Model Checking in variants of ATL
(Do agents make model checking explode?)
Jürgen Dix
(joint work with Wojtek Jamroga)
Department of Computer Science
Clausthal University of Technology
23th March, PANAM Seminaire, LIP6, Paris
Jürgen Dix (joint work with Wojtek Jamroga) · Clausthal University of Technology
23th March, PANAM Seminaire, LIP6, Paris 1/39
Abilities of Agents
CTL
1
Abilities of Agents
CTL
ATL
Model Checking
2
A closer look
Fine-grained Analysis
Imperfect Information
Table of Complexities
Jürgen Dix (joint work with Wojtek Jamroga) · Clausthal University of Technology
23th March, PANAM Seminaire, LIP6, Paris 2/39
Abilities of Agents
CTL
Reasoning about Time: CTL
CTL: Computational Tree Logic.
Reasoning about possible computations of a system
Models: states (time points, situations), transitions
Paths: courses of action, computations;
Path quantifiers: A (for all paths), E (there is a path);
Temporal operators: i(nexttime), ♦ (sometime), (always)
and U (until);
“Vanilla” CTL: each temporal operator must be immediately
preceded by exactly one path quantifier;
Reasoning in CTL can be automatized.
Jürgen Dix (joint work with Wojtek Jamroga) · Clausthal University of Technology
23th March, PANAM Seminaire, LIP6, Paris 3/39
Abilities of Agents
CTL
Example: Rocket and Cargo
A rocket and a cargo,
The rocket can be moved between London
(proposition roL) and Paris (proposition roP),
The cargo can be in London (caL), Paris (caP), or
inside the rocket (caR),
The rocket can be moved only if it has its fuel tank
full ( f uelOK ),
When it moves, it consumes fuel, and no f uel holds
after each flight.
Jürgen Dix (joint work with Wojtek Jamroga) · Clausthal University of Technology
23th March, PANAM Seminaire, LIP6, Paris 4/39
Abilities of Agents
CTL
Example: Rocket and Cargo
1
5
roL
nofuel
caL
roL 2
fuelOK
caL
roL
nofuel
caR
roL 6
fuelOK
caR
7
roP
nofuel
caL 3
roP
fuelOK
caL 4
roP
nofuel
caR
roP
fuelOK
caR 8
roP
nofuel
caP 11
roP
fuelOK
caP 12
roL → E♦roP
A(roL ∨ roP)
roL ∧ f uelOK → E
roL
nofuel
9 caP
roL
fuelOK
10 caP
Jürgen Dix (joint work with Wojtek Jamroga) · Clausthal University of Technology
roL → A
k(roP
kroP
→ no f uel)
23th March, PANAM Seminaire, LIP6, Paris 5/39
Abilities of Agents
ATL
ATL: What Agents Can Achieve
ATL: Agent Temporal Logic [Alur et al. 1997]
Temporal logic meets game theory
Main idea: cooperation modalities
“Vanilla” ATL: temporal operators are always
preceded by exactly one cooperation modality
hhAiiΦ
stands for
coalition A has a collective strategy
to enforce Φ
Jürgen Dix (joint work with Wojtek Jamroga) · Clausthal University of Technology
23th March, PANAM Seminaire, LIP6, Paris 6/39
Abilities of Agents
ATL
ATL: What Agents Can Achieve
hhjamesbondii♦blofeldKilled:
“James Bond can ensure that Blofeld is eventually killed”
hh∅ii hhblofeldii♦blofeldEscapes:
“Blofeld is always able to eventually escape”
¬hhjamesbondii♦worldSaved ∧ ¬hhblofeldii♦worldSaved ∧
hhjamesbond, blofeldii♦worldSaved:
“Only together they can make sure that the world will be saved”
Jürgen Dix (joint work with Wojtek Jamroga) · Clausthal University of Technology
23th March, PANAM Seminaire, LIP6, Paris 7/39
Abilities of Agents
ATL
The Rocket Example: Adding Agents and Actions
3 agents,
x can load the cargo, unload it, and move the rocket,
y can unload the cargo and move the rocket,
z can load the cargo and supply the rocket with fuel
(action fuel),
Each agent can also decide to do nothing (nop:
“no-operation”);
Jürgen Dix (joint work with Wojtek Jamroga) · Clausthal University of Technology
23th March, PANAM Seminaire, LIP6, Paris 8/39
Abilities of Agents
ATL
“Moving” action: highest priority,
“Loading” is affected when the rocket does not move
and more agents try to load than to unload,
“Unloading”: similarly,
“Fueling” can be accomplished only when the rocket
tank is empty (alone or in parallel with loading or
unloading).
