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of so-called linear degen-erations of Fl( V ): Let U be the variety of tuples ( f1, . . . , f n−1) ∈ End( V )n−1 of linear operators such that rk( fj−1 ◦ . . . ◦ fi) ≥ n + i − j for all i n ∏ > i=1 Gr i(V ) × U such that fi(Ui) ⊂ Ui+1 for all i = 1 , . . . , n − 1. Then πU : FU → U is a flat family with generic fibre ...	{ "page_id": null, "source": 7361, "title": "from dpo" }
, Math. Z. 287 (2017), no. 1-2, 615–654. X. Fang, M. Reineke, Supports for linear degenerations of flag varieties , Preprint 2018, arXiv:1805.09797. Types A and D quiver representation varieties Jenna Rajchgot (joint work with Ryan Kinser, Allen Knutson) Given a quiver Q with vertex set Q0, arrow set Q1, and dimension...	{ "page_id": null, "source": 7361, "title": "from dpo" }
locally by the vanishing of minors of a matrix. When φ is sufficiently general, an expression for its fundamental class in the cohomology ring of X is given by the Giambelli-Thom-Porteous formula. A. Buch and W. Fulton general-ized this to sequences of vector bundle maps V1 → V2 → · · · → Vn in . Related formulas were ...	{ "page_id": null, "source": 7361, "title": "from dpo" }
Type A In this section we discuss some geometric results in type A, including the bipartite Zelevinsky map from , and a degeneration from which was central to the proof of the K-theoretic type A quiver component formula. 1.1. Bipartite Zelevinsky map. Let Q be a bipartite (i.e. alternating) type A quiver and d a dimen...	{ "page_id": null, "source": 7361, "title": "from dpo" }
ranks is equivalent to a list of ranks of certain North-West justified submatrices of ζ(D, C, B, A ). Using this, one can show that the closure of the orbit through (D, C, B, A ) is isomorphic, via ζ, to a Schubert variety intersected with Y .The bipartite Zelevinsky map is useful beyond the bipartite setting because o...	{ "page_id": null, "source": 7361, "title": "from dpo" }
: K× → G be a group homomorphism and consider the right action of K× on G by g · t = μ(t−1)gμ (t) and the right action 2892 Oberwolfach Report 46/2019 of K× on X by x · t = x · μ(t). With this set-up, we get two families ˜H and ˜Y over A1 − { 0} where the fiber H · t in the first family is a subgroup which acts on the ...	{ "page_id": null, "source": 7361, "title": "from dpo" }
equioriented setting in . 2. Type D In recent joint work with Kinser , we obtain results in type D which are analo-gous to the type A results from . Indeed, we unify aspects of the equivariant ge-ometry of three classes of varieties: type D quiver representation varieties, double Grassmannians Gr (a, n ) × Gr (b, n ), ...	{ "page_id": null, "source": 7361, "title": "from dpo" }
in Representation Theory 2893 A. S. Buch, L. M. Feh´ er, and R. Rim´ anyi. Positivity of quiver coefficients through Thom polynomials. Adv. Math. , 197(1):306–320, 2005. A. S. Buch and W. Fulton. Chern class formulas for quiver varieties. Invent. Math. ,135(3):665–687, 1999. A. S. Buch and R. Rim´ anyi. A formula fo...	{ "page_id": null, "source": 7361, "title": "from dpo" }
Let X be a smooth algebraic variety, and E → X a vector bundle of rank two on X. Define the projective line fibration Y = P(E) as the variety of all lines in E. The natural projection π : Y → X induces pull-back π∗ : A∗(X) → A∗(Y ) and push-forward π∗ : A∗(Y ) → A∗− 1(X) (aka transfer or Gysin map ) in the (generalized...	{ "page_id": null, "source": 7361, "title": "from dpo" }
Schubert calculus [BGG, D] is an inductive description of Schubert cycles [ Xw ] ∈ CH ∗(Y ) for all elements w ∈ W in the Weyl group of G.Namely, if w = si1 si2 · · · siℓ is a reduced decomposition of w into the product of simple reflections, then [Xw ] = ∂iℓ . . . ∂ i2 ∂i1 [Xid ]. (1) A classical result in representat...	{ "page_id": null, "source": 7361, "title": "from dpo" }
the degrees of Bott–Samelson varieties corresponding to collections of simple roots (αi1 ), ( αi1 , α i2 ),. . . , ( αi1 , α i2 , . . . , α iℓ ). Let P ⊂ Rn be a convex polytope, and I ⊂ Rn a segment. Let Q ⊂ Rn be a polytope analogous to P (i.e., having the same normal fan). Define the push-pull polytope ∆( P, Q, I ) ...	{ "page_id": null, "source": 7361, "title": "from dpo" }
trapezoid with the vertices (0 , 0) , (2 , 0) , (0 , 1) , (1 , 1) . Take I = [(0 , 0) , (0 , 1)] , and Q = P . Then the push-pull polytope ∆( P, Q, I ) ⊂ R3 is the Minkowski sum of the FFLV polytope in type A2 corresponding to the weight ρ and the segment J = [(0 , 0, 0) , (1 , 0, 0)] (see [FFL] for the definition of F...	{ "page_id": null, "source": 7361, "title": "from dpo" }
