Formal Description of the Chord Protocol using ASM
Bojan Marinković1 , Paola Glavan2 , Zoran Ognjanović1
Mathematical Institute of the Serbian Academy of Sciences and Arts1
Belgrade, Serbia
[bojanm, zorano]@mi.sanu.ac.rs
Department of Mathematics and Descriptive Geometry2
Faculty od Mechanical Engineering and Naval Architecture
Zagreb, Croatia
[email protected]
Fourth Workshop on Formal and Automated
Theorem Proving and Applications
February 4-5, 2011,
Belgrade, Serbia
Marinković,Glavan,Ognjanović(MISANU, FSB)
ASM Description of Chord
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Outline
1
Introduction
2
(Informal) Description of Chord
3
Abstract State Machine
4
Related Work
5
ASM Formalization of Chord
Basic Notions
Chord Actions
6
Correctness of the Formalization
7
Conclusion
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Introduction
Outline
1
Introduction
2
(Informal) Description of Chord
3
Abstract State Machine
4
Related Work
5
ASM Formalization of Chord
Basic Notions
Chord Actions
6
Correctness of the Formalization
7
Conclusion
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Introduction
Peer-To-Peer Systems
No centralized control or hierarchical organization
Each node (peer) runs software with equivalent functionality
System’s states and tasks are dynamically allocated
Core operation: efficient retrieval of data items
No inherent bottlenecks and resistance to failures, attacks, etc.
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Introduction
Distributed Hash Tables
Lookup service similar to a hash table
Storing (key, value) pairs
Retrieval of the value associated with a given key
Any change in the set of nodes causes a minimal amount of
disruption
Chord: one of the first and simplest DHTs
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(Informal) Description of Chord
Outline
1
Introduction
2
(Informal) Description of Chord
3
Abstract State Machine
4
Related Work
5
ASM Formalization of Chord
Basic Notions
Chord Actions
6
Correctness of the Formalization
7
Conclusion
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(Informal) Description of Chord
Short Desctription of Chord
Number of nodes form a ring-shaped network
Supported operation: mapping the given key onto a node using
consistent hashing
Probability that two objects of the same type are assigned same
identifiers is negligible
One node is awared of a small number (maximum O(log N)) of
other nodes
Key is assigned to the first node in the circle whose identifier is
equal or greater than hash(key)
Properties:
Load-balance
All lookups are resolved via maximum O(log N) messages
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Abstract State Machine
Outline
1
Introduction
2
(Informal) Description of Chord
3
Abstract State Machine
4
Related Work
5
ASM Formalization of Chord
Basic Notions
Chord Actions
6
Correctness of the Formalization
7
Conclusion
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Abstract State Machine
Introduction
Introduced with: Y. Gurevich: Evolving Algebras 1993: Lipari Guide.
Definition
Abstract State Machine A is defined by a program Prog - consisting of
a finite number of transition rules, at most countable set of states and
initial states. A models the operational behavior of a real dynamic
system S in terms of evolution of states.
Definition
A state S is a first-order structure over a fixed signature (which is also
the signature of A), representing the instantaneous configuration of S.
The value of a term t at S is denoted by [t]S . The basic transition rule
is the following function update f (t1 , . . . , tn ) := t, where f is an arbitrary
n-ary function and t1 , . . . , tn , t are first-order terms.
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Abstract State Machine
Transition Rules
Conditional constructor
i f g then
R1
e l s e i f R2
endif
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Abstract State Machine
Transition Rules
Conditional constructor
i f g then
R1
e l s e i f R2
endif
Sequential constructor
seq
R1
...
Rn
endseq
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Abstract State Machine
Transition Rules
Conditional constructor
Parallel constructor
i f g then
R1
e l s e i f R2
endif
par
R1
...
Rn
endpar
Sequential constructor
seq
R1
...
Rn
endseq
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Abstract State Machine
Transition Rules
Conditional constructor
Parallel constructor
i f g then
R1
e l s e i f R2
endif
par
R1
...
Rn
endpar
Sequential constructor
Choose constructor
seq
R1
...
Rn
endseq
Marinković,Glavan,Ognjanović(MISANU, FSB)
choose v i n U
s a t i s f y i n g g(v )
R0
endchoose
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Abstract State Machine
Properties
Definition (ASM runs)
A run (or computation) of A is a finite or infinite sequence
S0 ; S1 ; S2 ; . . . where S0 is an initial state and every Si+1 is obtained
from Si executing a transition rule.
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Abstract State Machine
Properties
Definition (ASM runs)
A run (or computation) of A is a finite or infinite sequence
S0 ; S1 ; S2 ; . . . where S0 is an initial state and every Si+1 is obtained
