Computational Logic and Cognitive Science: An Overview Session 1: - - PowerPoint PPT Presentation

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

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

Cognitive Science in Osnabrück

 Institute of Cognitive Science  International Study Programs

 Bachelor Program  Master Program

 Joined degree with

Trento/Rovereto  PhD Program

 Doctorate Program

“Cognitive Science”

 Graduate School

“Adaptivity 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

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

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

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

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

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

• bservations: 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

Basic Types of Inferences: Abduction

 Abduction: From a known axiom (theory) and some

• bservation, 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

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

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

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  ∀ϕ, ψ ∈ Form: ϕ ∧ ψ ∈ Form  ∀ϕ, ψ ∈ Form: ϕ ∨ ψ ∈ Form  ∀ϕ ∈ Form:

¬ϕ ∈ Form

 Semantics:

 A formula ϕ is valid if ϕ is true for all possible assignments of the

atomic propositions occurring in ϕ

 A formula ϕ is satisfiable if ϕ is true for some assignment of the

atomic propositions occurring in ϕ

 Models of propositional logic are specified by Boolean algebras

(A model is a distribution of truth-values over AtProp making ϕ true)

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

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 ϕ and ϕ → ψ are provable then ψ

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

Propositional Logic

 Other calculi:

 Gentzen-type calculus

http://en.wikipedia.org/wiki/Sequent_calculus

 Tableaux-calculus

http://en.wikipedia.org/wiki/Method_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

First-Order Logic

 Syntactically well-formed first-order formulas for a signature

Σ = {c1,...,cn,f1,...,fm,R1,...,Rl} 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 ϕ and ψ: ϕ ∧ ψ, ϕ ∨ ψ, ¬ϕ, ϕ → ψ, ϕ ↔ ψ are formulas.  If x ∈ Var and ϕ is a formula, then ∀xϕ and ∃xϕ 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

First-order Logic

 Semantics (meaning) of FOL formulas.

 Expressions of FOL are interpreted using an interpretation function

I: Σ → A(U)

 I(ci) ∈ U  I(fi) : Uarity(fi) → U  I(Ri) : Uarity(Ri) → {true, false}  U is the called the universe or the domain

 A pair M = <U,I> is called a structure.

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

First-order Logic

 Semantics (meaning) of FOL formulas.

 Recursive definition for interpreting terms and evaluating truth values

• f formulas:

 For c ∈ {c1,...,cn}: [[ci]] = I(ci)  [[fi(t1,...,tr)]] = I(fI)([[t1]],...,[[tr]])  [[R(t1,...,tr)]] = true

iff <[[t1]],...,[[tr]]> ∈ I(R)

 [[ϕ ∧ ψ]] = true

iff [[ϕ]] = true and [[ψ]] = true

 [[ϕ ∨ ψ]] = true

iff [[ϕ]] = true or [[ψ]] = true

 [[¬ϕ]] = true

iff [[ϕ]] = false

 [[∀x ϕ(x)]] = true

iff for all d ∈ U: [[ϕ(x)]]x=d = true

 [[∃x ϕ(x)]] = true

iff there exists d ∈ U: [[ϕ(x)]]x=d = true

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

 Semantics

 Model  If the interpretation of a formula ϕ with respect to a structure M = <U,I>

results in the truth value true, M is called a model for ϕ (formal: M  ϕ)

 Validity  If every structure M = <U,I> is a model for ϕ we call ϕ valid ( ϕ)  Satisfiability  If there exists a model M = <U,I> for ϕ we call ϕ 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“

First-order Logic

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

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 ϕ is a logical consequence (or a logical entailment)

• f A = {A1,...,An}, if each model for A is also a model for ϕ.