N. Bernstein, I. M. Gelfand, S. I. Gelfand ,Schubert cells, and the cohomology of the spaces G/P , Russian Math. Surveys 28 (1973), no. 3, 1–26. [D] M. Demazure ,D´ esingularisation des vari´ et´ es de Schubert g´ en´ eralis´ ees , Collection of articles dedicated to Henri Cartan on the occasion of his 70th birthday, I...	{ "page_id": null, "source": 7361, "title": "from dpo" }
This talk gives a brief introduction to that story, and discusses a recent theorem of the author and Harada which uses the topology of Hessenberg varieties to give an inductive approach to the Stanley–Stembridge conjecture in a special case. Let n be a positive integer. A Hessenberg function is an increasing sequence h...	{ "page_id": null, "source": 7361, "title": "from dpo" }
1 2 3 4 5Let Γ be a (simple) graph with vertex set [ n]. A proper coloring of Γ is a function κ : [ n] → N such that κ(v) 6 = κ(w) whenever {v, w } is an edge. We define a monomial xκ in variables x = x1, x 2, . . . by xκ := ∏ > v∈[n] xκ(v). Stanley’s chromatic symmetric function is XΓ(x) := ∑ > κ xκ where the sum is t...	{ "page_id": null, "source": 7361, "title": "from dpo" }
the dot action represen-tation. For example, the Schur-basis expansion of XΓh (x) is known; there is a combinatorial formula for the coefficients due to Gasharov obtained by enumerat-ing Ph-tableaux . Gasharov’s formula determines the decomposition of the dot action representation into irreducible Sn-representations. I...	{ "page_id": null, "source": 7361, "title": "from dpo" }
is the set of all independent sets of vertices of size ℓ in Γh and for each I ∈ I ℓ(Γ h), the coefficients cIμ are the coefficients as in (1) associated to the semisimple Hessenberg variety B(S, h I ) in the flag variety of GL n−ℓ(C). As a corollary, we obtain a special case of the Stanley–Stembridge conjecture by indu...	{ "page_id": null, "source": 7361, "title": "from dpo" }
2017. M. Harada and M. Precup. Upper-triangular linear relations on multiplicities and the Stanley–Stembridge conjecture. preprint: 2018. M. Precup. The Betti numbers of regular Hessenberg varieties are palindromic. Trans. Groups. 23(2):491–499, 2018. Mini-Workshop: Degeneration Techniques in Representation Theory 2...	{ "page_id": null, "source": 7361, "title": "from dpo" }
To be precise, by the algebra An is a cluster algebra meaning that it can be generated recursively by seeds (certain maximally algebraically independent subsets of algebra generators for An) and an algebraic operation called mutation (a procedure to construct new seeds from a given one). For every seed s we have a ful...	{ "page_id": null, "source": 7361, "title": "from dpo" }
bounds and double Bruhat cells. Duke Math. J. 126 (2005), no. 1, 1–52. X. Fang, G. Fourier, P. Littelmann, Essential bases and toric degenerations arising from birational sequences. Adv. Math. 312 (2017), 107–149. E. Feigin, G. Fourier, P. Littelmann, Favourable modules: filtrations, polytopes, Newton-Okounkov bodies...	{ "page_id": null, "source": 7361, "title": "from dpo" }
×Smr . (3) For x = ( x1,1, . . . x r,m r ) ∈ Am+ ... +mr with pairwise distinct coordinates we have Wλ ⊗Aλ Cx ≃ > r ⊗ > i=1 > mi ⊗ > i=1 Wωi (xi,j ). (4) Wλ ⊗Aλ C0 ≃ Wλ (5) dim Wλ = ∏ri=1 (dim Wωi )mi . This theorem implies that the specialisation of Wλ as Aλ-module have equal di-mension every point. Hense, Wλ is proje...	{ "page_id": null, "source": 7361, "title": "from dpo" }
(h[t]) /Ann U(h[t]) v. (2) As algebra Al,λ is isomorphic to C[z1, . . . , z m1+... +mr ]Sm1 ×... ×Smr . (3) For x = ( xi, 1, . . . x r,m r ) ∈ Am+ ... +mr with pairwise distinct coordinates we have Dl,λ ⊗Al,λ Cx ≃ > r ⊗ > i=1 > mi ⊗ > i=1 Dl,ω i (xi,j ), Where Dl,λ is the sl r+1 -stable Demazure module of level l and h...	{ "page_id": null, "source": 7361, "title": "from dpo" }
of the form dixa(t) > dx i xb(t). Note that relations with derivatives first appeared in in the homogeneous co-ordinate ring of usual Drinfeld-Pl¨ ucker embedding of the semi-infinite flag variety in type A. References > A.Braverman, M.Finkelberg. Weyl modules and q-Whittaker functions , Math. Ann., vol. 359 (1), 201...	{ "page_id": null, "source": 7361, "title": "from dpo" }
study relations between these two constructions of toric degenerations. To be more precise, let A = ⋃ > t At = ⋃ > t Spec( C[x1(t)±1, . . . , x m(t)±1]) be a cluster variety, where t runs over a set of seeds which are all mutation equiv-alent, and x1(t), . . . , x m(t) are the corresponding cluster variables. Assuming ...	{ "page_id": null, "source": 7361, "title": "from dpo" }
Let i be a reduced word for w ∈ W . Then, there exists a seed t (resp., ˜t) such that ∆( X(w), Lλ, v t) ( resp., ∆( X(w), Lλ, v ˜t)) is unimodularly equivalent to ∆i(λ) ( resp., ˜∆i(λ)) for all λ ∈ P+. Analogous relations between string polytopes and cluster varieties were pre-viously given by Magee and Bossinger-Four...	{ "page_id": null, "source": 7361, "title": "from dpo" }
