[PPT] - Deciding regular grammar logics with converse through GF2 Stphane PowerPoint Presentation

SLIDE 1

Deciding regular grammar logics with converse through GF2

Stéphane Demri Laboratoire Spécification et Vérification (Cachan, France) Joint work with Hans de Nivelle (MPII, Saarbrücken, Germany)

Deciding regular grammar logics with converse through GF2 – p. 1

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Outline

1. Modal logics.
2. Guarded fragment GF.
3. Regular grammar logics with converse.
4. Translation into GF2 by simulating NDFA.

Deciding regular grammar logics with converse through GF2 – p. 2

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Modal languages

Simple and sufficiently expressive to talk about

relational structures

Local view for the description of structures Application domains:

Computer Science: temporal logics . . .
Knowledge Representation
Mathematics: arithmetics . . .
Linguistics

Deciding regular grammar logics with converse through GF2 – p. 3

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Basic modal languages and structures

Language:

✁

✂✄ ☎ ✂ ☎✆ ✝ ✞ ✝ ☎ ✟ ✠ ✡☞☛ ✠ ✡ ☛ ✌ ✡ ✌ ✆ ✍ ✎✏ ✑ ✒ ✓ ✒ ✔ ✒ ✔ ✒ ✕ ☎ ✖ ✠ ✡☞✗ ✘ ✞ ✗ ✟ ✆ ✝ ☎ ✟ ✍ ✎✏ ✑ ✙✛✚ ✜ ✒ ✢ ✚ ✣ ✚

Kripke models:

✁ ✙ ✤ ✥ ✦ ✧ ✦ ★ ✩ ✤ ✜

:

: non-empty set
✦

: binary relation on

✥

✧

: meaning function

Deciding regular grammar logics with converse through GF2 – p. 4

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Possible worlds semantics

✤ ✒ ✁

def

✥ ✧

.

✤ ✒ ✁ ✓

def

✤ ✒ ✁

.

✤ ✒ ✁ ✢ ✚ ✣

def for every

✔ ✦ ✥ ✧

,

✤ ✔ ✒ ✁

.

✤ ✒ ✁ ✙ ✚ ✜

def there is

✔ ✦ ✥ ✧

such that

✤ ✔ ✒ ✁

.

Deciding regular grammar logics with converse through GF2 – p. 5

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Standard decision problems

satisfi ability:

input: formula

,

question:

& s.t.

✤ ✒ ✁

? ( : class of models depending on application domain)

Model-checking.
validity, global
satisfi ability.

Deciding regular grammar logics with converse through GF2 – p. 6

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Decision procedures

Direct method:
fi ltration (abstraction),
proof-theoretical analysis (sequents),
automata (emptiness problem).
Translation into
richer modal logics (PDL,
calculus, . . . ),
fi rst-order fragments (FO2, GF, LGF, . . . ),
second-order logics (S2S,

LGF).

Deciding regular grammar logics with converse through GF2 – p. 7

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Translations into FOL

Syntactic translation (Morgan 76)

Hilbert-style system reifi cation provability predicate

Relational translation (van Benthem 76 + ...)

Exact encoding of the semantics

Functional translation (Ohlbach 88, Herzig 89)

Path terms and equational theories

+ other (sometimes ad hoc) translations

Deciding regular grammar logics with converse through GF2 – p. 8

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Relational translation

✥

✁ ✤ ✂ ✧ ✁

s

✥ ✂ ✧ ✁ ✄ ✤ ✓

✥

☎ ✆ ✤ ✂ ✧ ✁

✥

☎ ✤ ✂ ✧

✥

✆ ✤ ✂ ✧

✥

☎ ✆ ✤ ✂ ✧ ✁

✥

☎ ✤ ✂ ✧

✥

✆ ✤ ✂ ✧

✥

✙✛✚ ✜ ✤ ✂ ✧ ✁ ✦ ✥ ✂ ✤ ✧

✥

✤ ✧

, new

✥

✢ ✚ ✣ ✤ ✂ ✧ ✁ ✥ ✦ ✥ ✂ ✤ ✧

✥

✤ ✧ ✧

, new

✤ ✒ ✁

iff

✤ ✝ ✢ ✂ ✣ ✒ ✁ ✞ ✟ ✠

✥

✤ ✂ ✧

(

✝

is a valuation)

