Title: Logics for Data and Knowledge Representation
1Logics for Data and Knowledge Representation
- ClassL (part 1) syntax and semantics
2Outline
- Syntax
- Alphabet
- Formation rules
- Semantics
- Class-valuation
- Venn diagrams
- Satisfiability
- Validity
- Reasoning
- Comparing PL and ClassL
- ClassL reasoning using DPLL
3Language (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 ?
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4Formation 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)
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5Extensional 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
6Extensional 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
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7Class-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)
8Class-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)
9Venn 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
10Truth 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
11Model 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).
12Truth, 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
13Reasoning 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)
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14PL 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)
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15ClassL 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
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