from Si executing a transition rule.
External function can be understand as a (dynamic) oracle
ASM can model algorithms at the appropriate abstraction level to
verify properties of the system
Already modeled with ASMs: Bakery algorithm, Rail road crossing
problem, Kerberos algorithm, etc.
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Abstract State Machine
Distrubuted ASM
Multiple autonomous agents cooperatively model a concurrent
computation
Each agent executes its own single-agent program
The underlying semantic model ensures that the order that no
conflicts between the update sets computed for distinct agents
can arise
The global program is the union of all single-agent programs
Regular runs - state is global and move of an agent is atomic
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Related Work
Outline
1
Introduction
2
(Informal) Description of Chord
3
Abstract State Machine
4
Related Work
5
ASM Formalization of Chord
Basic Notions
Chord Actions
6
Correctness of the Formalization
7
Conclusion
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Related Work
Related Work
I. Stoica et al.: Chord: A Scalable Peer-to-Peer Lookup service for
Internet Applications
Introduces Chord and describes it using C++-like pseudo code
Several complexity results and consistency of entrance of a new
node
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Related Work
Related Work
I. Stoica et al.: Chord: A Scalable Peer-to-Peer Lookup service for
Internet Applications
Introduces Chord and describes it using C++-like pseudo code
Several complexity results and consistency of entrance of a new
node
R. Bakhshi, D. Gurov: Verification of Peer-to-peer Algorithms: A
Case Study
Specification of Chord in terms of the π-calculus
Verification of the corresponding stabilization algorithm
Consider maintaining of topological structure
Possible departures of nodes from a network are not examined
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Related Work
Related Work
I. Stoica et al.: Chord: A Scalable Peer-to-Peer Lookup service for
Internet Applications
Introduces Chord and describes it using C++-like pseudo code
Several complexity results and consistency of entrance of a new
node
R. Bakhshi, D. Gurov: Verification of Peer-to-peer Algorithms: A
Case Study
Specification of Chord in terms of the π-calculus
Verification of the corresponding stabilization algorithm
Consider maintaining of topological structure
Possible departures of nodes from a network are not examined
V. Tru’o’ng: Testing implementations of Distributed Hash Tables
An Erlang implementation of Chord
Analyzed using simulations
Consider maintaining of topological structure
It reports some failures and proposes modifications
Marinković,Glavan,Ognjanović(MISANU, FSB)
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ASM Formalization of Chord
Outline
1
Introduction
2
(Informal) Description of Chord
3
Abstract State Machine
4
Related Work
5
ASM Formalization of Chord
Basic Notions
Chord Actions
6
Correctness of the Formalization
7
Conclusion
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ASM Formalization of Chord
Basic Notions
Basic Notions
Motivation: errors in concurrent systems are difficult to reproduce
and find by program testing
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ASM Formalization of Chord
Basic Notions
Basic Notions
Motivation: errors in concurrent systems are difficult to reproduce
and find by program testing
M fixed positive integer, and N = 2M
Universe (Chord network): Chord := {0, 1, . . . , N − 1}
Indicator that x ∈ Chord is in Chord network Chord(x) = true
External function for the manipulations with the lists (list
constructor, add, remove, listitem)
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ASM Formalization of Chord
Basic Notions
Node Structure
successor : Chord → Chord
predecessor : Chord → Chord
finger : Chord → ListOfChord
next : Chord → {1, . . . , M}
keysvalues : Chord → ListOfKeysValues
Figure: Structure of Chord node
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ASM Formalization of Chord
Basic Notions
Node Structure
successor : Chord → Chord
predecessor : Chord → Chord
finger : Chord → ListOfChord
next : Chord → {1, . . . , M}
keysvalues : Chord → ListOfKeysValues
Figure: Structure of Chord node
Nodeid = hid, successor (id), predecessor (id), finger (id), next(id), keysvalues(id)i
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ASM Formalization of Chord
Basic Notions
Chord Functions
hash : Keys ∪ IPs → {0, 1, . . . , N − 1, undef }
ping : Chord → {true, false}
member of : Chord × Chord × Chord → {true, false}
(arg2 < arg1 6 arg3 )
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ASM Formalization of Chord
Basic Notions
Chord Functions
hash : Keys ∪ IPs → {0, 1, . . . , N − 1, undef }
ping : Chord → {true, false}
member of : Chord × Chord × Chord → {true, false}
(arg2 < arg1 6 arg3 )
find successor : Chord × {0, 1, . . . N − 1} → Chord
get : Chord × Keys → Values ∪ {undef }
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ASM Formalization of Chord
Basic Notions
Functions Definitions