(2017), 325–352. V. Genz, G. Koshevoy, and B. Schumann, Polyhedral parametrizations of canonical bases > &cluster duality , arXiv preprint arXiv:1711.07176, 2017. M. Gross, P. Hacking, S. Keel, and M. Kontsevich, Canonical bases for cluster algebras , J. Amer. Math. Soc. 31 (2018), 497–608. S.-J. Kang, M. Kashiwara,...	{ "page_id": null, "source": 7361, "title": "from dpo" }
it is unclear how to define one of the main ingredients: the degenerate flag variety. In the cases considered previously the associated graded algebra was always itself a universal enveloping algebra and one could define the variety as an orbit closure for the corresponding Lie group. In general, the associated graded ...	{ "page_id": null, "source": 7361, "title": "from dpo" }
2, 321–352. X. Fang, E. Feigin, G. Fourier, I. Makhlin, Weighted PBW degenerations and tropical flag varieties , Communications in Contemporary Mathematics 22 (2019), 1850016. I. Makhlin, Gelfand–Tsetlin degenerations of representations and flag varieties , > Flag varieties, Valuations and Standard Monomial Theory P...	{ "page_id": null, "source": 7361, "title": "from dpo" }
simplicity we assume that the Yp are projectively normal. In the proofs this condition can often be replaced by other properties like smooth in codimension 1, or the existence of open affine patches. Example 1. Let X = G/B ⊂ P(V (λ)) be the (generalized) flag variety, where G is a semisimple algebraic group and λ is a ...	{ "page_id": null, "source": 7361, "title": "from dpo" }
we choose a maximal chain such that VC(h) = min {V C′ (h) \| C′ maximal chain }. We replace then Qr+1 by Q\|A\| with basis {ep}p∈A and set: V : R → Q\|A\|≥0 , h 7 → ∑ > pi∈C (VC(h)) iepi 3. Results Theorem 1. i) The quasi-valuation V induces a filtration on R such that the associated graded algebra gr V R is finitely genera...	{ "page_id": null, "source": 7361, "title": "from dpo" }
0; ar , a r−1 + ar , . . . , a 0 + . . . + ar ) induces a bijection between the elements of ΓC and the LS-paths of shape nλ , n ∈ N, having support in C.ii) The Newton-Okounkov body ∆V (R) ⊂ Q\|A\| associated to V is the (gener-alized) polytope with integral structure described in . Conjecture 3. We conjecture that the ...	{ "page_id": null, "source": 7361, "title": "from dpo" }
Dumanski Department of Mathematics National Research University Higher School of Economics Usacheva str. 6 Moscow 119 048 RUSSIAN FEDERATION Dr. Xin Fang Mathematisches Institut Universit¨ at zu K¨ oln Weyertal 86 - 90 50931 K¨ oln GERMANY Prof. Dr. Evgeny Feigin Department of Mathematics National Research University H...	{ "page_id": null, "source": 7361, "title": "from dpo" }
Mathematics Smith College Northampton, MA 01063-0001 UNITED STATES	{ "page_id": null, "source": 7361, "title": "from dpo" }
Title: URL Source: Markdown Content: # Byzantine-fault tolerant algorithms in DistAlgo # Bachelor Thesis ## Roland Widmer from Bern, Switzerland Faculty of Science, University of Bern July 21th, 2020 Prof. Christian Cachin Orestis Alpos, Luca Zanolini Cryptology and Data Security Group Institute of Computer Science Un...	{ "page_id": null, "source": 7365, "title": "from dpo" }
. . . . . . . . . . . . . . . 93.1.1 Authenticated Echo Broadcast . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.1.2 Signed Echo Broadcast . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.2 Byzantine Reliable Broadcast . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.2.1 ...	{ "page_id": null, "source": 7365, "title": "from dpo" }
. . . . . . . . . . . . . . . . . . . . . . 19 4.2 Byzantine Reliable Broadcast . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 4.3 Byzantine Consistent Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 5 Conclusion 25 5.1 Discussion . . . . . . . . . . . . . . . . . . . . . . ...	{ "page_id": null, "source": 7365, "title": "from dpo" }
. . . 29 11Introduction Distributed algorithms are algorithms intended to achieve a certain degree of cooperation on a set of processes . They may be written in high-level pseudocode using certain assumptions, but practical implementations are usually much more extensive. DistAlgo is a high-level programming language t...	{ "page_id": null, "source": 7365, "title": "from dpo" }
that are able to perform computations in a distributed system are called processes. A process may for example represent a computer or a thread on a processor. In distributed computing, processes should be able to communicate with each other. The logical and physical network connecting processes is abstracted as a link....	{ "page_id": null, "source": 7365, "title": "from dpo" }
