühnberger Universität
Osnabrück
ICCL Summer School 2008 Technical University of
Dresden, August 25th August 29th, 2008
15
First-Order Logic

• Syntactically well-formed first-order formulas
  for a signature ? c1,...,cn,f1,...,fm,R1,...,R
  l are inductively defined.
• The set of Terms is the smallest class such that
• A variable x ? Var is a term, a constant ci ?
  c1,...,cn is a term.
• Var is a countable set of variables.
• If fi is a function symbol of arity r and
  t1,...,tr are terms, then fi(t1,...,tr) is a
  term.
• The set of Formulas is the smallest class such
  that
• If Rj is a predicate symbol of arity r and
  t1,...,tr are terms, then Rj(t1,...,tr) is a
  formula (atomic formula or literal).
• For all formulas ? and ? ? ? ?, ? ? ?, ??, ? ?
  ?, ? ? ? are formulas.
• If x ? Var and ? is a formula, then ?x? and ?x?
  are formulas.
• Notice that term and formula are rather
  different concepts.
• Terms are used to define formulas and not vice
  versa.

Helmar Gust Kai-Uwe Kühnberger Universität
Osnabrück
ICCL Summer School 2008 Technical University of
Dresden, August 25th August 29th, 2008
16
First-order Logic

• Semantics (meaning) of FOL formulas.
• Expressions of FOL are interpreted using an
  interpretation function I ? ? ?(?)
• I(ci) ? ?
• I(fi) ?arity(fi) ? ?
• I(Ri) ?arity(Ri) ? true, false
• ? is the called the universe or the domain
• A pair ? lt?,Igt is called a structure.

Helmar Gust Kai-Uwe Kühnberger Universität
Osnabrück
ICCL Summer School 2008 Technical University of
Dresden, August 25th August 29th, 2008
17
First-order Logic

• Semantics (meaning) of FOL formulas.
• Recursive definition for interpreting terms and
  evaluating truth values of formulas
• For c ? c1,...,cn ci I(ci)
• fi(t1,...,tr) I(fI)(t1,...,tr)
• R(t1,...,tr) true iff
  ltt1,...,trgt ? I(R)
• ? ? ? true iff ? true and ?
  true
• ? ? ? true iff ? true or ?
  true
• ?? true iff ? false
• ?x ?(x) true iff for all d ? ?
  ?(x)xd true
• ?x ?(x) true iff there exists d ? ?
  ?(x)xd true

Helmar Gust Kai-Uwe Kühnberger Universität
Osnabrück
ICCL Summer School 2008 Technical University of
Dresden, August 25th August 29th, 2008
18
First-order Logic

• Semantics
• Model
• If the interpretation of a formula ? with respect
  to a structure ? lt?,Igt results in the truth
  value true, ? is called a model for ? (formal ?
  ? ?)
• Validity
• If every structure ? lt?,Igt is a model for ? we
  call ? valid (? ?)
• Satisfiability
• If there exists a model ? lt?,Igt for ? we call ?
  satisfiable
• Example
• ?x?y (R(x) ? R(y) ? R(x) ? R(y)) valid
• If x and y are rich then either x is rich or y
  is rich
• If x and y are even then either x is even or y
  is even

Helmar Gust Kai-Uwe Kühnberger Universität
Osnabrück
ICCL Summer School 2008 Technical University of
Dresden, August 25th August 29th, 2008
19
First-order Logic

• Semantics
• An example
• ? x (N(x) ? P(x,c)) satisfiable
• There is a natural number that is smaller than
  17.
• There exists someone who is a student and likes
  logic.
• Notice that there are models which make the
  statement false
• Logical consequence
• A formula ? is a logical consequence (or a
  logical entailment) of A A1,...,An, if each
  model for A is also a model for ?.
• We write A ? ?
• Notice A ? ? can mean that A is a model for ? or
  that ? is a logical consequence of A
• Therefore people usually use different alphabets
  or fonts to make this difference visible

Helmar Gust Kai-Uwe Kühnberger Universität
Osnabrück
ICCL Summer School 2008 Technical University of
Dresden, August 25th August 29th, 2008
20
Theories

• The theory Th(A) of a set of formulas A Th(A)
  ? A ? ?
• Theories are closed under semantic entailment
• The operator Th A ? Th(A) is a so called
  closure operator
• X ? Th(X) extensive / inductive
• X ? Y ? Th(X) ? Th(Y) monotone
• Th(Th(X)) Th(X) idempotent

Helmar Gust Kai-Uwe Kühnberger Universität
Osnabrück
ICCL Summer School 2008 Technical University of
Dresden, August 25th August 29th, 2008
21
First-order Logic

• Semantic equivalences
• Two formulas ? and ? are semantically equivalent
  (we write ? ? ?) if for all interpretations of ?
  and ? it holds ? is a model for ? iff ? is a
  model for ?.
• A few examples
• ? ? ? ? ?
• ? ? ? ? ? ? ?
• ? ? (? ? ?) ? (? ? ?) ? (? ? ?)
• The following statements are equivalent (based on
  the deduction theorem)
• G is a logical consequence of A1,...,An
• A1 ? ... ? An ? G is valid
• Every structure is a model for this expression.
• A1 ? ... ? An ? ?G is not satisfiable.
• There is no structure making this expression true
• This can be used in the resolution calculus If
  an expression A1 ? ... ? An ? ?G is not
  satisfiable, then false can be derived
  syntactically.

