implementation of an interleaving semantics for tlda

7 May 2005

Implementation of anInterleaving Semantics for TLDA

Luhme XI

Niels Lohmann

[email protected]://www.informatik.hu-berlin.de/~nlohmann/arbeit

Luhme XI

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Introduction to TLDA

The semantic model of TLDA is a run.

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!

semantic model and basic concepts

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TLDA

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basic concepts:

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!


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TLDA

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basic concepts:

! transitions update values

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!


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basic concepts:


! …0f involved variables

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basic concepts:


! …0f involved variables

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!


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dom(t1)={agent1} dom(t2)={agent1, sem}


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basic concepts (continued):

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! C


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! a cut can be understood as a global state

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! C


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! a cut can be understood as a global state

! every run has an initial cut

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! CC0


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§

! C C'


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! when all possible transitions occur, the successor cut C' is reached

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! C C'


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

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! C C'


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

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! C C'


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this cut cannot be reached with steps

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syntactic elements:

! variables

syntax

constants


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syntactic elements:

! variables! rigid variables ( )

syntax

variables describingvalues in current cut Introduction to

TLDA

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syntactic elements:

! variables! rigid variables ( )! flexible variables ( )

syntax

variables describing values in successor cut


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syntactic elements:

! variables! rigid variables ( )! flexible variables ( )! primed flexible variables ( )

syntax

variables describinginvolvedness of setsof flexible variables


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Luhme XI

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syntactic elements:

! variables! rigid variables ( )! flexible variables ( )! primed flexible variables ( )! ~-variables ( )

syntax


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syntactic elements:


! formulas

syntax

formulas evaluated in steps of a run


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syntactic elements:


! formulas! step formulas

syntax

step formulas with only flexible variables


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syntactic elements:


! formulas! step formulas! state predicate

syntax

step formulas with only flexible variables


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syntactic elements:


! formulas! step formulas! state predicate! run formulas

syntax

formulas evaluated in a whole run


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evaluating step formulas:

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! C C'

evaluating formulas


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evaluating formulas


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Is there a transitionoccurring in SC involvingboth agent1 and sem?

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evaluating formulas


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Luhme XI

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! C C'There is no transitioninvolving agent2.

evaluating formulas


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evaluating run formulas:

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evaluating formulas


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Luhme XI

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idlet1

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! C0 C1 C2 C3

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evaluating formulas


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idlet1

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! C0 C1 C2 C3

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evaluating formulas


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Luhme XI

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Semantics

! TLDA: partial order semantics! transitions are partially ordered

u1 occurs before u2

! concurrent transitions have no order

t2idle pending

t1idle pending Which transition

occurs first?

partial ordered semantics

u11 0 u2 1


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Luhme XI

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Semantics

! other formalisms (e.g. TLA):interleaving semantics! all possible interleavings of transitions

! totally ordered! exponential number of interleavings

and intermediate states

First t1, then t2!

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t1 pendingidle

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t2 idlepending

t1 pendingpending

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{t1, t2} pendingpending

First t2, then t1!

t1 and t2 occur concurrently!

interleaving semantics


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Luhme XI

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Semantics

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

! model checking:! few experiences with partial order

semantics (complicated data structures)! interleavings can be easily represented by a

labelled graph (transition system)! explicit model checkers (LoLA, Spin) base

on graph search in a transition system

! Next subject: build a transition system

interleaving semantics for TLDA


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Transition System

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

sem ! 1 sem ! 0 sem ! 1

s0 s1 s2

definition


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Transition System

formally a quintuple TS = (S, S0, Act, R, L)! S: nonempty set of states

sem ! 1 sem ! 0 sem ! 1

s0 s1 s2

definition


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Transition System


! S0 " S: nonempty set of initial states

sem ! 1 sem ! 0 sem ! 1

s0 s1 s2

definition


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Transition System



! Act: finite set of actions

sem ! 1 sem ! 0 sem ! 1{a0} {a1}

{a2, a3}

s0 s1 s2

definition


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Transition System




! R " S # (Act) # S: transition relation, labeled with a nonempty set of actions

sem ! 1 sem ! 0 sem ! 1{a0} {a1}

{a2, a3}

s0 s1 s2

definition


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Transition System




! R " S # (Act) # S: transition relation, labeled with a nonempty set of actions

! L: S " (Var " Val): function to label states with a mapping variables/values

sem ! 1 sem ! 0 sem ! 1{a0} {a1}

{a2, a3}

s0 s1 s2

definition


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Transition System

We can construct a transition system TS!

for TLDA specifications ! in normal form:

! Definition: normal form:! , Init $ �Next $ Progress

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

state predicate step formula ignored right now

construction


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Transition Systemconstruction


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! C0

sC0

! States, initial states, labels

For each cut reachable with steps from C0,add a new state and label it.