successor (n), if ping(successor (n))∧




member of (h, n, successor (n)),







find successor (finger (n).listitem(i), h),
find successor (n, h) =
 otherwise where




 member of (finger (n).listitem(i), n, h)∧



(i = M∨


¬member of (finger (n).listitem(i + 1), n, h))
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ASM Formalization of Chord
Basic Notions
Functions Definitions

successor (n), if ping(successor (n))∧




member of (h, n, successor (n)),







find successor (finger (n).listitem(i), h),
find successor (n, h) =
 otherwise where




 member of (finger (n).listitem(i), n, h)∧



(i = M∨


¬member of (finger (n).listitem(i + 1), n, h))

value, if (key, value) ∈



keysvalues(find successor (id, hash(key))),
get(id, key) =



undef , otherwise.
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ASM Formalization of Chord
Chord Actions
Chord Actions
Start - Sε →
End - → Sε
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ASM Formalization of Chord
Chord Actions
Chord Actions
Start - Sε →
End - → Sε
Join - add a new node
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ASM Formalization of Chord
Chord Actions
Chord Actions
Start - Sε →
End - → Sε
Join - add a new node
Fairleave - one node less with information transfer
UnfairLeave - one node less without information transfer
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ASM Formalization of Chord
Chord Actions
Chord Actions
Start - Sε →
End - → Sε
Join - add a new node
Fairleave - one node less with information transfer
UnfairLeave - one node less without information transfer
Stabilize - update information on predecessor and successor
Update predecessor - check predecessor
Update fingers - update finger table
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ASM Formalization of Chord
Chord Actions
Chord Actions
Start - Sε →
End - → Sε
Join - add a new node
Fairleave - one node less with information transfer
UnfairLeave - one node less without information transfer
Stabilize - update information on predecessor and successor
Update predecessor - check predecessor
Update fingers - update finger table
Put - insert (key, value)
Get - call get
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ASM Formalization of Chord
Chord Actions
Start Rule
i f ∀j(j ∈ Chord ∧ Chord(j) = undef ) then
seq
i := hash(IP)
par
Chord(i) := true
predecessor (i) := undef
successor (i) := i
finger (i) := [ ]
next(i) := 0
keysvalues(i) := [ ]
stabilization(i)
endpar
endseq
e l s e i f Chord i s a l r e a d y s t a r t e d
endif
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ASM Formalization of Chord
Chord Actions
Join Rule
choose j i n Chord s a t i s f y i n g Chord(j) = true
i f j 6= undef then
seq
i := hash(IP)
i f Chord(i) 6= true then
i f i 6= undef then
seq
INITIALIZATION
TRANSFER KEYS
stabilization(i)
endseq
e l s e i f Chord i s f u l f i l l e d
endif
e l s e i f i i s a l r e a d y member o f Chord
endif
endseq
e l s e i f Chord i s empty
endif
endchoose
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ASM Formalization of Chord
Chord Actions
FairLeave Rule
i f Chord(i) = true then
seq
CHECK SUCCESSOR
i f successor (i) = predecessor (i) ∧ predecessor (i) = i then
seq
Chord(i) := undef
End
endseq
elseif
seq
par
TRANSFER KEYS
predecessor (successor (i)) := predecessor (i)
successor (predecessor (i)) := successor (i)
endpar
UNSET VALUES
endseq
endif
endseq
endif
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ASM Formalization of Chord
Chord Actions
Stabilization Rule
par
stabilize(i)
update predecessor (i)
update fingers(i)
endpar
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ASM Formalization of Chord
Chord Actions
Stabilize Rule
i f Chord(i) = true then
seq
i f ping(successor (i)) then
seq
x := predecessor (successor (i))
i f x 6= undef then
i f member of (x, i, successor (i)) then
successor (i) := x
endif
endif
i f x = undef ∨ member of (i, x, successor (i)) then
predecessor (successor (i)) := i
endif
endseq
elseif
UPDATE SUCCESSOR
endif
stabilize(i)
endseq
endif