process is faulty or correct. The simplest failure of a process is a crash, where a process stops to execute steps. With this crash-stop process abstraction, a process is considered faulty if it crashes at some time during the execution, otherwise it is called correct .3This process abstraction is simple, but it covers...	{ "page_id": null, "source": 7365, "title": "from dpo" }
among a set of processes, broadcast communication abstrac-tions are used. They differ according to the reliability of the dissemination. With the arbitrary-fault process abstraction, a process can deviate arbitrarily from its assigned algorithm. A process may also try to prevent the goals of the algorithm from being re...	{ "page_id": null, "source": 7365, "title": "from dpo" }
to DistAlgo The advantage of a pseudocode implementation is the high degree of abstraction, which is used to facili-tate the understanding of algorithms. However, such implementations may not be executed on a system. DistAlgo is a very high-level language for clear descriptions of distributed algorithms. It combines th...	{ "page_id": null, "source": 7365, "title": "from dpo" }
variable p is not defined, the sender of the received message is accessible in the following code block using this variable. The argument from is used since from is a reserved word in Python. > def receive(('Hi.', 'Greetings.'), from_ =pid): > #function body All messages sent and received by a process are kept in varia...	{ "page_id": null, "source": 7365, "title": "from dpo" }
to the queries used in the implementations of the algorithms. A comprehension is implemented using the setof statement. The following example returns the set of all processes from which the process received a Hi message. > setof(proc, received(('Hi',), from_= proc)) Aggregations use an aggregation operator, which may b...	{ "page_id": null, "source": 7365, "title": "from dpo" }
set of methods and handlers. A special setup method may be used to initialize data before the process is started. All arguments of the method automatically become instance variables of the process. If additional instance variables are required, it is possible to define them in the setup method. Another special method i...	{ "page_id": null, "source": 7365, "title": "from dpo" }
the corresponding processes. Then, as a third step, the processes may be started with the start statement. This executes the run method of the corresponding processes. The following example creates three Bob processes and one Alice process. We can see the benefits of this three-part process creation. The Bob processes ...	{ "page_id": null, "source": 7365, "title": "from dpo" }
bobs = new(Bob, num=3, at='BobNode') setup(alice, (n_bobs,)) setup(bobs, (alice,)) start(alice) start(bobs) To execute the program on the two nodes, two terminals are required. A node is named using the -n option. More information about running a DistAlgo program on multiple nodes can be found in the DistAlgo repositor...	{ "page_id": null, "source": 7365, "title": "from dpo" }
specified. An algorithm that implements this module must satisfy these properties. The validity property guarantees that if a process p broadcasts a message m, then every correct pro-cess eventually delivers m. The second property, no duplication , allows correct processes to deliver at most one message. This constrain...	{ "page_id": null, "source": 7365, "title": "from dpo" }
m. BCB2: No duplication: Every correct process delivers at most one message. BCB3: Integrity: If some correct process delivers a message m with sender p and process p is correct, then m was previously broadcast by p. BCB4: Consistency: If some correct process delivers a message m and another correct process delivers a ...	{ "page_id": null, "source": 7365, "title": "from dpo" }
been sent by another process, as there are also requirements for the links. The correct processes will receive enough ECHO messages to deliver the message m. This proves the validity property. No duplication follows directly from the algorithm. If the flag delivered is TRUE , no further message may be delivered. If a c...	{ "page_id": null, "source": 7365, "title": "from dpo" }
is required that provides two functions sign and verifysig . A process may get a signature of a message using the sign function with its process identifier. On the other hand, a process may verify a signature of an incoming message by calling the verifysig function, which returns a Boolean value. Algorithm overview. In...	{ "page_id": null, "source": 7365, "title": "from dpo" }
