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Knowledge From the Logical Point of View Tutorial by Marie Duží and Petr Jirků 2004

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Motto: Zeal without knowledge is a runaway horse. There is no royal road to learning.

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Overview Introduction The role of logic in understanding the concept of knowledge Logical languages and systems for knowledge representation Theoretical background of programming languages suitable to express knowledge

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Why Do We Need Knowledge? To understand the external world To communicate with each other (explicit knowledge) To be able to act in an adequate way even in critical situations (derivation of implied knowledge is necessary) Knowledge is power!

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What Is Knowledge? …každému soudu, který odpovídá pravdě, propůjčuji jméno poznatku. …. Bernard Bolzano Knowledge is a true justified belief Possible characteristics of knowledge (acquisition) (a priori knowledge vs. a posteriori knowledge, knowledge by acquiatance and knowledge by description, occurrence vs. disposition) Knowledge owner (agent/s) Knowledge (of P) is a justified belief of an agent that P is true

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Logic for Knowledge What we can expect from logic? Languages and logics for knowledge representation (classical: propositional logic, first-order logic, high-order logic; modal, many-valued and fuzzy logics) Intentional logics (Montague, transparent intentional logic) Epistemic logics (Kripkean semantics, syntactic approaches)

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Labyrinth of Knowledge Representation Tools Declarative and Procedural knowledge, Implicite and Explicit knowledge, Reasoning (forward, backward, abductive, monotonic vs. nonmonotonic), If-then rules, Cognitive structures, Production systems, Frames, Objects, Concepts, Neural networks, Semantic nets Planning, Intelligent search Unique vs. Multiple Representations

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Possible Definitions of Knowledge Knowledge is truth justifiable belief (analytical necessity) Knowledge is relation between agent A and a meaning of proposition P (nomic necessity) Knowledge is justifiable belief of agent A that proposition P is true (epistemic necessity)

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Different Systems Intentional Systems Hyperintensional Systems Extensional Systems

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Epistemic Logics Axioms All zero-order tautologies (K) K φ K (φ ψ ) K ψ axiom of logical rationality (epistemic modality K is closed wrt implication) Inference rules (MP) Modus ponens: From formulas φ and φ ψ derive ψ. (NEC) Necesitation: From a formula φ derive K φ.

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Stronger Epistemic Logics (T) K φ φ knowledge implies truth (D) K φ K φ logical racionality (4) K φ K K φ positive introspection (5) K φ K K φ negative introspection

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Other Nonclassical Logics Conditional Logic Deontic Logic Dynamic Logic Erotetic Logic (Logic of Questions) Intuitionistic Logic Modal Logic Many-valued and Fuzzy Logic Paraconsitent Logic Partial Logic Temporal Logic

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Reasoning Hide not your talents, they for use were made. Whats a Sun-dial in the Shade? (Benjamin Franklin)

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Classical Logical Derivability (A. Tarski) F set of formulas Cn: P (F) > P (F) operation on F such that Reflexivity X Cn(X) Monotonicity If X Y then Cn(X) Cn(Y) Transitivity Cn(Cn(X)) = Cn(X) hold.

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Various types of inferences De-duction In-duction Ab-duction …-duction

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Deduction All the rabbits in the hat are white. These rabbits are from the hat. Therefore: These rabbits are from the hat. Rule (premise) Fact (premise) Fact (conclusion)

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Induction These rabbits are white. All the rabbits are from the hat. These rabbits are from the hat. Fact (premise) Rule (conclusion)

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Abduction All the rabbits in the hat are white. These rabbits are white. That is so These rabbits are from the hat. Rule (premise) Fact (premise) Fact (conclusion)

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Abduction II B – Theoretical Background (deductively closed) G - Set of goals that should be explaned How to find a set H of hypotheses for which 1. ( B H) G, or G Cn(B H) 2. ( B H) is not inconsistent 3. H A a G H = 4. not ( B G ) 5. There is no set H´ H, such that B H´ G

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Abduction III Example (minimality of explanation) p => r p and q => r {p, q} is an explanation for r but it is not minimal for r while {p} is minimal Basicality of explanation (Feyrabend) Expanation is basic if is not explainable in terms of other explanations

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Abduction IV Example r => g w => g g => s Interpretation: s – shoes are wet, r - reined last night, w – watering-can was on, g – grass is wet

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Strengh of abductive closures How good is hypothesis H independently on alternatives How decisively H overcome alternatives How complete was searching in space of alternatives

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Logic and Programming Languages for Logically-Oriented Knowledge Bases consist of K - language of formulas describing a kb (facts and/or rules) Q - language of questions A - language of answers QA System answ : K x Q A

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Most Typical Examples First Order Theory Relational Data Bases Simple Deductive Data Bases Disjunctive Deductive Data Bases General Logic Programs

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First Order Theory K = Q = A, In = classical logical consequence operation Cn (or relation, which is monotonic relation on the set of wffs).