Deciding regular grammar logics with converse through GF2 – p. 9

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Recycling of variables

✂ ✤ ✁ ✂✁ ✤

☎
✥

✤ ✂ ✤ ✧ ✁ ✥ ✂ ✧

✥

✓ ✤ ✂ ✤ ✧ ✁ ✓ ✥ ✂ ✧

✥

☎ ✆ ✤ ✂ ✤ ✧ ✁

✥

☎ ✤ ✂ ✤ ✧

✥

✆ ✤ ✂ ✤ ✧

✥

✙ ✚ ✜ ✤ ✂ ✤ ✧ ✁ ✦ ✥ ✂ ✤ ✧

✥

✤ ✤ ✂ ✧

✥

✢ ✚ ✣ ✤ ✂ ✤ ✧ ✁ ✥ ✦ ✥ ✂ ✤ ✧

✥

✤ ✤ ✂ ✧ ✧ ✤ ✒ ✁

iff

✤ ✝ ✢

✁

✣ ✒ ✁ ✞ ✟ ✠

✥

✤

✁

✤

☎

✧

(

✝

✁

✤

☎

)

Deciding regular grammar logics with converse through GF2 – p. 10

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How to simply translate a large class of modal logics into the decidable GF2?

Deciding regular grammar logics with converse through GF2 – p. 11

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Guarded fragment (GF)

Restriction of FOL introduced in

(Andreka & van Benthem & Nemeti 98) to identify the “modal fragment of FOL ”.

✥

✧ ✤

✥

✧

: atomic & FreeVar

✥ ✧

FreeVar

✥ ✧

.

Relational translation of modal formulae

falls into GF.

Deciding regular grammar logics with converse through GF2 – p. 12

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GF - Complexity

GF: 2EXPTIME-complete (Grädel, JSL 99) GF

: EXPTIME-complete (Grädel, JSL 99)

Other decidable fragments of FOL:

GF + equality + constants.
GF2 with transitive guards.

2EXPTIME-complete (Szwast & Tendera 01, Kieronski 03).

Deciding regular grammar logics with converse through GF2 – p. 13

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GF - Theorem provers

Resolution-based decision procedure for GF

(Ganzinger & de Nivelle 99).

Tableaux-based decision procedure for GF

(Hadlik 02) and for FO2 (Marx et al. 00).

Deciding regular grammar logics with converse through GF2 – p. 14

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Main defect of GF

Simple frame conditions such as transitivity can’t be expressed in GF.

✤

✤

✥
✤

✧ ✥ ✥ ✤

✧

✥

✤
✧

✧

Deciding regular grammar logics with converse through GF2 – p. 15

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Relation inclusions as rules

transitivity:

✦

✦

✦ ✁ ✚ ✚ ✂ ✚

symmetry:

✄ ☎ ✦ ✦ ✁ ✚ ✚

(plus

✚ ✚

)

euclideanity:

✄ ☎ ✦

✦

✦ ✁ ✚ ✚ ✂ ✚

(plus

✚ ✚ ✂ ✚

)

CF condition:

✦ ☎

✂

✂ ✂

✦✝✆

✦ ✁ ✚ ✚ ☎ ✂

✂

✚ ✞

(plus

✚ ✚ ✞ ✂

✂

✚ ☎

)

Deciding regular grammar logics with converse through GF2 – p. 16

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Semi-Thue systems

Semi-Thue system : subset of

✁
.

✙✄✂ ✤ ✝ ✜

read as a rule

✂ ✝

.

One-step derivation relation

☎

:

✂ ☎ ✝

iff there exist 1.

✂ ☎ ✤ ✂ ✆

, and

2.

✂ ✔ ✝ ✔

, such that

✂ ✁ ✂ ☎ ✂ ✂ ✔ ✂ ✂ ✆

and

✝ ✁ ✂ ☎ ✂ ✝ ✔ ✂ ✂ ✆

.

Deciding regular grammar logics with converse through GF2 – p. 17

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Semi-Thue systems - Languages

Full derivation relation

☎

: reflexive and transitive closure of

☎

.

Language

☎ ✥ ✂ ✧ ✁ ✝

✂
☎

✝

for

✂

.

Example:

✁ ✚ ✄ ✤ ✚ ✚ ✤ ✚ ✚ ✂ ✚ ✤ ✚ ✚ ✂ ✚ ✤ ✚ ✚

.

