Bertram Ludäscher LUDAESCH@SDSC.EDU Department of Computer Science & Engineering University of California, San Diego CSE-291: Ontologies in Data Integration Spring 2004 Bertram Ludäscher LUDAESCH@SDSC.EDU

Introduction to Reasoning in First-order [Predicate] Logic and Description Logic(s) Introduction to FO (aka PL) Syntax, Semantics Two decidable fragments: propositional logic description logic(s) Reasoning w/ Tableaux Description Logic Reasoning … Discussion of Topics / Assignments

Syntax vs Semantics Syntax Semantics a.k.a. “formation rules”; “grammar”; … prescribes what a well-formed formula is (syntactically) Semantics the “meaning” of well-formed formulas defined via a mapping called interpretation

Propositional Logic: Syntax <logic> (or "propositional calculus") A system of symbolic logic using symbols to stand for whole propositions and logical connectives. Propositional logic only considers whether a proposition is true or false. In contrast to predicate logic, it does not consider the internal structure of propositions. http://wombat.doc.ic.ac.uk/foldoc/foldoc.cgi?propositional+logic Logical symbols: conjunction: , disjunction: , negation: , implication: , equivalence: , parentheses:   Non-logical symbols: propositional variables p, q, r, ... signature: set of propositional variables  = {p, q, r, ...} Formation rules for well-formed formulas (wff) an atomic formula (propositional variable) is a formula if F, G are formulas, so are: FG, F  G,  F, FG , FG,  F 

Propositional Logic: Semantics Propositions can be assigned a truth-value: either true or false (classical 2-valued logic: tertium non datur) other propositional logics exist: 3-valued, 4-valued, temporal, … (modal logics), …, fuzzy logic An interpretation I over a signature  is a mapping I:   {true, false} , associating a truth value to every propositional variable Truth tables describe how to extend I from atomic to composite formulas (Boolean Algebra): FG, F  G,  F, FG , FG

Boolean Algebra, Truth Tables http://wombat.doc.ic.ac.uk/foldoc/foldoc.cgi?two-valued+logic

Different Logical Bases Often: ,  ,  Alternatively: ,   ,  NAND NOR XOR What about: ite(A,B,C) … if A then B else C ?

Reasoning in Propositional Logic A formula F is … valid if it is true for all interpretations I satisfiable if it is true for some interpretation I unsatisfiable if it is true for no interpretation I Try these: p  q p  p p  p p  p p  p  p  p

Reasoning in Propositional Logic Def. “models” relationship “|=”: If a formula F evaluates to true for an interpretation I then I is called a model of F; written I |= F I is a model of {F1,…, Fk}, written I |= {F1,…, Fk},if I is a model of each Fj Automated deduction setting: Show that A1,,…, An (axioms) imply T (theorem), that is, every model of the axioms is also a model of the theorem: That is: if I |= {A1,,…, An } then I |= T Short: {A1,,…, An } |= T Often: Show that A1  …  An  T is unsatisfiable We need a procedure / reasoning algorithm: Predicate Calculus (in fact calculi: resolution, tableaux, …)

Example {p, p  q } |= q Truth table Resolution Tableaux

Example: Reasoning with Binary Decision Trees (see also: Binary Decision Diagrams, or BDDs) A  B A A A if-false if-true 1 1 B B false true 1 1  A …  B A  B A if-false if-true A A 1 true false B 1 B 1 1 1

Syntax of First-Order Logic (FO) Logical symbols: , , , , ,  ,  (“for all”),  (“exists”), ... Non-logical symbols: A FO signature  consists of constant symbols: a,b,c, ... function symbols: f, g, ... predicate (relation) symbols: p,q,r, .... function and predicate symbols have an associated arity; we can write, e.g., p/3, f/2 to denote the ternary predicate p and the function f with two arguments First-order variables: x, y, ... Formation rules for terms: constants and variables are terms if t1,…,tk are terms and f is a k-ary function symbols then f(t1,...,tk) is a term

Syntax of First-Order Logic (FO) Formation rules for formulas: if t1,…, tk are terms and p/k is a predicate symbol (of arity k) then p(t1, …, tk) is an atomic formula (short: atom) all variable occurrences in p(t1, …, tk) are free if F,G are formulas and x is a variable, then the following are formulas: FG, F  G,  F, FG , FG,  F , x: F (“for all x: F(x,...) is true”) x: F (“there exists x such that F(x,...) is true”) the occurrences of a variable x within the scope of a quantifier are called bound occurrences.

