Implementation of an Interleaving Semantics for TLDA

Implementation of an
Interleaving Semantics for TLDA
Luhme XI

Niels Lohmann
nlohmann@informatik.hu-berlin.de
http://www.informatik.hu-berlin.de/~nlohmann/arbeit

7 May 2005

Introduction to TLDA
semantic model and basic concepts

The semantic model of TLDA is a run.
! Luhme XI
idle pending critical idle
1 0 1 0
idle pending critical
Introduction to
§ TLDA
Semantics
Transition System
Implementation
Examples
Outlook

2


! Luhme XI
1 0 1 0
Introduction to
§ TLDA
Semantics
Transition System
basic concepts: Implementation
Examples
Outlook

2


! Luhme XI
1 0 1 0
Introduction to
§ TLDA
Semantics
Transition System
Examples
! transitions update values Outlook

2


! Luhme XI
idle t1 pending critical idle
t2
1 0 1 0
Introduction to
§ TLDA
Semantics
Transition System
Examples

! …0f involved variables

2


! Luhme XI
t2
1 0 1 0
Introduction to
§ TLDA
Semantics
Transition System
Examples

! …0f involved variables
dom(t1)={agent1} dom(t2)={agent1, sem}
2


C
! Luhme XI
1 0 1 0
Introduction to
§ TLDA
Semantics
Transition System
basic concepts (continued): Implementation
Examples
Outlook

3


C
! Luhme XI
1 0 1 0
Introduction to
§ TLDA
Semantics
Transition System
Examples
! a cut can be understood as a global state Outlook

3


C0 C
! Luhme XI
1 0 1 0
Introduction to
§ TLDA
Semantics
Transition System
Examples
! a cut can be understood as a global state Outlook

! every run has an initial cut

3


C C'
! Luhme XI
1 0 1 0
idle t2 pending critical
Introduction to
§ TLDA
Semantics
Transition System
Examples
Outlook

4


C C'
! Luhme XI
1 0 1 0
Introduction to
§ TLDA
Semantics
Transition System
Examples
! when all possible transitions occur, the Outlook

successor cut C' is reached

4


C C'
! Luhme XI
1 0 1 0
Introduction to
§ TLDA
Semantics
Transition System
Examples

! cuts C, C' and the occurred transitions
form a step 4


C C'
! Luhme XI
1 0 1 0
Introduction to
§
this cut cannot be reached with steps TLDA
Semantics
Transition System
Examples

! cuts C, C' and the occurred transitions
form a step 4

syntax

syntactic elements:
! variables Luhme XI

constants

Introduction to
TLDA
Semantics
Transition System
Implementation
Examples
Outlook

5

syntax

syntactic elements:

! rigid variables ( )
variables describing
values in current cut Introduction to
TLDA
Semantics
Transition System
Implementation
Examples
Outlook

5

syntax

syntactic elements:

! ﬂexible variables ( )
Introduction to
TLDA
Semantics
variables describing values in successor cut
Transition System
Implementation
Examples
Outlook

5

syntax

syntactic elements:

Introduction to
! primed ﬂexible variables ( ) TLDA
Semantics
variables describing Transition System
involvedness of sets Implementation
of ﬂexible variables
Examples
Outlook

5

syntax

syntactic elements:

Introduction to
Semantics
! ~-variables ( ) Transition System
Implementation
Examples
Outlook

5

syntax

syntactic elements:

Introduction to
Semantics
Implementation
! formulas Examples
Outlook
formulas evaluated in steps of a run

5

syntax

syntactic elements:

Introduction to
Semantics
Implementation
! formulas Examples

! step formulas Outlook

step formulas with only ﬂexible variables

5

syntax

syntactic elements:

Introduction to
Semantics
Implementation
! formulas Examples


! state predicate step formulas with only ﬂexible variables

5

syntax

syntactic elements:

Introduction to
Semantics
Implementation
! formulas Examples


! state predicate
! run formulas formulas evaluated in a whole run

5

evaluating formulas

evaluating step formulas:
C C'
! Luhme XI
t
1 0 1 0
Introduction to
§ TLDA
Semantics
Transition System
Implementation
Examples
Outlook