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Relational Data Bases K = set of ground atomic formulas (positive facts) represented by tables or relations Q = SQL A = {yes, no} It is non-monotonic since it is represented as set difference in relational algebra

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Simple Deductive Data Bases K = set of positive facts and rules of the form A :- B1, …, BN., where A is an atomic first-order formula Bi are literals and negation is treated as failure. Q = set of atomic formulas A = {yes, no} In = linear resolution with selected element

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Disjunctive Data Bases K = set of disjunctions of literals and rules as in simple deductive db Q = set of literals A = {yes, no} with substitution In = linear resolution

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General Logic Programs They are equivalent to closed first order theories. K – set of general clauses Q – set of general clauses In – classical logical consequence operation (or logical derivability relation)

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Hierarchy of Different Monotonic Derivabilities Theory is persistent if true/false formulas remain true/false after adding new formulas. Theory is reliable if truth/falsity of a formula in partial models entail its truth/falsity in every information completion. Theory is determined if each formula is determined, i.e. its truth/falsity is uniquely determined in the complete model. If the system is both determined and persistent, then it is reliable.

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Dynamics of Knowledge Expansion (T, φ) Contraction (T, φ) Revision (T, φ) Postulates of rationality Peter Gardenfors

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Hirarchy of Different Nonmonotonic Derivabilities I A formula φ is arguable when there is fixpoint f of nm-Cn operation that includes it. A formula φ is conceivable when its negation is not derivable from T. Formula is doubtless, if its negation is not arguable.

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Hierarchy of Different Nonmonotonic Derivabilities II Safe Forseable Plausible Uncontroversial Realizable Undeniable

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Major Nonmonotonic Logics Fixpoint logics (default logics, modal nonmonotonic logics, epistemic logics, TMS, RMS) Model preference logics (close-world assumptions, circumscription, conditional logic) Systems for abductive reasoning (dependency networks, assumption-based TMS, RMS)

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Default Logic R. Reiter, 1970 Default rules are rules of the form If α and if also β can be consistently assumed then γ. (α ; β / γ)

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Default Theories T = [F, D] F set of first-order formulas D finite set of (closed) defaults Extension(s) of default theories: fixpoints of nonmonotonic consequence operation which involves all facts and it is closed to both logical rules and defaults

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Examples I Theory with two extensions F = {b => ¬a and ¬c} D = {( ;a / a), ( ;b / b), ( ;c / c)} E1 = Cn( F {b}) E2 = Cn( F {a, c})

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Examples II Theory with just one extension F = { } D = {( ; a / ¬b); ( ;b / ¬c), ( ;c / d)} E = Cn({¬b, d})

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Examples III Theories without extension F = { } D = {( true ; a / ¬ a } F = {a} D = {(a ; b and c / c), (c ; ¬ b / ¬ b) }

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Various kinds of defaults General defaults (α ; β / γ) Seminormal defaults (true ; β and γ / γ) Normal defaults (true ; γ / γ) guarrantee existence of extension

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Lambda kalkul I Teory of algorithms, recursion, metalanguage for Lisp λ-calcul Alonzo Church 30ties in last century. Axioms and rules (Scott, Plotkin) Axioms for λ-abstraction (λ x. M) = (λy. M [x / y] ) if y is not free in M -rule ((λ x.M) N) = M [x / N ] -rule

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Lambda kalkul II Rules for equalities of terms Reflexivity Symetry Transitivity

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Lambda kalkul III Inference rules From M=M´ derive NM = NM From M=M´ odvoď MN = MN From M=M´ infer M = (λ x. M) = (λ x. M)

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Lambda kalkul IV When we understand binary relations of reduction () and equality (=) as primitive (not defined) terms, we can λ-calculus equivalently describe by the following axioms and rules (after Barendreght): (λ x. M) N M[x / N]) (β reduction) (λ x. Mx) M (η reduction)