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ASM Formalization of Chord
Chord Actions
Update predecessor Rule
i f Chord(i) = true then
seq
i f ping(predecessor (i)) 6= true then
seq
Chord(predecessor (i)) := undef
predecessor (i) := undef
endseq
endif
update predecessor (i)
endseq
endif
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ASM Formalization of Chord
Chord Actions
Update fingers Rule
i f Chord(i) = true then
seq
next(i) := next(i) + 1
i f (next(i) > M) then
next(i) := 1 / ∗ N = 2M ∗ /
endif
finger .listitem(next(i)) := find successor (i, i + 2next(i)−1 )
update fingers(i)
endseq
endif
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ASM Formalization of Chord
Chord Actions
Put Rule
i f Chord(i) = true then
seq
x := find successor (i, hash(key ))
i f x 6= undef
keysvalues(x).add(hhash(key), valuei)
e l s e i f Chord i s f u l f i l l e d
endif
endseq
endif
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ASM Formalization of Chord
Chord Actions
Empty Rules
End - go to Sε when last node leaves
UnfairLeave - caused by a node crash, communication problems,
etc.
Get - does not change the state of the network (call get function)
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Correctness of the Formalization
Outline
1
Introduction
2
(Informal) Description of Chord
3
Abstract State Machine
4
Related Work
5
ASM Formalization of Chord
Basic Notions
Chord Actions
6
Correctness of the Formalization
7
Conclusion
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Correctness of the Formalization
Correctness of find successor
Theorem
Let id ∈ {0, 1, . . . , N − 1}, n0 ∈ Chord and n = find successor (n0 , id). If
the successor pointers form a ring of all nodes in the Chord network,
then the following statement holds
(∀x ∈ Chord)(member of (x, id, n) = true∧x 6= n) ⇒ Chord(x) = undef .
Theorem
If the successor pointers form a ring of all nodes in the Chord network,
then the following statement holds
∀(n, n0 , id ∈ Chord)find successor (n, id) = find successor (n0 , id).
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Correctness of the Formalization
Rebuild of the Chord Ring
Theorem
Let the successor pointers form a ring of all nodes in a Chord network. Let a node join
a Chord network and break the ring of the successors pointers. Then, Stabilization
rule will rebuild the ring.
Figure: Join
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Correctness of the Formalization
Rebuild of the Chord Ring
Theorem
Let the successor pointers form a ring of all nodes in a Chord network. Let a node join
a Chord network and break the ring of the successors pointers. Then, Stabilization
rule will rebuild the ring.
Figure: Join
Figure: Unfair Leave
Theorem
Let the successor pointers form a ring of all nodes in a Chord network. Let a node
leaves a Chord network in an unfair way and breaks the ring of the successors
pointers. Then, Stabilization rule will rebuild the ring.
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Correctness of the Formalization
State Changing (1)
Definition (Node scenario)
For a node the following sequence of actions is allowed:
Start | Join, Stabilization, (Put | Get)∗ , FairLeave | UnfairLeave.
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Correctness of the Formalization
State Changing (1)
Definition (Node scenario)
For a node the following sequence of actions is allowed:
Start | Join, Stabilization, (Put | Get)∗ , FairLeave | UnfairLeave.
Definition (Network scenario)
For a Chord network the following sequence of actions is allowed:
Starti , Stabilizationi ,
((Joinij , Stabilizationij ) || Putik || Getil || FairLeaveim || UnfairLeavein )+ ,
End,
where i, ij , ik , il , im , in ∈ Chord, and for every s ∈ Chord, the
corresponding subsequence of actions is allowed for Nodes .
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Correctness of the Formalization
State Changing (2)
Theorem
When End rule is applied there are no other nodes in a Chord network
except the node which invokes End rule.
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Correctness of the Formalization
