Split (1)

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$\textsf{enbl}_{ \tau }$ ($\tau \in T^L \setminus \{ \textit{idle}\}$) and (b) that ${\mathfrak {A}}_n \models \bigwedge _{\langle q^0_n, \tau , q' \rangle \in \delta ^L} \lnot \textsf{enbl}_{ \tau }$. (The latter still holds when the conjunction is empty, in which case it evaluates to $\top $ by convention.) C...	{ "page_id": null, "source": 7332, "title": "from dpo" }
^i_n}(a_i)\), and !Image 189\) and (f) that ${\mathfrak {A}}_n \models \textsf{enbf}_{ \tau ^i_n }(a_i)$. Consequently !Image 190 satisfies the first disjunct of $\textsf {stepf}(a_i)$. If $ enb ( {\mathfrak {c}}^i_n ) = \emptyset $, we have $\tau ^i_n = \textit{idle}$ and $q^i_n = q^i_{n+1}$. Thus, from (e) ...	{ "page_id": null, "source": 7332, "title": "from dpo" }
196]( trivially. Therefore, the first conjunct of $\textsf {pastm}$ is true in ${\mathfrak {M}}$. Otherwise, let _t_ be the smallest number such that ${\mathfrak {A}}_t \models {{\textsf{t}}}{{\textsf{l}}}_{\vec {\mu }}$. Thus, from (d), \({\mathfrak {A}}_{n} \models \lnot {{\textsf{t}}}{{\textsf{l}}}_{\vec {\mu ...	{ "page_id": null, "source": 7332, "title": "from dpo" }
${\mathfrak {A}}_n \models \textsf {past}_{\vec {\mu }}$, there exists $n' 0$. Because !Image 203 or !Image 204, we have ${\mathfrak {A}}_0 \not \models \Box \lnot \textsf {past}_{\vec {\mu }}$, therefore !Image 205 be the smallest number such that \({\mathfrak {A}}_{n'} \models {{\textsf{t}}}{{\textsf{l}}}_{\ve...	{ "page_id": null, "source": 7332, "title": "from dpo" }
since !Image 213, it follows that ${\mathfrak {A}}_0 \models \Box \lnot {{\textsf{t}}}{{\textsf{l}}}_{\vec {\mu }}$, and, as a result, ${\mathfrak {A}}_{n'} \models \lnot {{\textsf{t}}}{{\textsf{l}}}_{\vec {\mu }}$ for all $n' 0$) and let \({\mathfrak {M}}= \{ {\mathfrak {A}}_n \}_{n \in	{ "page_id": null, "source": 7332, "title": "from dpo" }
{\mathbb {N}}}\) be a sequence, where each ${\mathfrak {A}}_n$ ($n \in {\mathbb {N}}$) is a $\Sigma ^M$-structure ($\Sigma ^M$ as in the beginning of this section) with universe _A_, such that $\textsf {Spec}(M)$ is true in ${\mathfrak {M}}$. We construct from ${\mathfrak {M}}$ a run \({\mathfrak {R}}= \l...	{ "page_id": null, "source": 7332, "title": "from dpo" }
next show that ${\mathfrak {R}}^0$ is a run for _L_ over _A_. Since ${\mathfrak {A}}_0 \models \textsf {initl}$ we have from (a) that $q^0_n \in Q^L_\text {init}$ and !Image 220). Now, it is a consequence of the assumption that $\textsf {Spec}(M)$ is true in ${\mathfrak {M}}$ that \({\mathfrak {A}}_n \models ...	{ "page_id": null, "source": 7332, "title": "from dpo" }
the latter case, it follows from (a) that $q^0_n = q^0_{n+1} = q$ and from (b) that $\tau ^0_n = \textit{idle}$. That is, $\langle \sigma ({\mathfrak {c}}^0_n), \tau ^0_n, \sigma ({\mathfrak {c}}^0_{n+1}) = \langle q^0_n, \tau ^0_n, q^0_{n+1} \rangle = \langle q, \tau , q' \rangle \in \delta ^L$ or \(\tau ^0_n = ...	{ "page_id": null, "source": 7332, "title": "from dpo" }
which, due to (d), implies that !Image 246\), !Image 248, implies that ${\mathcal {I}}^{i}_n \subseteq {\mathcal {I}}^{i}_{n+1}$; and if !Image 250, implies that !Image 251\)) !Image 252, !Image 258\), it follows that !Image 260, or !Image 261. For the former case, it follows from (c) that $q^i_n = q$ and \(q^i_{n+...	{ "page_id": null, "source": 7332, "title": "from dpo" }