. It is called totality . With this fifth property, we define Byzantine reliable broadcast (Module 2). As before, only 11 Algorithm 2 Signed Echo Broadcast. Implements: ByzantineConsistentBroadcast, instance bcb , with sender s. Uses: AuthPerfectPointToPointLinks, instance al . upon event 〈bcb , Init 〉 do sentecho := F...	{ "page_id": null, "source": 7365, "title": "from dpo" }
property arises. It states that if a message m is delivered by some correct process, then m is eventually delivered by every correct process. Module 2 Interface and properties of Byzantine reliable broadcast. > Module: Name: ByzantineReliableBroadcast, instance brb , with sender s. > Events: Request: 〈brb ,Broadcast \|m...	{ "page_id": null, "source": 7365, "title": "from dpo" }
process has received READY messages from more than f processes, it sends a READY message to all processes itself. This mechanism is required to satisfy the totality property. Correctness. To show the validity , no duplication , and integrity properties, the arguments of the proof of Authenticated echo broadcast may be ...	{ "page_id": null, "source": 7365, "title": "from dpo" }
echos [p] := m upon exists m 6 = ⊥ such that #( {p ∈ Π \| echos [p] = m}) > N+f > 2 and sentready = FALSE do sentready := TRUE forall q ∈ Π do trigger 〈al , Send \| q, [READY , m ]〉 upon exists m 6 = ⊥ such that #( {p ∈ Π \| readys [p] = m}) > f and sentready = FALSE do sentready := TRUE forall q ∈ Π do trigger 〈al , Send...	{ "page_id": null, "source": 7365, "title": "from dpo" }
message mwith label `broadcast by process p. > Properties: BCCH1: Validity: If a correct process pbroadcasts a message m, then every correct process eventually delivers m. > BCCH2: No duplication: For every process pand label `, every correct process eventually delivers at most one message with label `and sender p. > B...	{ "page_id": null, "source": 7365, "title": "from dpo" }
a ready flag is used. If an instance of Byzantine consistent broadcast delivers, the channel also delivers. Then, a new instance is created for this process and the corresponding sequence number is incremented. If the process is also the sender process of the new instance, then the ready flag is set to > TRUE , as the ...	{ "page_id": null, "source": 7365, "title": "from dpo" }
all processes in the system. This event is realized conveniently in DistAlgo by using the setup method. A Python dictionary is used as data structure for echos . To send messages to all processes later on, the identifiers of all processes are required. For this purpose, the argument procs is used. The number of process...	{ "page_id": null, "source": 7365, "title": "from dpo" }
DistAlgo, hence an if statement is used. > def receive(msg=('SEND', m), from_= self.sender): > if not (self.sentecho): self.sentecho =True > send(('ECHO', m), to= self.procs) To handle the ECHO messages another handler is defined as in pseudocode. In this handler, the from argument is no longer used to determine whethe...	{ "page_id": null, "source": 7365, "title": "from dpo" }
DistAlgo only receive handlers for messages may be defined, a direct translation is not possible. However, given that the condition only depends on the variables echos and delivered , it is sufficient to check the condition only in case the variables have been changed. The helper method call in the receive handler abov...	{ "page_id": null, "source": 7365, "title": "from dpo" }
sender, message)) start(procs) Alternative. Await statements can also be used as internal events in DistAlgo. The condition that is checked in the event handler in the pseudocode may also be placed in an await statement. Since an effort was made to implement the await statements efficiently, the performance is not cons...	{ "page_id": null, "source": 7365, "title": "from dpo" }
= True if sender == self else False self.quorumbound = (self.n + self.f) / 2 if self.is_sender: broadcast() def broadcast(): output('BROADCAST \' ', self.msg, '\' ') send(('SEND', self.msg), to= procs) To create a signature, the sign method of the Crypto class is used. The ECHO messages are no longer sent to all proces...	{ "page_id": null, "source": 7365, "title": "from dpo" }
same as in the Authenticated echo broadcast implementa-tion. 20 4.2 Byzantine Reliable Broadcast The description of the initialization, the broadcast method, and the receive handler for handling the > SEND and ECHO messages is omitted since these are almost identical to the corresponding structures in the Authenticated...	{ "page_id": null, "source": 7365, "title": "from dpo" }
three parts use the helper methods described above. > def run(): > while not (self.delivered): > if await (some(m in self.echos.values(), has=( len(setof(p, pin procs, self.echos[p] == m)) >self.quorumbound and > m!= False and not (self.sentready)) ): send_ready(m) > elif some(m in self.readys.values(), has=( len(setof...	{ "page_id": null, "source": 7365, "title": "from dpo" }