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Realization (languages for knowledge representation) Languages for AI (FRL, KRL, KL-One, …) Logic Programming Algorithmic Programming

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Two programming languages supported by well understandable mathematical theories (Horn) fragnent of predicate logic (implementation: Prolog with various extensions) Recursion theory and/or lambda calculus (implementation: Lisp and its clones)

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Lisp List processing language S-expressions; atoms, lists. The empty list; Manipulating lists (car, cdr, cons) M-expressions Recursive definitions Conditional expressions Lambda expressions

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S-expressions Symbolic expressions: atoms (natural numbers, words, lists of atoms or sublists) Examples (a b c) (1 3 5) (2 2 2) (2 2 2 ) ( ) nil ((Monday Tuesday) (a 1 b 2)) Extra blanks are ignored, not extra parentheses. They can completely change the meaning of expression!

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Manipulating lists (car, cdr, cons) Car, cdr are for splitting and/or constructing lists (car returns head i.e. the first element of a list, cdr returns its tail, cons joins a head to a tail) Examples (in M-notation) car `(a b c) ==> a cdr `(a b c) ==> (b c) cdr cdr `(a b c) ==> (c) cons `a nil ==> a cons `(a) `(b c) ==> (a b c)

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Conditional expressions Three-argument function (if predicate then-value else-value)

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Lambda expressions Lambda expressions are used to define functions (lambda (list-of-parameter-names) function-body) Examples ( lambda (x y) (cons y (cons x nil))) (`lambda (x y) cons y cons x nil A B) ==> (B A) (f A B) ==> (B A)

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How to bind a function symbol with the function definition? Define (f x y) cons y cons x nil (`lambda (f ) (f A B) `lambda (x y) cons y cons x nil)

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Prolog Programming in logic 1972 A. Colmerauer, P. Roussel (Université Marseille-Luminy) 1977 Warren (University of Edinburgh) Logické symboly ¬ λx

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Definite Clause Programs Logic Programming as Mechanized Deduction Resolution and Unification SLD Resolution and Procedural Semantics Herbrand Models Declarative and Fixpoint Semantics

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Some useful Logical Equivalences x ¬ P equiv ¬ x P x ( P if Q(x)) equiv P if x Q(x) A if (B and C) equiv (A if B) if C A if (B and C) equiv (A if C) and B ¬ (A and B) equiv ¬ A or ¬ B ¬ A or ¬ B equiv ¬ A if B ¬ A equiv false if A

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Logic Programming is Mechanized Reasoning Program = given assumptions A Output = desired conseuence C Computation = deduction of C from A Meaning of program = all consequences of A

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(Standard) Logic Programming Uses FOL to Describe Knowledge Uses Inferences to Proces Knowledge Uses Clausal Form Uses SLD Resolution as an Inference Method Definite Clause

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Herbrand Models P logic program Domain H(P)= ground terms of the language in question Herbrand Base B(P) = set of all atomic formulas constructed from H(P) and predicate symbol of the language Logic program then compute with terms from H(P) and ground instances in B(P)

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The not Predicate not(P) :- call(P), fail. not(P). Negation as failure in derivation. It is the source of non-monotonicity.

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Example x is a member of a list y member(X, [X | Y]). member(X, [ _ | Y]) :- member(X, Y).

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Description logic Concept and role descriptions Restrictions on role interpretations Concept constructors Role constructors Axioms

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References I Franz Baader – Diego Calvanese – Deborah McGuinness – Daniele Nardi – Peter Patel- Schneider (eds.): The Description Logic Handbook Theory, Implementation and Applications. Cambridge Univ. Press 2003. Gregory J. Chaitin: The Unknowable. Springer 1999. Melvin Fitting – Eva Orlowska (eds.): Beyond Two: Theory and Applications of Multiple- Valued Logic. Physica Verlag 2003. Peter Gärdenfors: Knowledge in Flux. Modelling the Dynamics of Epistemic States. The MIT Press 1988.

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References II Jaakko Hintikka: Knowledge and Belief. An Introduction to the Logic of the two notions. Cornell Univ. Press 1962. Raymond Turner: Truth and Modality for Knowledge Representation. Pitman 1990. Reasoning about Knowledge. The MIT Press Russell, Bertrand: Logic and Knowledge. (Essays 1901-1950), edited by Robert Charles Marsh, Routledge, London and New York 1956.

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Eventually End This tutorial has been prepared nonmonotonically by Mary and Peter distinctly from monotonical Mary and Paul.

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