${\mathfrak {R}}$. For (A1), we first show that if $\tau ^0_n = \vec {\mu }$ ($n \in {\mathbb {N}}$), then, for each _i_ ($1 \le i \le k$), there exists $n' > n$, such that !Image 264 and suppose $\tau ^0_n = \tau = \vec {\mu }$. From (b), it must be \({\mathfrak {A}}_n \models {{\textsf{t}}}{{\textsf{l}}}_...	{ "page_id": null, "source": 7332, "title": "from dpo" }
is satisfiable over a finite model. ### Remark 3 It should be clear from the proofs of Lemma3\) over finite domains are in one-to-one correspondence. It should also be clear that all runs of _M_ have a property ${\mathcal {P}}$ (expressible in $\textsf {MFOTL}$) if and only if \(\textsf {Spec}(M) \wedge \lnot \varp...	{ "page_id": null, "source": 7332, "title": "from dpo" }
We now proceed to define the formulas describing _N_’s network. The following two formulas (short for _broadcast leader synchronous_ and _send follower synchronous_ respectively) are a translation of conditions (S1) and (S2) respectively from Sect.3.4 and $\textsf {sendfa}$, the symbol !Image 278 exists because \(T^L...	{ "page_id": null, "source": 7332, "title": "from dpo" }
{P}}\) is the formula (over $\Sigma ^N$) expressing ${\mathcal {P}}$ in $\textsf {MFOTL}$, is unsatisfiable over finite domains. Relevant to Rem.1 synchronous distributed machine $N = \langle L, F \rangle $ and a property ${\mathcal {P}}$ expressed in $\textsf {MFOTL}$ (over $\Sigma ^N$) by the formula \(...	{ "page_id": null, "source": 7332, "title": "from dpo" }
$t^{\vec {\mu }}_m$ holds true and if !Image 292 respectively then $t^{\vec {\mu }}_m$ also holds true at time $n+1$ ($n \in {\mathbb {N}}$), for each follower _x_. Similarly, for the followers, one introduces _m_ new unary predicates !Image 293 a follower may send to the leader, and enforces the condition that...	{ "page_id": null, "source": 7332, "title": "from dpo" }
so far. The correctness criterion for this protocol is that, eventually, the output bits of all processes will be the same. We now give a formal specification of the above protocol as a network of a leader and followers. Intuitively, each follower corresponds to one of the aforementioned _k_ processes. Since our model ...	{ "page_id": null, "source": 7332, "title": "from dpo" }
a distributed machine out of _L_ and _F_ we must satisfy the constraints $T^L_\text {in} \cap T^F_\text {in} = \emptyset $, $T^L_\text {in} \backsimeq T^F_\text {out}$ and $T^F_\text {in} \backsimeq T^L_\text {out}$. To satisfy the first restriction we could have typeset either $T^L_\text {out}$ or \(T^F_\text ...	{ "page_id": null, "source": 7332, "title": "from dpo" }
receipt of a _prepare_ message from the leader, each follower enters a _preparing_ state. In this state the follower can choose to (a) abort the transaction (for its own reason); or (b) prepare to receive (from the leader) a command to commit or abort. If the follower aborts, it sends an _aborted_ message to the leader...	{ "page_id": null, "source": 7332, "title": "from dpo" }
= \{ \overleftarrow{prepared }, \overleftarrow{aborted } \}\), $T^L_\text {out} = \{ \overrightarrow{prepare }, \overrightarrow{commit }, \overrightarrow{abort } \}$, and $T^L_\text {local} = \emptyset $; thus !Image 303. Let $Q^L = \{ working ,\, waiting ,\, committing ,\, aborting ,\, committed ,\, aborted \}$,...	{ "page_id": null, "source": 7332, "title": "from dpo" }
if and only if $\textsf {Spec}(M^\text {2PC})$ is satisfiable. According to the informal correctness criterion stated above, we need to verify that the following FOTL-sentence is unsatisfiable over finite domains: $$\begin{aligned} \textsf {Spec}(M^\text {2PC}) \,\wedge \, \lnot \, \Diamond ( \forall x \,.\, {{\texts...	{ "page_id": null, "source": 7332, "title": "from dpo" }
