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 ICCL Summer School 2008 Universität Osnabrück Technical University of Dresden, August 25th – August 29th, 2008

Who we are… Helmar Gust Kai-Uwe Kühnberger Interests: Analogical Interests: Analogical Reasoning, Logic Reasoning, Ontologies, Programming, E-Learning Neuro-Symbolic Systems, Neuro-Symbolic Integration Integration Where we work: University of Osnabrück Institute of Cognitive Science Working Group: Artificial Intelligence Helmar Gust & Kai-Uwe Kühnberger ICCL Summer School 2008 Universität Osnabrück 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 ICCL Summer School 2008 Universität Osnabrück 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 ICCL Summer School 2008 Universität Osnabrück 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 ICCL Summer School 2008 Universität Osnabrück 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 ICCL Summer School 2008 Universität Osnabrück 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 ICCL Summer School 2008 Universität Osnabrück 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 observations: Induction is the inference mechanism for (machine) learning. Helmar Gust & Kai-Uwe Kühnberger ICCL Summer School 2008 Universität Osnabrück Technical University of Dresden, August 25th – August 29th, 2008

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 ICCL Summer School 2008 Universität Osnabrück 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 ICCL Summer School 2008 Universität Osnabrück Technical University of Dresden, August 25th – August 29th, 2008

Propositional Logic and First-Order Logic Some rather Abstract Stuff… Helmar Gust & Kai-Uwe Kühnberger ICCL Summer School 2008 Universität Osnabrück 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 ICCL Summer School 2008 Universität Osnabrück 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  (p ∧ q) → q and   (r → p) → ((r → q) → (r → p ∧ q))   p → (p ∨ q)  q → (p ∨ q) and   (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 ICCL Summer School 2008 Universität Osnabrück 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 ICCL Summer School 2008 Universität Osnabrück Technical University of Dresden, August 25th – August 29th, 2008

First-Order Logic  Syntactically well-formed first-order formulas for a signature Σ = { c 1 ,..., c n , f 1 ,..., f m , R 1 ,..., R l } are inductively defined.  The set of Terms is the smallest class such that:  A variable x ∈ Var is a term, a constant c i ∈ { c 1 ,..., c n } is a term.  Var is a countable set of variables.  If f i is a function symbol of arity r and t 1 ,..., t r are terms, then f i ( t 1 ,..., t r ) is a term.  The set of Formulas is the smallest class such that:  If R j is a predicate symbol of arity r and t 1 ,..., t r are terms, then R j ( t 1 ,..., t r ) 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 ∀ x ϕ and ∃ x ϕ 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 ICCL Summer School 2008 Universität Osnabrück 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 ( c i ) ∈ U  I ( f i ) : U arity( f i ) → U  I ( R i ) : U arity( R i ) → { 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 ICCL Summer School 2008 Universität Osnabrück Technical University of Dresden, August 25th – August 29th, 2008