Jürgen Dix (joint work with Wojtek Jamroga) · Clausthal University of Technology
23th March, PANAM Seminaire, LIP6, Paris 9/39
Abilities of Agents
ATL
Rocket Example: Adding Agents and Actions
<load1,nop2,fuel>
<nop1,nop2,fuel>
<load1,unload2,fuel>
<nop1,nop2,nop3>
<unload1,nop2,fuel>
<load1,unload2,nop3> <unload ,unload ,fuel>
1
2
<unload1,nop2,nop3>
<nop1,unload2,fuel>
<unload1,unload2,nop3>
<nop1,unload2,load3>
1
5
9
roL
nofuel
caL
roL 2
fuelOK
caL
roL
nofuel
caR
roL 6
fuelOK
caR
roL
nofuel
caP
roL
fuelOK
caP
10
Jürgen Dix (joint work with Wojtek Jamroga) · Clausthal University of Technology
<load1,nop2,nop3>
<nop1,nop2,load3>
<load1,nop2,load3>
<load1,unload2,load3>
7
roP
nofuel
caL 3
roP
fuelOK
caL 4
roP
nofuel
caR
roP
fuelOK
caR 8
roP
nofuel
caP 11
roP
fuelOK
caP 12
23th March, PANAM Seminaire, LIP6, Paris 10/39
Abilities of Agents
ATL
Models:
Concurrent Game Structures
(CGS)
M = hAgt, Q, Π, π, Act, d, oi
Agt: a finite set of all agents
Q: a set of states
Π: a set of atomic propositions
π : Q → P (Π): a valuation of propositions
Act : a finite set of (atomic) actions
d : Agt × Q → P (Act) defines actions available to
an agent in a state
o: a (deterministic) transition function that
assigns outcome states q0 = o(q, α1 , . . . , αk ) to
states and tuples of actions
Jürgen Dix (joint work with Wojtek Jamroga) · Clausthal University of Technology
23th March, PANAM Seminaire, LIP6, Paris 11/39
Abilities of Agents
ATL
Strategies and Paths
Strategy is a conditional plan
sa : Q → Act (memoryless)
Path is an infinite sequence of states that can be
affected by subsequent transitions
Paths refer to possible courses of action
SA = hsa1 , sa2 , . . . , sar i collective strategy for
A = {a1 , a2 , . . . , ar }
Function out(q, SA ) returns the set of all paths that
may result from agents A executing strategy SA from
state q onward.
Jürgen Dix (joint work with Wojtek Jamroga) · Clausthal University of Technology
23th March, PANAM Seminaire, LIP6, Paris 12/39
Abilities of Agents
ATL
Semantics (1)
M, q |= p iff p ∈ π(q)
(where p ∈ Π);
M, q |= ¬ϕ iff M, q 6|= ϕ;
M, q |= ϕ ∨ ψ iff M, q |= ϕ or M, q |= ψ;
Jürgen Dix (joint work with Wojtek Jamroga) · Clausthal University of Technology
23th March, PANAM Seminaire, LIP6, Paris 13/39
Abilities of Agents
ATL
Semantics (2)
M, q |= hhAii
kϕ
iff there is a collective strategy SA
such that, for every path Λ ∈ out(q, SA ), we have
M, Λ[1] |= ϕ;
M, q |= hhAiiϕ iff there exists SA such that, for
every Λ ∈ out(q, SA ), we have M, Λ[i] |= ϕ for every
i ≥ 0;
M, q |= hhAiiϕ U ψ iff there exists SA such that, for
every Λ ∈ out(q, SA ), we have M, Λ[i] |= ψ for some
i ≥ 0, and M, Λ[ j] |= ϕ for every 0 ≤ j < i.