for symmetry, if it receives no bid it broadcasts a _nogo_ message. If a sensor that has bid for the transmission slot receives a _go_ message, it sends its data and moves to its initial state; otherwise, if it receives a _nogo_ message simply moves to its initial state. If a sensor that has not bid for the transmissio...	{ "page_id": null, "source": 7332, "title": "from dpo" }
\overleftarrow{bid } \}\), $T^L_\text {out} = \{ \overrightarrow{go }, \overrightarrow{nogo } \}$, and $T^L_\text {local} = \{ wait \}$; thus !Image 309. Let $Q^L = \{ q_0, \ldots , q_4 \}$, $Q^L_\text {init} = \{ q_0 \}$, and !Image 310. _L_ is depicted in Fig.7, \(T^F_\text {out} = \{ \overrightarrow{bid } \}...	{ "page_id": null, "source": 7332, "title": "from dpo" }
stated above, we need to verify that the following FOTL-sentence is unsatisfiable over finite domains: $$\begin{aligned}&\textsf {Spec}(N^\text {MAC}) \\&\quad \wedge \, \quad \Box \forall x(({{\textsf{t}}}{{\textsf{f}}}_{\overrightarrow{bid}}(x) \wedge \exists y (x \ne y \wedge {{\textsf{t}}}{{\textsf{f}}}_{\overright...	{ "page_id": null, "source": 7332, "title": "from dpo" }
(2003) Constraint-based verification of parameterized cache coherence protocols. Formal Methods Syst Design 23(3):257–301"), 11 Automatic verification of parameterized cache coherence protocols. International Conference on Computer Aided Verification. Springer International Publishing, Cham, pp 353–360")] and [14 On th...	{ "page_id": null, "source": 7332, "title": "from dpo" }
possibly sacrificing completeness for constraints requiring equality). At the same time, $\textsf {MFOTL}$ remains decidable (sacrificing completeness when it comes to properties requiring equality) and has relatively low worst-case computational complexity. The decidability and computational complexity of \(\textsf ...	{ "page_id": null, "source": 7332, "title": "from dpo" }
have shown that protocols specified in this framework can automatically be translated into monodic first-order temporal logic, maintaining their semantics. Automated theorem provers for said logic can then be used to check whether that translation (and thus the original protocol) satisfies a property of interest. Thus,...	{ "page_id": null, "source": 7332, "title": "from dpo" }
Title: Parameterized verification of leader/follower systems via first-order temporal logic URL Source: Markdown Content: Formal Methods in System Design (2021) 58:440–468 # Parameterized verification of leader/follower systems via first-order temporal logic G. Kourtis 1 · C. Dixon 1 · M. Fisher 1 · A. Lisitsa 2 Rece...	{ "page_id": null, "source": 7332, "title": "from dpo" }
of Computer Science, University of Liverpool, Liverpool, UK # 123 Formal Methods in System Design (2021) 58:440–468 441 ## 1 Introduction Parameterized verification is becoming increasingly important nowadays, with technologies like the Internet of Things (IoT), sensor networks, robot swarms, and satellite constellatio...	{ "page_id": null, "source": 7332, "title": "from dpo" }
to automatically translate any suitable protocol to a monodic first-order temporal formula that encapsulates its behaviour. One is then able to check whether that protocol has a certain property by checking whether said formula logically entails the property. (Of course the property must be expressible in monodic first...	{ "page_id": null, "source": 7332, "title": "from dpo" }
