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Title: Logics for Data and Knowledge Representation


1
Logics for Data and Knowledge Representation
  • ClassL (part 1) syntax and semantics

2
Outline
  • Syntax
  • Alphabet
  • Formation rules
  • Semantics
  • Class-valuation
  • Venn diagrams
  • Satisfiability
  • Validity
  • Reasoning
  • Comparing PL and ClassL
  • ClassL reasoning using DPLL

3
Language (Syntax)
INTRODUCTION SYNTAX SEMANTICS REASONING
PL AND CLASSL CLASSL REASONING USING DPLL
  • The syntax of ClassL is similar to PL
  • Alphabet of symbols S0
  • Auxiliary symbols parentheses ( )
  • Defined symbols
  • ? (falsehood symbol, false, bottom) ? df P ? P
  • T (truth symbol, true, top) T df ?

3
4
Formation Rules (FR) well formed formulas
INTRODUCTION SYNTAX SEMANTICS REASONING
PL AND CLASSL CLASSL REASONING USING DPLL
  • Well formed formulas (wff) in ClassL can be
    described by the following BNF () grammar
    (codifying the rules)
  • ltAtomic Formulagt A B ... P Q ...
    ? ?
  • ltwffgt ltAtomic Formulagt ltwffgt ltwffgt ?
    ltwffgt ltwffgt ? ltwffgt
  • Atomic formulas are also called atomic
    propositions
  • Wff are class-propositional formulas (or just
    propositions)
  • A formula is correct if and only if it is a wff
  • S0 FR define a propositional language
  • () BNF BackusNaur form (formal grammar)

4
5
Extensional Semantics Extensions
INTRODUCTION SYNTAX SEMANTICS REASONING
PL AND CLASSL CLASSL REASONING USING DPLL
  • The meanings which are intended to be attached to
    the symbols and propositions form the intended
    interpretation s (sigma) of the language
  • The semantics of a propositional language of
    classes L are extensional (semantics)
  • The extensional semantics of L is based on the
    notion of extension of a formula (proposition)
    in L
  • The extension of a proposition is the totality,
    or class, or set of all objects D (domain
    elements) to which the proposition applies

6
Extensional interpretation
INTRODUCTION SYNTAX SEMANTICS REASONING
PL AND CLASSL CLASSL REASONING USING DPLL
D Cita, Kimba, Simba
BeingLion
Monkey
Tree
Kimba
.
Cita
.
Simba
.
Lion2
Lion1
The World
The Mental Model
The Formal Model
6
7
Class-valuation s
INTRODUCTION SYNTAX SEMANTICS REASONING
PL AND CLASSL CLASSL REASONING USING DPLL
  • In extensional semantics, the first central
    semantic notion is that of class-valuation (the
    interpretation function)
  • Given a Class Language L
  • Given a domain of interpretation U
  • A class valuation s of a propositional language
    of classes L is a mapping (function) assigning to
    each formula ? of L a set s(?) of objects
    (truth-set) in U
  • s L ? pow(U)

8
Class-valuation s
INTRODUCTION SYNTAX SEMANTICS REASONING
PL AND CLASSL CLASSL REASONING USING DPLL
  • s(?) Ø
  • s(?) U (Universal Class, or Universe)
  • s(P) ? U, as defined by s
  • s(P) a ? U a ? s(P) comp(s(P))
    (Complement)
  • s(P ? Q) s(P) n s(Q) (Intersection)
  • s(P ? Q) s(P) ? s(Q) (Union)

9
Venn Diagrams and Class-Values
INTRODUCTION SYNTAX SEMANTICS REASONING
PL AND CLASSL CLASSL REASONING USING DPLL
  • By regarding propositions as classes, it is very
    convenient to use Venn diagrams

s(P)
s(P)
P
P
s(?)
s(?)
s(P ? Q)
s(P ? Q)
P
Q
P
Q
10
Truth Relation (Satisfaction Relation)
INTRODUCTION SYNTAX SEMANTICS REASONING
PL AND CLASSL CLASSL REASONING USING DPLL
  • Let s be a class-valuation on language L, we
    define the truth-relation (or class-satisfaction
    relation) ? and write
  • s ? P
  • (read s satisfies P) iff s(P) ? Ø
  • Given a set of propositions G, we define
  • s ? G
  • iff s ? ? for all formulas ? ? G

11
Model and Satisfiability
INTRODUCTION SYNTAX SEMANTICS REASONING
PL AND CLASSL CLASSL REASONING USING DPLL
  • Let s be a class valuation on language L. s is a
    model of a proposition P (set of propositions G)
    iff s satisfies P (G).
  • P (G) is class-satisfiable if there is a class
    valuation s such that s ? P (s ? G).

12
Truth, satisfiability and validity
INTRODUCTION SYNTAX SEMANTICS REASONING
PL AND CLASSL CLASSL REASONING USING DPLL
  • Let s be a class valuation on language L.
  • P is true under s if P is satisfiable by s (s ?
    P)
  • P is valid if s ? P for all s (notation ? P)
  • In this case, P is called a tautology (always
    true)
  • NOTE the notions of true and false are
    relative to some truth valuation.
  • NOTE A proposition is true iff it is
    satisfiable

13
Reasoning on Class-Propositions
INTRODUCTION SYNTAX SEMANTICS REASONING
PL AND CLASSL CLASSL REASONING USING DPLL
  • Given a class-propositions P we want to reason
    about the following
  • Model checking Does s satisfy P? (s ? P?)
  • Satisfiability Is there any s such that s ? P?
  • Unsatisfiability Is it true that there are no s
    satisfying P?
  • Validity Is P a tautology? (true for all s)

13
14
PL and ClassL are notational variants
INTRODUCTION SYNTAX SEMANTICS REASONING
PL AND CLASSL CLASSL REASONING USING DPLL
  • Theorem P is satisfiable w.r.t. an intensional
    interpretation ? if and only if P is
    satifisfiable w.r.t. an extensional
    interpretation s

PL ClassL
Syntax ? ?
? ?
? ?
? ?
? ?
P, Q... P, Q...
Semantics ?true, false ?o, (compare models)
14
15
ClassL reasoning using DPLL
INTRODUCTION SYNTAX SEMANTICS REASONING
PL AND CLASSL CLASSL REASONING USING DPLL
  • Given the theorem and the correspondences above,
    ClassL reasoning can be implemented using DPLL.
  • The first step consists in translating P into P
    expressed in PL
  • Model checking Does s satisfy P? (s ? P?)
  • Find the corresponding model ? and check that
    v(P) true
  • Satisfiability Is there any s such that s ? P?
  • Check that DPLL(P) succeeds and returns a ?
  • Unsatisfiability Is it true that there are no s
    satisfying P?
  • Check that DPLL(P) fails
  • Validity Is P a tautology? (true for all s)
  • Check that DPLL(?P) fails

15
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