Examples x man(x)  person(x). man(bill). child(marriage(bill,hillary),chelsea). Variable: x Constants (0-ary function symbols): bill/0, hillary/0, chelsea/0 Function symbols: marriage/2 Predicate symbols: man/1, person/1, child/2

Semantics of Predicate Logic Let D be a non-empty domain (a.k.a. universe of discourse). A structure is a pair I = (D,I), with an interpretation I that maps ... each constant symbols c to an element I(c) D each predicate symbol p/k to a k-ary relation I(p)  Dk, each function symbol f/k to a k-ary function I(f): DkD Let I be a structure,  : Vars  D a variable assignment. A valuation valI, maps Term to D and Fml to {true, false} valI, (x) =  (x) ; for x  Vars valI, (f(t1,...,tk)) = I(f)( valI, (t1),..., valI, (tk) ) ; for f(t1,...,tk)  Term valI, (p(t1,...,tk)) = I(p)( valI, (t1),..., valI, (tk) ) ; for p(t1,...,tk)  At valI, (F  G) = valI, (F) and valI, (G) are true ; for F,G Fml valI, (F  G) = valI, (F) or valI, (G) is true ; for F,G Fml valI, ( F) = true (false) if valI, (F) is false (true) ; for FFml valI, ( x F) = valI,[x/t] (F) is true for some t  D ; for FFml valI, ( x F) = valI,[x/t] (F) is true for all t  D ; for FFml

Example Let’s pick an interpretation I: Formula F = x man(x)  person(x). Domain D = {b, h, c, d, e} Let’s pick an interpretation I: I(bill) = b, I(hillary) = h, I(chelsea) = c I(person) = {b, h, c} I(man) = {b} Under this I, the formula F evaluates to true. If we choose I’ like I but I’(man) = {b,d}, then F evaluates to false Thus, I is a model of F, while I’ is not: I |= F I’ |=/= F

FO Semantics (cont’d) F entails G (G is a logical consequence of F) if every model of F is also a model of G: F |= G F is consistent or satisfiable if it has at least one model F is valid or a tautology if every interpretation of F is a model Proof Theory: Let F,G, ... be FO sentences (no free variables). Then the following are equivalent: F_1, ..., F_k |= G F_1  ...  F_k  G is valid F_1  ...  F_k   G is unsatisfiable (inconsistent)

Proof Theory A calculus is formal proof system to establish F1,…, Fk |= T via formal (syntactic) derivations F1,…, Fk |– ... |– T, where the “|–” denotes allowed proof steps Examples: Hilbert Calculus, Gentzen Calculus, Tableaux Calculus, Natural Deduction, Resolution, ... First-order logic is “semi-decidable”: the set of valid sentences is recursively enumerable, but not recursive (decidable) Some inference engines: http://www.semanticweb.org/inference.html

Querying vs. Reasoning Querying: Reasoning: given a DB instance I (= logic interpretation), evaluate a query expression (e.g. SQL, FO formula, Prolog program, ...) boolean query: check if I |=  (i.e., if I is a model of ) (ternary) query: { (X, Y, Z) | I |=  (X,Y,Z) } => check happyFathers in a given database Reasoning: check if I |=  implies I |=  for all databases I, i.e., if  =>  undecidable for FO, F-logic, etc. Descriptions Logics are decidable fragments concept subsumption, concept hierarchy, classification semantic tableaux, resolution, specialized algorithms

Reasoning Example (1) p(0) (2) x p(x)  p(s(x)) (3) p(s(s(0))). We want to show that (1) & ... & (2) implies (3) Approach: assume negation of (3) and show that it leads to a contradiction with {(1), (2)} Question: Why is this sound?

“Types” of Formulas () rule for F = A  B (and other disjunctions) (and other conjunctions) () rule for F = x: A(X,...) substitute a -variable X with an arbitrary term t () rules for F = x: A(X,...) substitute a -variable X with a new constant c

(Semantic) Tableaux Rules t arbitrary c new () rule for F = A  B () rule for F = A  B () rule for F = x: A(X,...) substitute a -variable X with an arbitrary term t () rules for F = x: A(X,...) substitute a -variable X with a new constant c A branch is closed if it contains complementary formulas A tableaux is closed if every branch is closed

FO Tableaux Calculus Theorem (Soundness, Completeness of Tableaux calculus): Let A1,..., Ak and F be first-order logic sentences. (Recall: a sentence is a closed formula, i.e., has no free variables) Then the following are equivalent: A1, ..., Ak |= F A1  ...  Ak  F is unsatisfiable (inconsistent) There is a closed tableaux for {A1, ..., Ak ,  F}

Reasoning with DLs (Shawn Bowers)