☎ ✥ ✚ ✧ ✁ ☎ ✥ ✚ ✧ ✁ ✚ ✤ ✚

.

Deciding regular grammar logics with converse through GF2 – p. 18

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Regular semi-Thue sys. with converse

✁
such that

1.

✁

✄

(disjoint subsets) based on

✂

satisfying

✂ ✥

✧

✁ ✄

,

✂ ✥ ✄ ✧ ✁

,

and

✚ ✁ ✚

for every

✚

, 2. is fi nite and

✁

(context-freeness),

3.

✂ ✝

iff

✂ ✝

(

✚ ✂ ✂ ✁ ✂ ✂ ✚

and

✄ ✁ ✄

),

4. each

☎ ✥ ✚ ✧

is a regular language.

Deciding regular grammar logics with converse through GF2 – p. 19

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Example of semi-Thue systems

✁

✚

✤

✚

✤

✂
✤
✤
✄

plus the converse rules:

✚

✤

✚

✤

✂
✤
✤
✄

☎ ✥ ✚ ✧ ✁ ✚

,

☎ ✥

✧

✁ ✚ ✤ ✚ ✤

✤
.

☎ ✥ ✚ ✧ ✁ ✚

,

☎ ✥

✧

✁ ✚ ✤ ✚ ✤

✤
.

✁

modal axioms for K

✁

+ universal modality

✢ ✣

.

Deciding regular grammar logics with converse through GF2 – p. 20

SLIDE 21

✁
frames
✙

✤ ✂ ✜

frame

✁ ✙ ✤ ✜

: for

✚

,

✦ ✁

and

✦ ✁ ✄ ☎ ✦

.

satisfi es

def for

✂ ✝

,

✂ ✄

with

☎ ✁ ✙ ✤ ✜

,

✦✝✆ ✄ ✁ ✦

✄

.

Deciding regular grammar logics with converse through GF2 – p. 21

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Regular grammar logics with converse

Def. Logic characterized by a class
f

✙ ✤ ✂ ✜

frames defi ned by a regular semi-Thue
sys. with converse.

They are Sahlqvist’s modal logics with frame conditions in

☎

. Relational translation works but not into GF.

Deciding regular grammar logics with converse through GF2 – p. 22

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Standard modal logics

logic

✠

✁

✦ ✂

frame condition K

✄ ✦ ☎

(none) KT

✄ ✦✝✆ ☎ ☎

reflexivity KB

✄ ✦✝✆ ✦ ☎

symmetry KTB

✄ ✦✝✆ ✦✝✆ ☎ ☎

refl. and sym.

K4

✄ ✦ ☎ ✆ ✄ ✦ ☎ ✞

transitivity KT4 = S4

✄ ✦ ☎ ✞

refl. and trans.

KB4

✄ ✦ ✆ ✦ ☎ ✆ ✄ ✦✝✆ ✦ ☎ ✞

sym. and trans.

K5

✁ ✄ ✦ ☎ ✆ ✄ ✦✝✆ ✦ ☎ ✞ ✆ ✄ ✦ ☎ ✂ ✟ ✄ ✦ ☎

euclideanity KT5 = S5

✄ ✦ ✆ ✦ ☎ ✞

equivalence rel. K45

✁ ✄ ✦ ☎ ✞ ✆ ✄ ✦ ☎ ✂ ✞

trans. and eucl.

Deciding regular grammar logics with converse through GF2 – p. 23

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Numerous other logics in the class

Description logics (with role hierarchy,

transitive roles),

Knowledge logics, e.g. S5
(DE),
Bimodal logics for intuitionistic modal logics

IntK

✁

,

Extensions with the universal modality,
Information logics.

Deciding regular grammar logics with converse through GF2 – p. 24

SLIDE 25

From frames to -frames

☎ ✥ ✧

def

✁ ✙ ✤ ✔ ✜

with

✔ ✦ ✁ ✄ ★ ✠

✁

✦ ✂ ✄

.

☎ ✁

inverse substitution

✄ ☎

(Caucal 96) with

✚

✁ ☎ ✥ ✚ ✧

.

Lemma.

☎ ✥ ✧

is the smallest

✙ ✤ ✂ ✜

frame

including and satisfying (CF semi-Thue

sys. with converse).

Corollary.

satisfi es iff

☎ ✥ ✧ ✁

.