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! C0 C1

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Luhme XI

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! C0 C1 C2

sC0sC1

sC2


Repeat this for all runs of !.


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idle

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! C0 C1 C2

sC0sC1

sC2

! Transition relation:

Repeat this for all runs of !.

Each step fulfils a set of clauses Ai of Next.Label the new transition relation with Ai (Ai " Act).

A0 A1

Luhme XI

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Transition System

properties of TS!:

! For all runs of !, for all cuts C reachable with steps from the initial cut, there exists a corresponding state sC in TS!

reachable from an initial state.

! For all states sC in TS! reachable from

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

properties


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"

Transition System

Do we reach all cuts with steps from C0?

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C0 C1 C2 C3

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u1 " u2 " "

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!*

properties


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"

Transition System


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C0 C1 C2 C3

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u1 " u2 " "

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!*

properties


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"

Transition System


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C0 C1 C2

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u1 " u2 " "

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Luhme XI

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"

Transition System


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C0 C1 C2

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!properties


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Luhme XI

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Transition System

Does "# ⊨ ! hold?

But it always holds for environment invariant formulas.

Can be understood literally:A formula ! is called environment invariant i! for all runs " with " ⊨ ! holds:when "# has the same restrictions on the system variables as ", then "# ⊨ !.

Not necessarily!properties


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Luhme XI

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Transition System

Properties of TS! (! environment

invariant):

! For all runs of !, for all cuts C reachable with steps from the initial cut, there exists a corresponding state sC in TS!

reachable from an initial state.

! For all states sC in TS! reachable from

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

properties revised


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Luhme XI

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Transition System

Properties of TS! (! environment

invariant):1. Every cut is represented by a state.

2. Every run is represented by a sequence of states beginning at an initial state.

3. Every state represents a cut.

4. Every sequence of states beginning at an initial state represents a run.

All interleavings of ! are contained in TS!.

! TS! is a TLDA interleaving semantics.

propositions


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Implementation

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

! present features:! parse a TLDA specification! build a DNF-tree! discard irrelevant/contradicting actions! create C-code to build the transition system

! future:! better heuristics! TLDA model checking! reduction techniques

overview


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Implementation

creation of actions/pruning of contradictions

" "

" "

DNF-tree (n=4)

pruning heuristics


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Implementation


" "

" "

DNF-tree (n=4)

pruning heuristics


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#

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Implementation


" "

" "

DNF-tree (n=4)

pruning heuristics


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continue here…

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Implementation

treating ~-variables

! most actions contradict because of~-variables

! ! “the more, the better”! a lot of knowledge can be gained:

! d ! ! ! ! "

gaining knowledge


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Examples

requesting1

idle1

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requesting2

idle2

semaphore

mutual exclusion


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First example: mutual exclusion

! Petri net

! TLDA: three components/variables

Examples

requesting1

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requesting2

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semaphore

mutual exclusion


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Examplesmutual exclusion


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Tool demonstration #1

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Examplesmutual exclusion

state inscriptions:

agent1 agent2 semaphore


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Examplescrossing the river


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! Second example:! farmer bought a wolf, a goat and cabbage! all have to cross a river with a small boat! the farmer can only take one thing with him! left unattended, the goat eats the cabbage! left unattended, the wolf eats the goat

Can they cross the river safely?

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Tool demonstration #2

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state inscriptions:

farmer goat wolf cabbage


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initial state (all at east side of the river)

state inscriptions:



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initial state (all at east side of the river)

goal state(all at westside of the river)

state inscriptions:



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Luhme XI

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Outlook

! many exponential problems:! disjunctive normal form:

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

! ~-variables:Dining philosophers’ problem could not yet be modelled: 5 forks and 5 philosophers lead to 1023 ~-variables and 2046 variables in C++. (a new data structure might help)

! actions:step-explosion (Stephan Roch’s work)

problems


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Outlook

! TLDA inapplicable for explicit model checking?

! Would symbolic model checking help?

! Any solutions for open problems?

! Any questions about my Studienarbeit?

! Ideas for subjects for a Diplomarbeit?

future/further work


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7 May 2005

Implementation of anInterleaving Semantics for TLDA

Thank you!

Niels Lohmann

[email protected]://www.informatik.hu-berlin.de/~nlohmann/arbeit

implementation of an interleaving semantics for tlda

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