6

evaluating formulas

C C'
! Luhme XI
t
1 0 1 0
Introduction to
§ TLDA
Semantics
Transition System
Implementation
Examples
Outlook

Is there a transition
occurring in SC involving
both agent1 and sem?
6

evaluating formulas

There is no transition
C C'
! involving agent2. Luhme XI
t
1 0 1 0
Introduction to
§ TLDA
Semantics
Transition System
Implementation
Examples
Outlook

6

evaluating formulas

evaluating run formulas:
! Luhme XI
t1 t2
1 0 1
idle pending
Introduction to
§ TLDA
Semantics
Transition System
? Implementation
Examples
Outlook

7

evaluating formulas

C0 C1 C2 C3
! Luhme XI
t1 t2
1 0 1
idle pending
Introduction to
§ TLDA
Semantics
Transition System
? Implementation
Examples
Outlook

7

evaluating formulas

C0 C1 C2 C3
! Luhme XI
t1 t2
1 0 1
idle pending
Introduction to
§ TLDA
Semantics

!
?
Transition System
Implementation
Examples
Outlook

7

Semantics
partial ordered semantics

! TLDA: partial order semantics
! transitions are partially ordered Luhme XI

1 u1 0 u2 1

Introduction to
u1 occurs before u2 TLDA
Semantics
Transition System

! concurrent transitions have no order Implementation
Examples
Outlook
idle pending Which
t1
transition
idle t2 pending occurs ﬁrst?

8

Semantics
interleaving semantics

! other formalisms (e.g. TLA):
interleaving semantics Luhme XI

! all possible interleavings of transitions
idle t1 pending t2 pending First t1,
idle idle pending then t2! Introduction to
TLDA

idle t2 idle t1 pending Semantics
idle pending pending First t2, Transition System
then t1! Implementation
idle {t1, t2} pending Examples
idle pending Outlook
t1 and t2 occur
concurrently!
! totally ordered
! exponential number of interleavings
and intermediate states 9

Semantics for TLDA
interleaving semantics

ﬁrst subject of my work: development of
an interleaving semantics for TLDA Luhme XI

! model checking:
! few experiences with partial order
semantics (complicated data structures) Introduction to
TLDA

! interleavings can be easily represented by a Semantics
Transition System
labelled graph (transition system) Implementation
Examples
! explicit model checkers (LoLA, Spin) base Outlook
on graph search in a transition system

! Next subject: build a transition system 10

Transition System
deﬁnition

formally a quintuple TS = (S, S0, Act, R, L)
Luhme XI

Introduction to
TLDA
Semantics
Transition System
Implementation
Examples
Outlook

sem ! 1 sem ! 0 sem ! 1
s0 s1 s2
11

Transition System
deﬁnition

! S: nonempty set of states Luhme XI

Introduction to
TLDA
Semantics
Transition System
Implementation
Examples
Outlook

sem ! 1 sem ! 0 sem ! 1
s0 s1 s2
11

Transition System
deﬁnition


! S0 " S: nonempty set of initial states

Introduction to
TLDA
Semantics
Transition System
Implementation
Examples
Outlook

sem ! 1 sem ! 0 sem ! 1
s0 s1 s2
11

Transition System
deﬁnition


! Act: ﬁnite set of actions Introduction to
TLDA
Semantics
Transition System
Implementation
Examples
Outlook
{a2, a3}

sem ! 1 sem ! 0 sem ! 1
{a0} {a1}
s0 s1 s2
11

Transition System
deﬁnition


TLDA
! R " S # (Act) # S: transition relation, labeled Semantics
with a nonempty set of actions Transition System
Implementation
Examples
Outlook
{a2, a3}

sem ! 1 sem ! 0 sem ! 1
{a0} {a1}
s0 s1 s2
11

Transition System
deﬁnition


TLDA
! R " S # (Act) # S: transition relation, labeled Semantics
with a nonempty set of actions Transition System
Implementation
! L: S " (Var " Val): function to label states Examples

with a mapping variables/values Outlook
{a2, a3}

sem ! 1 sem ! 0 sem ! 1
{a0} {a1}
s0 s1 s2
11

Transition System
construction

We can construct a transition system TS!
for TLDA speciﬁcations ! in normal form: Luhme XI