 We write A  ϕ  Notice: A  ϕ can mean that A is a model for ϕ or that ϕ 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

Theories

 The theory Th(A) of a set of formulas A:

Th(A) := {ϕ | A  ϕ}

 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

First-order Logic

 Semantic equivalences

 Two formulas ϕ and ψ are semantically equivalent (we write ϕ ≡ ψ) if for all

interpretations of ϕ and ψ it holds: M is a model for ϕ iff M is a model for ψ.

 A few examples:  ϕ ∧ ϕ ≡ ϕ  ϕ ∧ ψ ≡ ψ ∧ ϕ  ϕ ∧ (ψ ∨ χ) ≡ (ϕ ∧ ψ) ∨ (ϕ ∧ χ)

 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

Repetition: Semantic Equivalences



Here is a list of semantic equivalences



(ϕ ∧ ψ) ≡ (ψ ∧ ϕ), (ϕ ∨ ψ) ≡ (ψ ∨ ϕ) (commutativity)



(ϕ ∧ ψ) ∧ χ ≡ ϕ ∧ (ψ ∧ χ), (ϕ ∨ ψ) ∨ χ ≡ ϕ ∨ (ψ ∨ χ) (associativity)



(ϕ ∧ (ϕ ∨ ψ)) ≡ ϕ, (ϕ ∨ (ϕ ∧ ψ)) ≡ ϕ (absorption)



(ϕ ∧ (ψ ∨ χ)) ≡ (ϕ ∧ ψ) ∨ (ϕ ∧ χ) (distributivity)



(ϕ ∨ (ψ ∧ χ)) ≡ (ϕ ∨ ψ) ∧ (ϕ ∨ χ) (distributivity)



¬¬ϕ ≡ ϕ (double negation)



¬(ϕ ∧ ψ) ≡ (¬ϕ ∨ ¬ψ), ¬(ϕ ∨ ψ) ≡ (¬ϕ ∧ ¬ψ) (deMorgan)



(⊥ ∧ ϕ) ≡ ⊥, (⊥ ∨ ϕ) ≡ ϕ



( ∧ ϕ) ≡ ϕ, ( ∨ ϕ) ≡  

Here are some more semantic equivalences



(ϕ ∧ ϕ) ≡ ϕ, (ϕ ∨ ϕ) ≡ ϕ (idempotency)



ϕ ∨ ¬ϕ ≡  (tautology)



ϕ ∧ ¬ϕ ≡ ⊥ (contradiction)



¬∀xϕ ≡ ∃x¬ϕ, ¬∃xϕ ≡ ∀x¬ϕ (quantifiers)



(∀x ϕ ∧ ψ) ≡ ∀x (ϕ ∧ ψ), (∀x ϕ ∨ ψ) ≡ ∀x (ϕ ∨ ψ)



∀x(ϕ ∧ ψ) ≡ (∀xϕ ∧ ∀xψ)



Etc.

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

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

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

Boolean Algebras

 P ⇒ [[P]] ⊆ U

 if arity is 1 (or [[P]] ⊆ U... U if arity > 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 space

can 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

Boolean Algebras

 The power set ℘(U) 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 = ∅

 U is the largest set in ℘(U), such that X ⊆ U for all X ∈℘(U)

 ∅ is the smallest set in ℘(U), such that ∅ ⊆ X for all X ∈℘(U)

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

Boolean Algebras

 The concept of a lattice

 Definition: A partial order D = <D,≤> 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 M = <D,≤,¬,,> (or

alternatively <D,∧,∨,¬,,>) such that

 <D,≤> = <D,∧,∨> 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

Lindenbaum Algebras

 The Linbebaum algebra for propositional logic with atomic propositions

p and q

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

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

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) y:q(x,y)))

 Negation normal form

x:(p(x) y:¬q(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

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

 ∀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))

What is the ‘meaning’ of these Axioms?

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

 ∀x,y,z: NTPP(x,y) ∧ NTPP(y,z) → NTPP(x,z)  Easy to see if we look at models!

Is This a Theorem?

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

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

Thank you very much!!

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