DistAlgo process is able to determine the virtual instance of the received message. In this implementation, the Authenticated echo broadcast implementation is used, but another algorithm implementing Byzantine consistent broadcast would also be possible. The first part of the setup method initializes the data structure...	{ "page_id": null, "source": 7365, "title": "from dpo" }
the identifier of the sender process is also the sender of the SEND message and may be obtained by using the from argument. > def receive(msg=('BCCH', 'BROADCAST', m)): > if self.ready: bcb_broadcast(m) self.ready =False def bcb_broadcast(m: str): send(('BCB', 'SEND', m, self.n[self]), to =procs) The following receive ...	{ "page_id": null, "source": 7365, "title": "from dpo" }
in self.delivered) and > len(setof(tup, tup in self.echos.keys(), self.echos[tup] == item)) >self.quorumbound and > item != False )): bcb_deliver(item, item) The bcb deliver method adds the label to the set delivered . This has the same effect as setting the flag delivered to TRUE in the Authenticated echo broadcast . ...	{ "page_id": null, "source": 7365, "title": "from dpo" }
23 setup(procs, (procs, nprocesses, f)) start(procs) sender = procs.pop() sender2 = procs.pop() send(('BCCH', 'BROADCAST', 'test'), to= sender) send(('BCCH', 'BROADCAST', 'test2'), to= sender) # not ready yet work() send(('BCCH', 'BROADCAST', 'test3'), to= sender) send(('BCCH', 'BROADCAST', 'test4'), to= sender2) 24 5C...	{ "page_id": null, "source": 7365, "title": "from dpo" }
above. On the one hand, helper methods may be used that check the condition and are always called, if a variable on which the condition depends changes. Another possibility, as in the implementation of the Authenticated double-echo broadcast , is to use an await statement. Broadcast event. Since DistAlgo processes may ...	{ "page_id": null, "source": 7365, "title": "from dpo" }
class of the sender process would contain a lot of duplicated code, since the sender process would only be an extension of a normal process. ## 5.2 Future work It will be interesting to improve the implementations. On the one hand, we could provide authenticity and thus make the correctness of the algorithms complete. ...	{ "page_id": null, "source": 7365, "title": "from dpo" }
(process): def setup(alice:da.common.ProcessId): self.greeted = False def run(): send(('Hi Alice',), to=alice) if await (self.greeted == True ): send(('Bye',), to=alice) def receive(msg=('Hi Bob',), from_=p): output("Hi Bob received from", p) self.greeted = True ; class Alice (process): def setup(n_bobs:int): pass def ...	{ "page_id": null, "source": 7365, "title": "from dpo" }
of the 21st International Sympo-sium on Principles and Practice of Programming Languages 2019 , Mar. 2019, arXiv:1704.00082v4. Y. Liu, B. Lin, and S. Stoller, DistAlgo Language Description , liu/distalgo/language.pdf, Mar. 2017. DistAlgo Readme.md on GitHub , Mar. 2020. RFC 793 - Transmission Control Protocol , Se...	{ "page_id": null, "source": 7365, "title": "from dpo" }
Title: Reliable Broadcast despite Mobile Byzantine Faults URL Source: Markdown Content: # HAL Id: hal-04277831 Preprint submitted on 9 Nov 2023 HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may com...	{ "page_id": null, "source": 7365, "title": "from dpo" }
investigated since then, thanks to its abil-ity to prevent arbitrarily (i.e., Byzantine) faulty processes from equivocating by sending different messages to different processes. It has been introduced as a one-shot primitive that allows a pre-defined process in the system to spread a single message and generalized as a...	{ "page_id": null, "source": 7365, "title": "from dpo" }
rather than internal misbehavior, and tools such as software rejuvenation techniques , intrusion detection systems , and trusted exe-cution environments are available. Despite several fundamental distributed problems have been analyzed in the literature considering the MBF model (i.e., Byzantine agreement [17, 6], app...	{ "page_id": null, "source": 7365, "title": "from dpo" }
consider relatively strong assumptions in our system model, the same as those considered in related work, in order determine fun-damental solvability conditions. Relaxation of most of these assumptions has already been partially investigated . The rest of the paper is structured as follows. After reviewing related work...	{ "page_id": null, "source": 7365, "title": "from dpo" }