! Deﬁnition: normal form: Introduction to
TLDA
! , Init $ Next $ Progress Semantics
Transition System
state predicate step formula ignored right now Implementation
Examples
Outlook

! Actions:
Act is the set of the clauses of the
disjunctive normal form of Next. 12

Transition System
construction

! States, initial states, labels
C0
! Luhme XI
1 0 1 0
Introduction to
§ TLDA
Semantics

For each cut reachable with steps from C0, Transition System
Implementation
add a new state and label it. Examples
sC Outlook
0

13

Transition System
construction

C0 C1
! Luhme XI
1 0 1 0
Introduction to
§ TLDA
Semantics

Implementation
sC sC Outlook
0 1

13

Transition System
construction

C0 C1 C2
! Luhme XI
1 0 1 0
Introduction to
§ TLDA
Semantics

Implementation
sC sC sC Outlook
0 1 2

Repeat this for all runs of !. 13

Transition System
construction

! Transition relation:
C0 C1 C2
! Luhme XI
1 0 1 0
Introduction to
§ TLDA
Semantics

Each step fulﬁls a set of clauses Ai of Next. Transition System
Implementation
Label the new transition relation with Ai (Ai " Act). Examples
sC sC sC Outlook
0 1 2

A0 A1

Repeat this for all runs of !. 14

Transition System
properties

properties of TS!:
Luhme XI

! For all runs of !, for all cuts C reachable
with steps from the initial cut, there Introduction to

exists a corresponding state sC in TS! TLDA
Semantics

reachable from an initial state. Transition System
Implementation

! For all states sC in TS! reachable from Examples
Outlook

an initial state, there exists a run of !
with a corresponding cut C reachable
with steps from the initial cut.
15

Transition System
properties

Do we reach all cuts with steps from C0?
Luhme XI

!* C0 C1 C2 C3

t2 t4 Introduction to
1 0 1 0 TLDA
t5
idle t3 pending critical Semantics

§ Transition System
" u1 " u2 " " Implementation
Examples
Outlook

16

Transition System
properties

Luhme XI

!* C0 C1 C2 C* C3

1 0 1 0 TLDA
t5

§ Transition System
Examples
Outlook

16

Transition System
properties

Luhme XI

!* C0 C1 C2 C*

1 0 1 0 TLDA
t5
Transition System
Examples
§
Outlook

16

Transition System
properties

Do we reach all cuts with steps from C0? !
Luhme XI

!* C0 C1 C2 C*

1 0 1 0 TLDA
t5
Transition System
Examples
§
Outlook

16

Transition System
properties

Does "# ⊨ ! hold? Not necessarily!
Luhme XI

But it always holds for environment
invariant formulas. Introduction to
TLDA
Semantics
Transition System
Can be understood literally: Implementation

A formula ! is called environment invariant Examples
Outlook

i! for all runs " with " ⊨ ! holds:
when "# has the same restrictions on the
system variables as ", then "# ⊨ !.
17

Transition System
properties revised

Properties of TS! (! environment
invariant): Luhme XI

! For all runs of !, for all cuts C reachable
with steps from the initial cut, there Introduction to

exists a corresponding state sC in TS! TLDA
Semantics

reachable from an initial state. Transition System
Implementation

! For all states sC in TS! reachable from Examples
Outlook

an initial state, there exists a run of !
with a corresponding cut C reachable
with steps from the initial cut.
18

Transition System
propositions

Properties of TS! (! environment
invariant): Luhme XI

1. Every cut is represented by a state.
2. Every run is represented by a sequence of Introduction to
states beginning at an initial state. TLDA
Semantics
3. Every state represents a cut. Transition System
Implementation
4. Every sequence of states beginning at an Examples
initial state represents a run. Outlook

All interleavings of ! are contained in TS!.
! TS! is a TLDA interleaving semantics. 19

Implementation
overview

second subject of my work:
the prototypic implementation “TLDC” Luhme XI

! present features:
! parse a TLDA speciﬁcation
Introduction to
! build a DNF-tree TLDA
Semantics
! discard irrelevant/contradicting actions Transition System