the leader has not received any “no” messages). To # 123 442 Formal Methods in System Design (2021) 58:440–468 > Fig. 1 A simple protocol in which the leader ( a0) has to record the messages it receives from its followers ( a1,a2,a3) be able to make a decision about the consensus in the future, the leader has to mainta...	{ "page_id": null, "source": 7332, "title": "from dpo" }
2 we review some relevant concepts from first-order temporal logic. In Sect. 3 we give a formal definition of our model. In Sect. 4 we show how our model automatically translates to (monodic) first-order temporal logic. In Sect. 5 we specify with our framework three protocols appearing in practice, namely the FloodSet ...	{ "page_id": null, "source": 7332, "title": "from dpo" }
ϕ W ψ.If ϕ is an FOTL -formula, we denote its length by ‖ϕ‖. FOTL -formulae are interpreted in first-order temporal structures , i.e. sequences M = A0, A1, . . . of first-order structures over the same domain A. That is, given a non-empty set A, An = 〈 A, I n 〉 (n ∈ N), where is I n an interpretation of predicate and c...	{ "page_id": null, "source": 7332, "title": "from dpo" }
formula ∀x∀y(P(x, y) → Q(x, y)) is not monodic. The set of all monodic formulae form the monodic fragment of FOTL (abbr. MFOTL ). FOTL is incomplete (not recursively axiomatizable) and undecidable . In contrast, MFOTL is finitely axiomatizable , and, if its first-order part is restricted to a decidable fragment of fir...	{ "page_id": null, "source": 7332, "title": "from dpo" }
domains using MFOTL . In particular, reports that each of the following three axioms enforces finite domains (and is derivable from each of the other two): (F1) ♦∀x(♦P(x) → P(x)) (F2) [∀ x (P(x) → ¬P(x)) ] → [ ♦ ∀x ¬P(x)] (F3) [∀x(P(x) → P(x)) ] → [ ♦ ∀x( P(x) → P(x)) ] ( is interpreted as “sometime in the past”....	{ "page_id": null, "source": 7332, "title": "from dpo" }
∀x y (E(x, y) → E(y, x)) (E3) ∀x yz (E(x, y) ∧ E(y, z) → E(x, z)) (E4) ∀x1 · · · x n y1 · · · yn (∧ni=1 E(x i , yi ) ∧ P(x1, . . . , x n ) → P(y1, . . . , yn )) In a temporal setting, one needs in addition axiom (E5) below, enforcing the condition that equal objects remain equal across time: # 123 Formal Methods in Sys...	{ "page_id": null, "source": 7332, "title": "from dpo" }
have a collection of k + 1 ( k > 0) finite-state machines in a network, k of which are replicas of each other and the remaining machine is a distinguished machine orchestrating the operation of the replicas. Each replica is referred to as a follower ,the distinguished machine is referred to as a leader , and the whole ...	{ "page_id": null, "source": 7332, "title": "from dpo" }
to an inbox (i) being empty ; (ii) having exactly one element; (iii) having one or more elements; (iv) having more than one element; or (v) being full , i.e. containing the identities of all followers in the network. Type (a)(ii)-(v) transitions for the leader delete the contents of the corresponding inbox. Each follow...	{ "page_id": null, "source": 7332, "title": "from dpo" }
leader’s three different types of transitions (reacting to an inbox, broad-casting a message, and local) we introduce the following notation. Transitions in which the leader reacts to an inbox Iμ are denoted with the message μ followed by a quantifier , i.e. =0 , > =1 , ≥1 , >1 , or all (written as a superscript). Th...	{ "page_id": null, "source": 7332, "title": "from dpo" }