both Byzantine faulty processes and fail-stop failures, the latter distinguishing between two dif-ferent kinds of Byzantine behaviors, i.e. those attempting to prevent the liveness and those attempting to prevent the safety of the BRB. Recently, Guerraoui et al. and Li et al. extended BRB to distributed systems with ...	{ "page_id": null, "source": 7365, "title": "from dpo" }
under constrained mobility the adversary can move agents only when pro-tocol messages are sent (similarly to how viruses would propagate), while under unconstrained mobility [4, 6, 17, 24, 31, 27] agents do not move with messages but rather during specific phases of the round. More in detail, Reischuk con-sidered mal...	{ "page_id": null, "source": 7365, "title": "from dpo" }
Byzantine-tolerant registers [10, 8, 11]. To the best of our knowledge, no efforts have been made to investigate the BRB problem in the presence of MBFs. All existing works that assume MBFs rely on some kind of best-effort communication subsystem (i.e., no guarantees exist when a process is controlled by a Mobile Byzan...	{ "page_id": null, "source": 7365, "title": "from dpo" }
executes a distributed protocol P consisting of a set of local al-gorithms. Each algorithm in P is represented by a finite state automaton whose transitions correspond to computation and communication steps. A computa-tion step denotes a computation executed locally by a given process, while a communication step denote...	{ "page_id": null, "source": 7365, "title": "from dpo" }
process for a period of time lasting at least ∆ s ∈ Q+ (rational positive numbers), i.e., once arrived, an agent compromises a node for at least ∆s consecutive time units, and when ∆ s i B(ti) for b ≤ i < e ). Definition 3 (Correct process) . A process pi is correct when it is	{ "page_id": null, "source": 7365, "title": "from dpo" }
not faulty, that is, pi is correct at time tk if it is not controlled by a Byzantine agent at time tk. Similarly, a process pi is correct in the period Tb,e if it remains correct between times tb and te. Let us remark that when a process pi is correct, it executes P but potentially it may start its execution from a com...	{ "page_id": null, "source": 7365, "title": "from dpo" }
perform local computation on the processes and an upper bound on the time required by a message to be delivered via a P2P link, both of them known by all processes. In addition, we assume that the computation evolves in sequential synchronous rounds r1, r 2, . . . , r j , . . . . Every round rj is divided into three ph...	{ "page_id": null, "source": 7365, "title": "from dpo" }
in terms of round, and mobile Byzantine agents can move only between two consecutive rounds, i.e. after the computation phase of a round ri and before the send phase of round ri+1 1(see Garay’s MBF model ). Let us stress that in the S-MOB + setting every process is either faulty or cor-rect for an entire round. Therefo...	{ "page_id": null, "source": 7365, "title": "from dpo" }
compromised information). However, OBFA does not provide any information about the length of the period pi was faulty. Definition 6 (Full Failure Awareness Oracle OFFA ). If a mobile Byzantine agent takes control of a process pi at time tj and leaves pi at time tk, then the full failure awareness oracle OFFA generates ...	{ "page_id": null, "source": 7365, "title": "from dpo" }
message disseminated by a pre-defined process called the source , while the Byzantine Broadcast Channel (BBC) primitive extends BRB allowing all processes to disseminate an arbitrary number of messages so that all correct processes eventually deliver the same set of messages 2.Let us note that in the original BRB and ...	{ "page_id": null, "source": 7365, "title": "from dpo" }
time units), m is reliably delivered by any ∆ c-infinitely often correct process pj in a period where pj is correct. Similarly to BRB, this primitive is specified by considering an instance for every message generated by the identified source. More formally, a MBRB (∆ b, ∆c) communication primitive must guarantee the f...	{ "page_id": null, "source": 7365, "title": "from dpo" }
process pj that is ∆ c-infinitely often correct in a period in which pj is correct. More formally, a MBBC (∆ b,∆ c) communication primitive must guarantee the following properties: • (∆ b, ∆c)-Validity : If there exists a period Ti,j lasting at least ∆ b where a process ps is correct in Ti,j and executes MBRB.Broadcast...	{ "page_id": null, "source": 7365, "title": "from dpo" }
specific to mobile Byzantine faults in the following sections. # 5 Impossibility Results This section presents several impossibility results for the MBRB and MBBC problems. In particular, Theorems 7 and 9 prove the impossibility of solv-ing both MBRB and MBBC if the system is asynchronous, or if the agents’ 10 movement...	{ "page_id": null, "source": 7365, "title": "from dpo" }