Jürgen Dix (joint work with Wojtek Jamroga) · Clausthal University of Technology
23th March, PANAM Seminaire, LIP6, Paris 14/39
Abilities of Agents
ATL
Rocket Agents Again
<load1,nop2,fuel>
<nop1,nop2,fuel>
<load1,unload2,fuel>
<nop1,nop2,nop3>
<unload1,nop2,fuel>
<load1,unload2,nop3> <unload ,unload
1
2,fuel>
<unload1,nop2,nop3>
<nop1,unload2,fuel>
<unload1,unload2,nop3>
<nop1,unload2,load3>
1
5
9
roL
nofuel
caL
roL 2
fuelOK
caL
roL
nofuel
caR
roL 6
fuelOK
caR
roL
nofuel
caP
roL
fuelOK
caP
10
<load1,nop2,nop3>
<nop1,nop2,load3>
<load1,nop2,load3>
<load1,unload2,load3>
7
roP
nofuel
caL 3
roP
fuelOK
caL 4
roP
nofuel
caR
roP
fuelOK
caR 8
roP
nofuel
caP 11
roP
fuelOK
caP 12
no f uel → hh3iino f uel
Jürgen Dix (joint work with Wojtek Jamroga) · Clausthal University of Technology
23th March, PANAM Seminaire, LIP6, Paris 15/39
Abilities of Agents
ATL
<load1,nop2,fuel>
<nop1,nop2,fuel>
<load1,unload2,fuel>
<nop1,nop2,nop3>
<unload1,nop2,fuel>
<load1,unload2,nop3> <unload ,unload
1
2,fuel>
<unload1,nop2,nop3>
<nop1,unload2,fuel>
<unload1,unload2,nop3>
<nop1,unload2,load3>
1
5
9
roL
nofuel
caL
roL 2
fuelOK
caL
roL
nofuel
caR
roL 6
fuelOK
caR
roL
nofuel
caP
roL
fuelOK
caP
10
<load1,nop2,nop3>
<nop1,nop2,load3>
<load1,nop2,load3>
<load1,unload2,load3>
7
roP
nofuel
caL 3
roP
fuelOK
caL 4
roP
nofuel
caR
roP
fuelOK
caR 8
roP
nofuel
caP 11
roP
fuelOK
caP 12
no f uel → hh3iino f uel
Jürgen Dix (joint work with Wojtek Jamroga) · Clausthal University of Technology
23th March, PANAM Seminaire, LIP6, Paris 16/39
Abilities of Agents
ATL
<load1,nop2,fuel>
<nop1,nop2,fuel>
<load1,unload2,fuel>
<nop1,nop2,nop3>
<unload1,nop2,fuel>
<load1,unload2,nop3> <unload ,unload
1
2,fuel>
<unload1,nop2,nop3>
<nop1,unload2,fuel>
<unload1,unload2,nop3>
<nop1,unload2,load3>
1
5
9
roL
nofuel
caL
roL 2
fuelOK
caL
roL
nofuel
caR
roL 6
fuelOK
caR
roL
nofuel
caP
roL
fuelOK
caP
10
<load1,nop2,nop3>
<nop1,nop2,load3>
<load1,nop2,load3>
<load1,unload2,load3>
7
roP
nofuel
caL 3
roP
fuelOK
caL 4
roP
nofuel
caR
roP
fuelOK
caR 8
roP
nofuel
caP 11
roP
fuelOK
caP 12
no f uel → hh3iino f uel
Jürgen Dix (joint work with Wojtek Jamroga) · Clausthal University of Technology
23th March, PANAM Seminaire, LIP6, Paris 17/39
Abilities of Agents
ATL
<load1,nop2,fuel>
<nop1,nop2,fuel>
<load1,unload2,fuel>
<nop1,nop2,nop3>
<unload1,nop2,fuel>
<load1,unload2,nop3> <unload ,unload
1
2,fuel>
<unload1,nop2,nop3>
<nop1,unload2,fuel>
<unload1,unload2,nop3>
<nop1,unload2,load3>
1
5
9
<load1,nop2,nop3>
<nop1,nop2,load3>
<load1,nop2,load3>
<load1,unload2,load3>
roL
nofuel
caL
roP
fuelOK
caL 4
roL
nofuel
caR
roL
nofuel
caP
roP
fuelOK
caP 12
no f uel → hh3iino f uel
Jürgen Dix (joint work with Wojtek Jamroga) · Clausthal University of Technology
23th March, PANAM Seminaire, LIP6, Paris 18/39
Abilities of Agents
ATL
<load1,nop2,fuel>
<nop1,nop2,fuel>
<load1,unload2,fuel>
<nop1,nop2,nop3>
<unload1,nop2,fuel>
<load1,unload2,nop3> <unload ,unload
1
2,fuel>
<unload1,nop2,nop3>
<nop1,unload2,fuel>
<unload1,unload2,nop3>
<nop1,unload2,load3>
1
5
9
roL
nofuel
caL
roL 2
fuelOK
caL
roL
nofuel
caR
roL 6
fuelOK
caR
roL
nofuel
caP
roL
fuelOK
caP
10
<load1,nop2,nop3>