State Changing (2)
Theorem
When End rule is applied there are no other nodes in a Chord network
except the node which invokes End rule.
Theorem
An arbitrary sequence σ1 of actions in a Chord network can be
reduced to an allowed sequence σ0 , such that σ0 and σ1 cause the
same sequence of state transitions of the network.
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Correctness of the Formalization
State Changing (2)
Theorem
When End rule is applied there are no other nodes in a Chord network
except the node which invokes End rule.
Theorem
An arbitrary sequence σ1 of actions in a Chord network can be
reduced to an allowed sequence σ0 , such that σ0 and σ1 cause the
same sequence of state transitions of the network.
Theorem
If a nonempty prefix of an allowed sequence of actions produces the
current state of a Chord network, then the successor pointers will
eventually form a ring of all nodes in the network.
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Correctness of the Formalization
Key Manipulation
Theorem (Golden rule)
∀((key, value)∈ Keys × Values)
((key, value) ∈ keysvalues(i) ⇒ hash(key) 6 i),
where 6 respects that 0 is the first successor of N − 1.
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Correctness of the Formalization
Key Manipulation
Theorem (Golden rule)
∀((key, value)∈ Keys × Values)
((key, value) ∈ keysvalues(i) ⇒ hash(key) 6 i),
where 6 respects that 0 is the first successor of N − 1.
Corollary
If get returns undef for some key ∈ Keys, then there is no
value ∈ Values such that (key, value) pair is stored in the Chord
network.
Marinković,Glavan,Ognjanović(MISANU, FSB)
ASM Description of Chord
4th ARGO Workshop
35 / 38
Conclusion
Outline
1
Introduction
2
(Informal) Description of Chord
3
Abstract State Machine
4
Related Work
5
ASM Formalization of Chord
Basic Notions
Chord Actions
6
Correctness of the Formalization
7
Conclusion
Marinković,Glavan,Ognjanović(MISANU, FSB)
ASM Description of Chord
4th ARGO Workshop
36 / 38
Conclusion
Conclusion and Furthure Work
Presented an ASM-based formalization of the Chord protocol
Marinković,Glavan,Ognjanović(MISANU, FSB)
ASM Description of Chord
4th ARGO Workshop
37 / 38
Conclusion
Conclusion and Furthure Work
Presented an ASM-based formalization of the Chord protocol
First comprehensive formal analysis of Chord
Marinković,Glavan,Ognjanović(MISANU, FSB)
ASM Description of Chord
4th ARGO Workshop
37 / 38
Conclusion
Conclusion and Furthure Work
Presented an ASM-based formalization of the Chord protocol
First comprehensive formal analysis of Chord
Proves with respect to the regular runs and execution of the rules
UnfairLeave and Put in stable states
Regular runs eliminate several observed shortages
Practice vs Assumptions - Challenges to modify Chord
Marinković,Glavan,Ognjanović(MISANU, FSB)
ASM Description of Chord
4th ARGO Workshop
37 / 38
Conclusion
Conclusion and Furthure Work
Presented an ASM-based formalization of the Chord protocol
First comprehensive formal analysis of Chord
Proves with respect to the regular runs and execution of the rules
UnfairLeave and Put in stable states
Regular runs eliminate several observed shortages
Practice vs Assumptions - Challenges to modify Chord
Apply similar technique to describe some other DHT protocols
Marinković,Glavan,Ognjanović(MISANU, FSB)
ASM Description of Chord
4th ARGO Workshop
37 / 38
Conclusion
Conclusion and Furthure Work
Presented an ASM-based formalization of the Chord protocol
First comprehensive formal analysis of Chord
Proves with respect to the regular runs and execution of the rules
UnfairLeave and Put in stable states
Regular runs eliminate several observed shortages
Practice vs Assumptions - Challenges to modify Chord
Apply similar technique to describe some other DHT protocols
Starting point for a formal proof assistant (Coq, Isabelle/HOL)
Marinković,Glavan,Ognjanović(MISANU, FSB)
ASM Description of Chord
4th ARGO Workshop
37 / 38
Thank you for your attention
Any question?
Marinković,Glavan,Ognjanović(MISANU, FSB)
ASM Description of Chord
4th ARGO Workshop
38 / 38