Computational Logic and Cognitive Science: An Overview

Computational Logic and Cognitive Science:

An Overview

Session 1: Logical Foundations

ICCL Summer School 2008Technical University of Dresden25th 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





http://www.cogsci.uni-osnabrueck.de/o3/admin/o3f.php?type=dirs/users/fotos/&id=hgust.jpg&mode=file

http://www.cogsci.uni-osnabrueck.de/o3/admin/o3f.php?type=dirs/users/fotos/&id=kkuehnbe.jpg&mode=file

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





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?





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





Forms of Reasoning: Deduction, Abduction,

Induction

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





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.





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.





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 causationHelmar Gust & Kai-Uwe Kühnberger




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.





Propositional Logic and First-Order Logic

Some rather Abstract Stuff…





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




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.





Propositional Logic

Other calculi:

Gentzen-type calculushttp://en.wikipedia.org/wiki/Sequent_calculus

Tableaux-calculushttp://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





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




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

First-Order Logic

Syntactically well-formed first-order formulas for a signature = {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 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.





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 = <,I> is called a structure.





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 <[[t1]],...,[[tr]]> I(R) [[ ]] = true iff [[]] = true and [[]] = true [[ ]] = true iff [[]] = true or [[]] = true [[]] = true iff [[]] = false [[x (x)]] = true iff for all d : [[(x)]]x=d = true [[x (x)]] = true iff there exists d : [[(x)]]x=d = true





Semantics Model

If the interpretation of a formula with respect to a structure = <,I> results in the truth value true, is called a model for (formal: )

Validity If every structure = <,I> is a model for we call valid ( )

Satisfiability If there exists a model = <,I> for we call satisfiable Example:

xy (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





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





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





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.





Repetition: Semantic Equivalences

Here is a list of semantic equivalences

( ) ( ), ( ) ( ) (commutativity) ( ) ( ), ( ) ( ) (associativity) ( ( )) , ( ( )) (absorption) ( ( )) ( ) ( ) (distributivity) ( ( )) ( ) ( ) (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) Etc.





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





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





Boolean Algebras P [[P]]

if arity is 1 (or [[P]] ... 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 spacecan be interpreted as a Boolean algebra

QP





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 ()





Boolean Algebras

The concept of a lattice Definition: A partial order = <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 = <D,,,,> (or

alternatively <D,,,,,>) such that <D,> = <D,,> is a distributive lattice is the top and the bottom element is a complement operation





Lindenbaum Algebras The Linbebaum algebra for propositional logic with atomic propositions

p and q





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





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 xy:(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





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





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?





x,y,z: NTPP(x,y) NTPP(y,z) NTPP(x,z)

Easy to see if we look at models!

Is This a Theorem?





Relations of Regions of the RCC-8(a canonical model: n-dimensional closed discs)





Thank you very much!!





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Computational Logic and Cognitive Science: An Overview