<nop1,nop2,load3>
<load1,nop2,load3>
<load1,unload2,load3>
7
roP
nofuel
caL 3
roP
fuelOK
caL 4
roP
nofuel
caR
roP
fuelOK
caR 8
roP
nofuel
caP 11
roP
fuelOK
caP 12
caL → hh1, 3ii♦caP
caL → ¬hh1ii♦caP
Jürgen Dix (joint work with Wojtek Jamroga) · Clausthal University of Technology
23th March, PANAM Seminaire, LIP6, Paris 19/39
Abilities of Agents
ATL
Alternative:
Alternating Transition Models
(ATS)
Actions as sets of states that can be affected.
In a way, actions represented in a “pre-compiled”
way.
Problems with ATS
:
Difficult to extend to indeterministic transitions.
They are usually larger than CGS.
Jürgen Dix (joint work with Wojtek Jamroga) · Clausthal University of Technology
23th March, PANAM Seminaire, LIP6, Paris 20/39
Abilities of Agents
Model Checking
Model Checking CTL and ATL: Without agents, with
perfect information
Model checking:
Does
ϕ hold in model M
(CGS) and state
q?
Nice results: model checking CTL and ATL is
tractable!
Jürgen Dix (joint work with Wojtek Jamroga) · Clausthal University of Technology
23th March, PANAM Seminaire, LIP6, Paris 21/39
Abilities of Agents
Model Checking
Model Checking CTL and ATL (no
agents
)
Theorem (Clarke, Emerson & Sistla 1986)
CTL model checking is P-complete, and can be done in
time linear in the size of the model and the length of the
formula.
Theorem (Alur, Kupferman & Henzinger 1998)
ATL model checking is P-complete, and can be done in
time linear in the size of the model and the length of the
formula.
Jürgen Dix (joint work with Wojtek Jamroga) · Clausthal University of Technology
23th March, PANAM Seminaire, LIP6, Paris 22/39
A closer look
Model Checking ATL without
agents
Nice results: model checking CTL and ATL without
agents is tractable, when the size m is measured as
the number of transitions
More natural: the number n of states.
Without agents: m is bounded by n2 .
Jürgen Dix (joint work with Wojtek Jamroga) · Clausthal University of Technology
23th March, PANAM Seminaire, LIP6, Paris 23/39
A closer look
Model Checking ATL with
agents
m: transitions, n: states, d : actions (decisions),
k: agents. How does m depend on n and k?
m = O(nd k )
m is not polynomially bounded in n when
agents are present.
Jürgen Dix (joint work with Wojtek Jamroga) · Clausthal University of Technology
23th March, PANAM Seminaire, LIP6, Paris 24/39
A closer look
Fine-grained Analysis
3 agents, 3 attributes . . . 12 states and 216 transitions
<load1,nop2,fuel>
<nop1,nop2,fuel>
<load1,unload2,fuel>
<nop1,nop2,nop3>
<unload1,nop2,fuel>
<load1,unload2,nop3> <unload ,unload
1
2,fuel>
<unload1,nop2,nop3>
<nop1,unload2,fuel>
<unload1,unload2,nop3>
<nop1,unload2,load3>
1
5
9
roL
nofuel
caL
roL 2
fuelOK
caL
roL
nofuel
caR
roL 6
fuelOK
caR
roL
nofuel
caP
roL
fuelOK
caP
10
Jürgen Dix (joint work with Wojtek Jamroga) · Clausthal University of Technology
<load1,nop2,nop3>
<nop1,nop2,load3>
<load1,nop2,load3>
<load1,unload2,load3>
7
roP
nofuel
caL 3
roP
fuelOK
caL 4
roP
nofuel
caR
roP
fuelOK
caR 8
roP
nofuel
caP 11
roP
fuelOK
caP 12
23th March, PANAM Seminaire, LIP6, Paris 25/39
A closer look
Fine-grained Analysis
Is Model Checking with agents exponential?
