1. 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

2. 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

3. 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

4. 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.
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 

5. 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

6. Boolean Algebra, Truth Tables

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

8. 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

9. 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, …)

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

11. 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

12. 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

13. 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.

14. 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

15. 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

16. 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

17. 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)

18. 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:

19. 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

20. 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?

21. “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

22. (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

23. 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}

24. Reasoning with DLs (Shawn Bowers)