Deciding regular grammar logics with converse through GF2 – p. 25

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Principles of the translation

Previous work: S4, S5, K5 into GF2 with ad hoc

method (de Nivelle 2001).

Simulation of the propagation of formulae as in

analytic proof systems.

Differences with the relational translation:

1. not an exact encoding of the semantics;
2. built-in encoding of the frame conditions.

Deciding regular grammar logics with converse through GF2 – p. 26

SLIDE 27

Input formula

W.l.o.g.

in NNF (

✓

nly in front of ).

For

✚

,

☎ ✥ ✚ ✧

is regular.

Finite state automaton

✦ ✁ ✙ ✤

✁

✤ ✂✁ ✤ ✜

,

✁

✥ ✄ ✧ ✥ ✧

(with

✄

transitions)

✥ ✦ ✧ ✁ ☎ ✥ ✚ ✧

.

✄

✦ ✁ ✙ ✤

✤
✁

✤ ✜

(

).

Deciding regular grammar logics with converse through GF2 – p. 27

SLIDE 28

Predicate symbols

accessibility relation: binary

✦

(

✚

).

proposition: unary

for .

automaton predicates: unary

for

✢ ✚ ✣ ✁ ✂

✥

✧

and

in

✦

. If

✁

initial state of

✦

,

✁ ✆

✥

✂ ✧

intends to mean: “

✢ ✚ ✣

holds in the state assigned to

✂

”

✁ ✆

✥

✂ ✧ ✁

renaming of

✥

✢ ✚ ✣ ✤ ✂ ✤ ✧

.

Deciding regular grammar logics with converse through GF2 – p. 28

SLIDE 29

Interpreting

✢ ✚ ✣ ✁ ✂

✥

✧

,

in

✦

.

model

with

✤ ✝ ✢ ✂ ✁ ✣ ✒ ✁ ✞ ✟ ✠

✥

✂ ✧

iff whenever

✁ ✦ ☎ ☎ ✦

✦✝✆

✞

&

✚ ☎

✚

✞ ✥ ✄ ✦ ✧

,

✤ ✝ ✢ ✂ ✞ ✣ ✒ ✁

.

Deciding regular grammar logics with converse through GF2 – p. 29

SLIDE 30

Encoding the automata

✢ ✚ ✣ ✁ ✂

✥

✧

&

✦ ✁ ✙ ✤

✁

✤

✁

✤ ✜

.

✦

✁

is the conjunction of (

✂ ✤ ✁

✁

✤

☎

)

✂

✦ ✥ ✂ ✤ ✧ ✥

✥

✂ ✧ ✂

✥

✧ ✧

if

✦

✄

and

✚

,
✂

✦ ✥ ✤ ✂ ✧ ✥

✥

✂ ✧ ✂

✥

✧ ✧

if

✦

✄

and

✚

,
✂
✥

✂ ✧ ✂

✥

✂ ✧

if

☎

✄

,

✂

✁ ✆

✥

✂ ✧

✥

✤ ✂ ✤ ✧

(

✥
✤
✤
✧

to be defi ned).

Deciding regular grammar logics with converse through GF2 – p. 30

SLIDE 31 ✢ ✚ ✣

for K5

✦

✁
☎
✁

✚ ✚ ✤ ✚ ✚ ✚ ✂ ✥ ✂ ✤ ✧ ✁ ✆ ✂ ✥ ✂ ✧ ✁ ✆ ✂ ✥ ✧ ✂ ✥ ✤ ✂ ✧ ✁ ✆ ✂ ✥ ✂ ✧ ☎ ✆ ✂ ✥ ✧ ✂ ✥ ✂ ✤ ✧ ☎ ✆ ✂ ✥ ✂ ✧ ☎ ✆ ✂ ✥ ✧ ✂ ✥ ✤ ✂ ✧ ☎ ✆ ✂ ✥ ✂ ✧ ☎ ✆ ✂ ✥ ✧ ✂ ✥ ✂ ✤ ✧ ☎ ✆ ✂ ✥ ✂ ✧ ✁ ✆ ✂ ✥ ✧ ✂ ✁ ✆ ✂ ✥ ✂ ✧ ✥ ✂ ✧

Deciding regular grammar logics with converse through GF2 – p. 31

SLIDE 32

Local propagation of constraints

✂ ✥ ✂ ✤ ✧ ✁ ✆ ✂ ✥ ✂ ✧ ✁ ✆ ✂ ✥ ✧ ✂ ✥ ✤ ✂ ✧ ✁ ✆ ✂ ✥ ✂ ✧ ☎ ✆ ✂ ✥ ✧

. . .