Agents make models explode!
Do agents make model checking explode?
Jürgen Dix (joint work with Wojtek Jamroga) · Clausthal University of Technology
23th March, PANAM Seminaire, LIP6, Paris 26/39
A closer look
Fine-grained Analysis
Model Checking CTL and ATL (with
agents
)
Proposition (Jamroga & Dix 2005)
ATL model checking (wrt ATS) is NP-complete with
respect to the number of states and agents.
Proposition (Jamroga & Dix 2005), (Laroussinie et al. 2006)
ATL model checking (wrt CGS) is ∆P
3 -complete with
respect to the number of states and agents.
For positive ATL model checking (wrt CGS) is even
ΣP2 -complete.
Jürgen Dix (joint work with Wojtek Jamroga) · Clausthal University of Technology
23th March, PANAM Seminaire, LIP6, Paris 27/39
A closer look
Imperfect Information
Example: Robots and Carriage
1
2
pos0
2
1
Jürgen Dix (joint work with Wojtek Jamroga) · Clausthal University of Technology
pos1
2
1
pos2
23th March, PANAM Seminaire, LIP6, Paris 28/39
A closer look
Imperfect Information
The CGS model
1
wait,wait
push,push
2
us
h
it,p
wait,wait
push,push
wa
2
pu
sh
,w
a
2
1
pos1
q2
pos2
Jürgen Dix (joint work with Wojtek Jamroga) · Clausthal University of Technology
pos0
wa
1
pos2
q0
it
wa
sh,
h
pu
us
it,p
it
pos0
wait,push
push,wait
q1
wait,wait
push,push
pos1
23th March, PANAM Seminaire, LIP6, Paris 29/39
A closer look
Imperfect Information
Example: Robots and Carriage
pos0
2
1
Jürgen Dix (joint work with Wojtek Jamroga) · Clausthal University of Technology
push,wait
it
pos1
q2
pos2
us
,w
a
q1
wait,wait wait,wait
push,push push,push
push
it,p
sh
us
h
it,p
sh
wa
pu
pos2
wait,push
it
2
q2
h
us
pos1
wa
1
pos2
it,p
wait,wait
push,push
sh,
,w
a
it
pu
2
pos0
wa
1
pu
q0
wait,wait
push,push
wait
wa
wait,wait
push,push
w
p
23th March, PANAM Seminaire, LIP6, Paris 30/39
A closer look
Imperfect Information
ATL with perfect imperfect Information
1
2
1
pos0
2
2
1
pos2
Jürgen Dix (joint work with Wojtek Jamroga) · Clausthal University of Technology
pos1
2
2
1
pos1
1
1
pos2
2
pos0
23th March, PANAM Seminaire, LIP6, Paris 31/39
A closer look
Imperfect Information
ATL with Imperfect Information
wait,wait
push,push
1
2
q0
pos0
h
wa 1
it,p
us
sh
,w
pu
2
1
Jürgen Dix (joint work with Wojtek Jamroga) · Clausthal University of Technology
it
pos2
h
us
q2
it,p
2
wait,wait
push,push
2
pos1
wa
1
pos2
wa
sh,
pu
ait
pos0
wait,push
push,wait
q1
wait,wait
push,push
pos1
23th March, PANAM Seminaire, LIP6, Paris 32/39
A closer look
Imperfect Information
Imperfect Information (i), memoryless (r)
We extend CGS with epistemic relations ∼a , one per
agent
Uniform strategies per agent:
q ∼a q0 ⇒ sa (q) = sa (q0 )
Uniform strategies for group of agents:
q ∼A q0 ⇒ sa (q) = sa (q0 ), where q ∼A q0 is defined
by
there is an agent a ∈ A with q ∼a q0
Jürgen Dix (joint work with Wojtek Jamroga) · Clausthal University of Technology
23th March, PANAM Seminaire, LIP6, Paris 33/39
A closer look
Imperfect Information
ATL with Imperfect Information: ATLir
M, q |= hhAiiir
kϕ
iff there exists uniform SA such
S
that, for every path Λ ∈ q0 ∼A q out(q0 , SA ), we have
M, Λ[1] |= ϕ;
M, q |= hhAiiir ϕ iff there exists uniform SA such
S
that, for every Λ ∈ q0 ∼A q out(q0 , SA ), we have
M, Λ[i] |= ϕ for every i ≥ 0;
M, q |= hhAiiir ϕ U ψ iff there exists uniform SA such
S
that, for every Λ ∈ q0 ∼A q out(q0 , SA ), we have
M, Λ[i] |= ψ for some i ≥ 0, and M, Λ[ j] |= ϕ for
every 0 ≤ j < i.