s, m ) event is generated from pdest . To guarantee both (∆ b, ∆c)-Validity and ∆ c-Totality pdest must be different from ps. It is therefore necessary that in P psrc sends the message m through the reliable authenticated links at least once to allow a ∆c-infinitely often correct process pdest to become aware of the me...	{ "page_id": null, "source": 7365, "title": "from dpo" }
exists no protocol P implementing the Mobile Byzan-tine Reliable Broadcast (resp. Mobile Byzantine Broadcast Channel) in 〈ASYNC , M, O 〉, with M ∈ { A-MOB , S-MOB } and O ∈ {O FFA , OBFA }.Proof. The claim follows from the same argument provided as in Theorem 7, given that the 〈ASYNC , S-MOB , OFFA 〉 setting is the str...	{ "page_id": null, "source": 7365, "title": "from dpo" }
) in round r∆b+∆ 1+1 . We remark that the failure state of any process may change unexpectedly due to the move-ment of a Byzantine agent. Let us consider another local execution history H′′ > s of process ps where the failure state of ps evolves in the opposite way from H′ > s , that is process ps is faulty in rounds r...	{ "page_id": null, "source": 7365, "title": "from dpo" }
rΔ1 > XXXXXXX (b) Graphical representations for Theorem 12. Figure 1: Graphical representations for Theorems’ proof. obtained from OFFA with process p1 through the point-to-point primitive, pro-cess p1 cannot distinguish an execution of A where ps is correct and reveals a previous faulty state, from another where ps is...	{ "page_id": null, "source": 7365, "title": "from dpo" }
of contradiction, let us assume that such a protocol P ex-ists. Let us assume a permanently correct process ps (i.e. ∀rj , p s ∈ C(rj )) that 13 ASYNC SYNC S-MOB + > OBFA OFFA > 77 > (Cor. 11) (Th. 10) S-MOB 7OBFA OFFA > 77 > (Cor. 8) (Cor. 11) (Cor. 11) A-MOB 77 > (Cor. 8) (Th. 9) > (a) MBRB > ASYNC SYNC S-MOB + > OBF...	{ "page_id": null, "source": 7365, "title": "from dpo" }
previous rounds. It is even defining an algorithm A that allows any process pi to share and retrieve the state and events occurred on the process through the point-to-point primitive: process pi can execute such a protocol either as correct or as faulty, and the two executions would be indistinguishable by any other pr...	{ "page_id": null, "source": 7365, "title": "from dpo" }
on the deliveries of messages that depend on the actual and previous failure states of the processes, generating thus symmetry conditions that are impossible to break without violating one of the properties charac-terizing the specification. In particular, the main challenge is to ensure that a single broadcast instanc...	{ "page_id": null, "source": 7365, "title": "from dpo" }
〈SYNC, S-MOB +, # OFFA 〉 Theorem 12 and Corollary 11 motivate the definition of a stronger local ora-cle than those considered in related work dealing with mobile Byzantine faults, OF F A : both MBRB and MBBC are impossible to solve in the ( 〈SYNC , S-MOB +, ONFA /BFA 〉) settings. Theorem 10 states the impossibility in...	{ "page_id": null, "source": 7365, "title": "from dpo" }
all processes in the subsequent round. Every process that P2P -receives a SEND message in round rb+1 from ps computes the ECHO protocol’s message for 〈s, r b, m 〉 and enqueues it to P2P -send to all peers. In round rb+2 , the processes that receive sufficiently many ECHO messages (more than ( n + f )/2) for an MBBC-bro...	{ "page_id": null, "source": 7365, "title": "from dpo" }
the variable rc in Algorithm 1) during compute phase. We stress the fact that protocol’s messages in PM BBC −RB (SEND , ECHO , READY , and ABORT ) must be propagated in specific rounds with respect to the beginning of the MBBC-broadcast, in order to progress till the delivery of the associated message m.For ease of bet...	{ "page_id": null, "source": 7365, "title": "from dpo" }
〈s, r b, m 〉] ∪ { q} 22: if Type = ABORT then 23: Aborts[ 〈s, r b, m 〉] ← Aborts[ 〈s, r b, m 〉] ∪ { q} 24: upon P2P .deliver (q , 〈ROUND , j〉) do 25: RC[ q] ← j Compute Phase 26: To send ← ∅ , rc ← getMajority (RC. values ) 27: for 〈s, r b, m 〉 ∈ Sends do 28: if rc = rb+1 then 29: To send ← To send ∪ {〈 ECHO , s, r b, ...	{ "page_id": null, "source": 7365, "title": "from dpo" }
can move only between the compute and send phase of two consecutive rounds. This implies that ∆ s is assumed greater than or equal to one round. Such mobility model has the following effects to the agents’ capabilities: at the beginning of a round rj , mobile agents can potentially control the messages that are diffuse...	{ "page_id": null, "source": 7365, "title": "from dpo" }

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