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Computational Logic and Cognitive Science: An Overview

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.

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Computational Logic and Cognitive Science: An Overview

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  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 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

  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 besoundand should becomplete. • 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/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

  15. First-Order Logic • Syntacticallywell-formed first-order formulasfor a signature  = {c1,...,cn,f1,...,fm,R1,...,Rl} are inductively defined. • The set of Termsis 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 Formulasis 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 functionI:   () • 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. 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 <[[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 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  = <,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: • 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 xis 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 falsecan 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) • (  (  ))  (  )  (  ) (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. 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 > 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  = <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 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) 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

  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 asetof 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

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