1 / 35

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.

fairly
Download Presentation

Computational Logic and Cognitive Science: An Overview

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related