Introduction to Propositional and First-order Logic: Syntax, Semantics, and Reasoning

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

Introduction to Reasoning inFirst-order [Predicate] Logicand 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 • 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 propositional logic <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 Iis 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) …  B A  B A A A A if-false if-true 0 1 0 1 B 0 B true false 0 1 0 1  A …  B A  B A if-false if-true A A 1 0 B 0 1 B 1 false true 0 1 0 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 Ibe 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 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: • 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 happyFathersin a given database • Reasoning: • check if I |=  impliesI |=  for all databases I, • i.e., if =>  • undecidable for FO, F-logic, etc. • Descriptions Logics aredecidable 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) () rule for F = A  B (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

c new t arbitrary (Semantic) Tableaux Rules • A branch is closed if it contains complementary formulas • A tableaux is closed if every branch is closed • () 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

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)

Introduction to Propositional and First-order Logic: Syntax, Semantics, and Reasoning