Helmar Gust Kai-Uwe Kühnberger Universität
Osnabrück
ICCL Summer School 2008 Technical University of
Dresden, August 25th August 29th, 2008
22
Repetition Semantic Equivalences

• Here is a list of semantic equivalences
• (? ? ?) ? (? ? ?), (? ? ?) ? (? ?
  ?) (commutativity)
• (? ? ?) ? ? ? ? ? (? ? ?), (? ? ?) ? ? ? ? ? (? ?
  ?) (associativity)
• (? ? (? ? ?)) ? ?, (? ? (? ? ?)) ? ?
  (absorption)
• (? ? (? ? ?)) ? (? ? ?) ? (? ? ?) (distributivit
  y)
• (? ? (? ? ?)) ? (? ? ?) ? (? ? ?)
  (distributivity)
• ??? ? ? (double negation)
• ?(? ? ?) ? (?? ? ??), ?(? ? ?) ? (?? ? ??)
  (deMorgan)
• (? ? ?) ? ?, (? ? ?) ? ?
• (? ? ?) ? ?, (? ? ?) ? ?
• Here are some more semantic equivalences
• (? ? ?) ? ?, (? ? ?) ? ? (idempotency)
• ? ? ?? ? ? (tautology)
• ? ? ?? ? ? (contradiction)
• ??x? ? ?x??, ??x? ? ?x?? (quantifiers)
• (?x ? ? ?) ? ?x (? ? ?), (?x ? ? ?) ? ?x (? ? ?)
• ?x(? ? ?) ? (?x? ? ?x?)

ICCL Summer School 2008 Technical University of
Dresden, August 25th August 29th, 2008
Helmar Gust Kai-Uwe Kühnberger Universität
Osnabrück
23
Properties of Logical Systems

• Soundness
• A calculus is sound, if only such conclusions can
  be derived which also hold in the model
• In other words Everything that can be derived is
  semantically true
• Completeness
• A calculus is complete, if all conclusions can be
  derived which hold in the models
• In other words Everything that is semantically
  true can syntactically be derived
• Decidability
• A calculus is decidable if there is an algorithm
  that calculates effectively for every formula
  whether such a formula is a theorem or not
• Usually people are interested in completeness
  results and decidability results
• We say a logic is sound/complete/decidable if
  there exists a calculus with these properties

Helmar Gust Kai-Uwe Kühnberger Universität
Osnabrück
ICCL Summer School 2008 Technical University of
Dresden, August 25th August 29th, 2008
24
Some Properties of Classical Logic

• Propositional Logic
• Sound and Complete, i.e. everything that can be
  proven is valid and everything that is valid can
  be proven
• Decidable, i.e. there is an algorithm that
  decides for every input whether this input is a
  theorem or not
• First-order logic
• Complete (Gödel 1930)
• Undecidable, i.e. no algorithm exists that
  decides for every input whether this input is a
  theorem or not (Church 1936)
• More precisely FOL is semi-decidable
• Models
• The classical model for FOL are Boolean algebras

Helmar Gust Kai-Uwe Kühnberger Universität
Osnabrück
ICCL Summer School 2008 Technical University of
Dresden, August 25th August 29th, 2008
25
Boolean Algebras

• P ? P ? ?
• if arity is 1 (or P ? ??... ?? if arity gt 1)
• ? x1,...,xn P(x1,...,xn) ? Q(x1,...,xn) ? P
  ? Q
• We can draw Venn diagrams
• Regions (e.g. arbitrary subsets) of the
  n-dimensional real spacecan be interpreted as a
  Boolean algebra

Q
P
Helmar Gust Kai-Uwe Kühnberger Universität
Osnabrück
ICCL Summer School 2008 Technical University of
Dresden, August 25th August 29th, 2008
26
Boolean Algebras

• The power set ?(?) has the following properties
• It is a partially ordered set with order ?
• A ? B is the largest set X with X ? A and X ? B
• A ? B is the smallest set X with A ? X and B ? X
• comp(A) is the largest set X with A ? X ?
• ? is the largest set in ?(?), such that X ? ? for
  all X ??(?)
• ? is the smallest set in ?(?), such that ? ? X
  for all X ??(?)