! create C-code to build the transition system Implementation
Examples

! future: Outlook

! better heuristics
! TLDA model checking
! reduction techniques 20

Implementation
pruning heuristics

creation of actions/pruning of contradictions
Luhme XI

DNF-tree (n=4)

Introduction to
TLDA
" "
Semantics
Transition System
" " Implementation
Examples
Outlook
" "

21

Implementation
pruning heuristics

Luhme XI

DNF-tree (n=4)

# " "
Introduction to
TLDA
Semantics
Transition System
" " Implementation
Examples
Outlook
" "

21

Implementation
pruning heuristics

Luhme XI

DNF-tree (n=4)

Introduction to
TLDA
" "
Semantics
Transition System
" " Implementation
Examples
continue here… Outlook
" "

21

Implementation
gaining knowledge

treating ~-variables
! most actions contradict because of Luhme XI

~-variables
! ! “the more, the better”
Introduction to

! a lot of knowledge can be gained: TLDA
Semantics

! d Transition System
Implementation
! Examples
Outlook
!
!
! "
22

Examples
mutual exclusion

Luhme XI

requesting1 requesting2

Introduction to
TLDA
critical1 critical2 Semantics
Transition System
semaphore Implementation

idle1 idle2 Examples
Outlook

23

Examples
mutual exclusion

First example: mutual exclusion
! Petri net Luhme XI

requesting1 requesting2

Introduction to
TLDA
critical1 critical2 Semantics
Transition System
semaphore Implementation

idle1 idle2 Examples
Outlook

! TLDA: three components/variables 23

Examples
mutual exclusion

Luhme XI

Introduction to
TLDA
Semantics
Transition System

Tool demonstration #1 Implementation
Examples
Outlook

24

Examples
mutual exclusion

Luhme XI

Introduction to
TLDA
Semantics
Transition System
Implementation
Examples
Outlook

state inscriptions:

agent1 agent2 semaphore
25

Examples
crossing the river

! Second example:
! farmer bought a wolf, a goat and cabbage Luhme XI

! all have to cross a river with a small boat
! the farmer can only take one thing with him
Introduction to
! left unattended, the goat eats the cabbage TLDA
Semantics
! left unattended, the wolf eats the goat Transition System
Implementation
Examples
Outlook

Can they cross the river safely? 26

Examples
crossing the river

Luhme XI

Introduction to
TLDA
Semantics
Transition System

Tool demonstration #2 Implementation
Examples
Outlook

27

Examples
crossing the river

state inscriptions:
Luhme XI
farmer goat wolf cabbage

Introduction to
TLDA
Semantics
Transition System
Implementation
Examples
Outlook

28

Examples
crossing the river
initial state (all at east side of the river)

state inscriptions:
Luhme XI

Introduction to
TLDA
Semantics
Transition System
Implementation
Examples
Outlook

28

Examples
crossing the river
initial state (all at east side of the river)

state inscriptions:
Luhme XI

Introduction to
TLDA
Semantics
Transition System
Implementation
Examples
Outlook

goal state
(all at west
side of the river) 28

Outlook
problems

! many exponential problems:
! disjunctive normal form: Luhme XI

Peterson’s Mutex-algorithm could not yet
be modelled: few lines of TLDA end up in
milliards of actions. (pruning at its limits) Introduction to
TLDA
! ~-variables: Semantics

Dining philosophers’ problem could not Transition System
Implementation
yet be modelled: 5 forks and 5 philosophers Examples
lead to 1023 ~-variables and 2046 variables Outlook

in C++. (a new data structure might help)
! actions:
step-explosion (Stephan Roch’s work)
29

Outlook
future/further work

! TLDA inapplicable for explicit model
checking? Luhme XI

! Would symbolic model checking help?
Introduction to
TLDA
Semantics

! Any solutions for open problems? Transition System
Implementation
Examples
Outlook

! Any questions about my Studienarbeit?

! Ideas for subjects for a Diplomarbeit? 30

Implementation of an
Interleaving Semantics for TLDA

Thank you!

Niels Lohmann
nlohmann@informatik.hu-berlin.de
http://www.informatik.hu-berlin.de/~nlohmann/arbeit

7 May 2005

Implementation of an Interleaving Semantics for TLDA

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