Jürgen Dix (joint work with Wojtek Jamroga) · Clausthal University of Technology
23th March, PANAM Seminaire, LIP6, Paris 34/39
A closer look
Imperfect Information
Example: Robots and Carriage
2
1
push,wait
ait
wa
it,p
u
,w
pos1
1
sh
q1
wait,wait wait,wait
push,push push,push
push
pu
sh
pu
it
wait,push
Jürgen Dix (joint work with Wojtek Jamroga) · Clausthal University of Technology
wa
h
pos2
us
2
q2
sh,
pos1
pu
1
pos2
it,p
wait,wait
push,push
2
2
pos0
wait,wait
push,push
wait
pos0
wa
1
q0
,w
ait
wa 1
it,p
us
h
wait,wait
push,push
q2
pos2
wai
pus
23th March, PANAM Seminaire, LIP6, Paris 35/39
A closer look
Imperfect Information
Model checking ATLir
Model checking ATLir is ∆P
2 -complete in the number
of transitions
NP-hard: Schobbens 2004,
∆P2 -complete: Jamroga & Dix 2005.
What if the numbers of states and agents are
parameters?
Jürgen Dix (joint work with Wojtek Jamroga) · Clausthal University of Technology
23th March, PANAM Seminaire, LIP6, Paris 36/39
A closer look
Imperfect Information
Model checking ATLir
Proposition
Model checking ATLir is ∆P
3 -complete wrt the number of
states (n), decisions (d ) and agents (k) in the model,
and the length of the formula.
Surprise:
the same complexity for perfect
and imperfect information!
Jürgen Dix (joint work with Wojtek Jamroga) · Clausthal University of Technology
23th March, PANAM Seminaire, LIP6, Paris 37/39
A closer look
Table of Complexities
Complexity Results for Temporal and Strategic Logics
m, l
n, k , l
nlocal , k, l
CTL
P-complete [1]
P-complete [1]
PSPACE-complete [2]
ATL
P-complete [3]
ATLir ∆P
2 -complete [4,7]
∆P3 -complete [5,6] EXPTIME-complete [8,9]
∆P3 -complete [7]
PSPACE-complete [9]
[1] Clarke, Emerson & Sistla (1986). Automatic verification of finite-state .... ACM Prog. Lang. Syst
[2] Kupferman, Vardi & Wolper (2000). An automata-theoretic approach to .... J. ACM
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[5] Jamroga & Dix (CEEMAS 2005). Do agents make model checking explode?
[6] Laroussinie, Markey & Oreiby (FORMATS 2006). Model-Checking Timed.
[7] Jamroga & Dix (2007). Model checking abilities of agents. Theory of Computing systems
[8] Hoek, Lomuscio & Wooldridge (AAMAS 2006). On the complexity of practical ATL model checking.
[9] Jamroga & Ågotnes (AAMAS 2007). Modular Interpreted Systems
Jürgen Dix (joint work with Wojtek Jamroga) · Clausthal University of Technology
23th March, PANAM Seminaire, LIP6, Paris 38/39
A closer look
Table of Complexities
Some variants: Information vs. recall
ATLIr : Perfect information, no memory (imperfect recall).
This is classical ATL, see [3].
ATLIR : Perfect information, full memory (perfect recall).
Same complexity as ATL: polynomial in m, l . See [3].
ATLir : Imperfect information, imperfect recall: see last slide.
ATLiR : Imperfect information, perfect recall: undecidable
(currently working on a proof).
Jürgen Dix (joint work with Wojtek Jamroga) · Clausthal University of Technology
23th March, PANAM Seminaire, LIP6, Paris 39/39