Helmar Gust Kai-Uwe Kühnberger Universität
Osnabrück
ICCL Summer School 2008 Technical University of
Dresden, August 25th August 29th, 2008
27
Boolean Algebras

• The concept of a lattice
• Definition A partial order ? ltD,?gt is called a
  lattice if for each two elements x,y ? D it
  holds sup(x,y) exists and inf(x,y) exists
• sup(x,y) is the least upper bound of elements x
  and y
• inf(x,y) is the greatest lower bound of x and y
• The concept of a Boolean Algebra
• Definition A Boolean algebra is a tuple ?
  ltD,?,?,?,?gt (or alternatively ltD,?,?,?,?,?gt) such
  that
• ltD,?gt ltD,?,?gt is a distributive lattice
• ? is the top and ? the bottom element
• ? is a complement operation

Helmar Gust Kai-Uwe Kühnberger Universität
Osnabrück
ICCL Summer School 2008 Technical University of
Dresden, August 25th August 29th, 2008
28
Lindenbaum Algebras

• The Linbebaum algebra for propositional logic
  with atomic propositionsp and q

Helmar Gust Kai-Uwe Kühnberger Universität
Osnabrück
ICCL Summer School 2008 Technical University of
Dresden, August 25th August 29th, 2008
29
Normal Forms

• If there are a lot of different representations
  of the same statement
• Are there simple ones?
• Are there normal forms?
• Different normal forms for FOL
• Negation normal form
• Only negations of atomic formulas
• Prenex normal form
• No embedded Quantifiers
• Conjunctive normal form
• Only conjunctions of disjunctions
• Disjunctive normal form
• Only disjunctions of conjunctions
• Gentzen normal form
• Only implications where the condition is an
  atomic conjunction and the conclusion is an
  atomic disjunction

Helmar Gust Kai-Uwe Kühnberger Universität
Osnabrück
ICCL Summer School 2008 Technical University of
Dresden, August 25th August 29th, 2008
30
Normal Forms

• If there are a lot of different representations
  of the same statement
• Are there simple ones?
• Are there normal forms?
• Different normal forms for FOL (?x(p(x)
  ??yq(x,y)))
• Negation normal form ?x(p(x) ??yq(x,y))
• Only negations of atomic formulas
• Prenex normal form ?x?y(p(x) ?q(x,y))
• No embedded Quantifiers
• Conjunctive normal form p(cx) ?q(cx,y)
• Only conjunctions of disjunctions
• Disjunctive normal form
• Only disjunctions of conjunctions
• Gentzen normal form q(cx,y) ? p(cx)
• Only implications where the condition is an
  atomic conjunction and the conclusion is an
  atomic disjunction

Helmar Gust Kai-Uwe Kühnberger Universität
Osnabrück
ICCL Summer School 2008 Technical University of
Dresden, August 25th August 29th, 2008
31
Clause Form

• Conjunctive normal form.
• We know Every formula of propositional logic can
  be rewritten as a conjunction of disjunctions of
  atomic propositions.
• Similarly every formula of predicate logic can be
  rewritten as a conjunction of disjunctions of
  literals (modulo the quantifiers).
• A formula is in clause form if it is rewritten as
  a set of disjunctions of (possibly negative)
  literals.
• Example p(cx) ,q(cx,y)
• Theorem Every FOL formula F can be transformed
  into clause form F such that
• F is satisfiable iff F is satisfiable

Helmar Gust Kai-Uwe Kühnberger Universität
Osnabrück
ICCL Summer School 2008 Technical University of
Dresden, August 25th August 29th, 2008
32
What is the meaning of these Axioms?

• ?x C(x,x)
• ?x,y C(x,y) ? C(y,x)
• ?x,y P(x,y) ? ?z (C(z,x) ? C(z,y))
• ?x,y O(x,y) ? ?z (P(z,x) ? P(z,y))
• ?x,y DC(x,y) ? ?C(x,y)
• ?x,y EC(x,y) ? C(x,y) ? ?O(x,y)
• ?x,y PO(x,y) ? O(x,y) ? ?P(x,y) ? ?P(y,x)
• ?x,y EQ(x,y) ? P(x,y) ? P(y,x)
• ?x,y PP(x,y) ? P(x,y) ? ?P(y,x)
• ?x,y TPP(x,y) ? PP(x,y) ? ?z(EC(z,x) ? EC(z,y))
• ?x,y TPPI(x,y) ? PP(y,x) ? ?z(EC(z,y) ? EC(z,x))
• ?x,y NTPP(x,y) ? PP(x,y) ? ??z(EC(z,x) ?
  EC(z,y))
• ?x,y NTPPI(x,y) ? PP(y,x) ? ??z(EC(z,y) ?
  EC(z,x))

Helmar Gust Kai-Uwe Kühnberger Universität
Osnabrück
ICCL Summer School 2008 Technical University of
Dresden, August 25th August 29th, 2008
33
Is This a Theorem?

• ?x,y,z NTPP(x,y) ? NTPP(y,z) ? NTPP(x,z)
• Easy to see if we look at models!

Helmar Gust Kai-Uwe Kühnberger Universität
Osnabrück
ICCL Summer School 2008 Technical University of
Dresden, August 25th August 29th, 2008
34
Relations of Regions of the RCC-8
(a canonical model n-dimensional closed discs)
Helmar Gust Kai-Uwe Kühnberger Universität
Osnabrück
ICCL Summer School 2008 Technical University of
Dresden, August 25th August 29th, 2008
35
Thank you very much!!

Helmar Gust Kai-Uwe Kühnberger Universität
Osnabrück
ICCL Summer School 2008 Technical University of
Dresden, August 25th August 29th, 2008