✂ ✥ ✂ ✤ ✧ ☎ ✆ ✂ ✥ ✂ ✧ ✁ ✆ ✂ ✥ ✧ ✂ ✁ ✆ ✂ ✥ ✂ ✧ ✥ ✂ ✧

If

✤ ✝ ✒ ✁ ✞ ✟ ✠

✦

✁ ✂ ✁ ✆ ✂ ✥ ✂ ✧

and,

✝ ✥ ✂ ✧ ✦ ☎ ☎

✦✝✆

✞

&

✚ ☎

✚

✞ ☎ ✥ ✚ ✧

then

✤ ✝ ✢ ✂ ✞ ✣ ✒ ✁ ✞ ✟ ✠ ✥ ✂ ✧

.

Deciding regular grammar logics with converse through GF2 – p. 32

SLIDE 33

Translation

✥

✁ ✤ ✂ ✤ ✧ ✁ ✁ ✥ ✂ ✧ ✁ ✄ ✤ ✓

✥

☎ ✆ ✤ ✂ ✤ ✧ ✁

✥

☎ ✤ ✂ ✤ ✧

✥

✆ ✤ ✂ ✤ ✧

✥

✙✛✚ ✜ ✤ ✂ ✤ ✧ ✁ ✦ ✥ ✂ ✤ ✧

✥

✤ ✤ ✂ ✧

for

✚

(forward witness)
✥

✙✛✚ ✜ ✤ ✂ ✤ ✧ ✁ ✦ ✥ ✤ ✂ ✧

✥

✤ ✤ ✂ ✧

for

✚ ✄

(backward wit.)

✥

✢ ✚ ✣ ✤ ✂ ✤ ✧ ✁

✆
✥

✂ ✧

✁

initial state of

✦

Deciding regular grammar logics with converse through GF2 – p. 33

SLIDE 34

Translation of

✥ ✧ ✁ ✥

✦

✁

★✁

✄ ✂ ✁ ✄ ✂

✦

✁

✧
✥

✤

✁

✤

☎

✧

✂

is the only free variable in

✥

✤ ✂ ✤ ✧

.

✥

✧

is in GF with variables

✁

✤

☎

.

Size of

✥ ✧

in

✥ ✒ ✒ ✧

.

✥ ✧

can be computed in logspace in

✒ ✒

( is the size of the largest

✦

).

Deciding regular grammar logics with converse through GF2 – p. 34

SLIDE 35

Satisfiability preservation

Theorem.

is -satisfi able iff

✥ ✧

is satisfi able in GF2.

Important points in the proof:

regularity of the languages

☎ ✥ ✚ ✧

s,

existence of a closure operator on

✙ ✤ ✂ ✜

frames.

Extensions: nominals, universal modality,

intuitionistic (modal) logics.

Deciding regular grammar logics with converse through GF2 – p. 35

SLIDE 36

Regular languages and FOL

✁

✚

✚

✤ ✚ ✄ ✤ ✚

✚

✤ ✚ ✄ ☎ ✥ ✚ ✧ ✁ ✥

✧
✥

✚ ✄ ✧

is not star-free.

There is no CF semi-Thue sys.

such that

☎ ✥ ✚ ✧ ✁ ✥ ✚

✧
and

✥ ✚

✧
is star-free.
No encoding of regular languages into GF2

but rather encoding of modal logics whose frame conditions satisfy some regularity conditions.

Deciding regular grammar logics with converse through GF2 – p. 36

SLIDE 37

Some open questions

1. Extension to decidable CF grammar logics

(undecidability in full generality).

✞ ✁ ✚ ✚ ✞ ✚ ✤ ✚ ✚ ✚ ✞ ☎ ✆ ✥ ✚ ✧

is not regular for

✁

but

✞

satisfi ability is decidable (Gabbay 75).
2. Characterize the complexity of
satisfi ability by analyzing the

☎ ✥ ✚ ✧

s?

3. Design a PSPACE fragment of GF.

Deciding regular grammar logics with converse through GF2 – p. 37