on top). For examples using the above notation see Sect. 5 and figures therein. 3.2 Leaders and their execution Let T L > in = { μ1, . . . , μr }, T L > out = { μ1, . . . , μs }, and T L > local = { 1, . . . , t } be sets of symbols (note that μ and μ are different symbols); let T L > in = { μ=0,μ=1,μ≥1,μ...	{ "page_id": null, "source": 7332, "title": "from dpo" }
special transition the leader takes when it is unable to take any other transition. We define the size of L, denoted ‖L‖, to be the quantity \|Q L \| + \| T L \| + \| δL \|. For some example leaders see Sect. 5. We describe the execution of a leader as a sequence of certain configurations in the context of a (non-empty) set ...	{ "page_id": null, "source": 7332, "title": "from dpo" }
(c) (short for enabled )the set of transitions τ ∈ T L \ { idle } such that 〈q, τ, q′ 〉 ∈ δL , for some q′ ∈ Q L , and either τ ∈ T L > out ∪ T L > local or one of the following holds: (a) τ =μ=0 (μ ∈ T L > in ) and Iμ = ∅ ;(b) τ =μ=1 (μ ∈ T L > in ) and \|Iμ\| = 1; (c) τ =μ≥1 (μ ∈ T L > in ) and \|Iμ\| ≥ 1; (d) τ...	{ "page_id": null, "source": 7332, "title": "from dpo" }
,μ≥1 ,μ>1 ,μall for all μ ∈ T L > in , 〈I 1, . . . , I r 〉 τ 〈J 1, . . . , J r 〉 iff I j ⊆ J j (for all 1 ≤ j ≤ r); (b) if τ =μ=1 ,μ≥1 ,μ >1 , or μall for some μ =μi ∈ T L > in , 〈I 1, . . . , I r 〉 τ 〈J 1, . . . , J r 〉 iff J i = ∅ and I j ⊆ J j (for all 1 ≤ j ≤ r, j = i). A run for L over A is a seque...	{ "page_id": null, "source": 7332, "title": "from dpo" }
of new messages arriving, except when their content is being deleted. The following simple lemma should be clear from the above. It will be useful when proving that the logical formula in Sect. 4.1 describing the leader’s execution is correct. Lemma 1 Let R = {〈 cn , τ n 〉} n∈N be a run for L over a set A. For all n ∈ ...	{ "page_id": null, "source": 7332, "title": "from dpo" }
3.1, T F > in , T F > out , and T F > local are respectively the sets of labels for the transitions in which F reacts to a message in its inbox, sends a message to the leader, and the set of F’s local actions. The transition idle is a special transition each follower takes when it is unable to take any other transition...	{ "page_id": null, "source": 7332, "title": "from dpo" }
enabled, whereas transitions in which the follower reacts to a message are enabled only if the corresponding message is present in I.As in the leader’s case, we want to enforce at each moment of time a form of persistence for the inbox I, i.e. that I can grow (when a new message arrives from the leader) but no message ...	{ "page_id": null, "source": 7332, "title": "from dpo" }
unless explicitly deleted when a transition in T F > in is taken. The following simple lemma should be clear from the above. It will be useful when proving that the logical formula in Sect. 4.2 describing the followers’ execution is correct. Lemma 2 Let R = {〈 cn , τ n 〉} n∈N be a run for F. For all n ∈ N, if τn =μ ...	{ "page_id": null, "source": 7332, "title": "from dpo" }
as the universe of M. Intuitively, each a ∈ A is identified with a (replicated) follower F. A run for M over A is a (k + 1)-tuple R = 〈 R0, R1, . . . , Rk 〉,where R0 = {〈 c0 > n , τ 0 > n 〉} n∈N (c0 > n = 〈 q0 > n , I > μ1 > n , . . . , I > μrn 〉, {μ1, . . . , μr } = T L > in ) is a run for L over A and each Ri = {...	{ "page_id": null, "source": 7332, "title": "from dpo" }
by the leader at some point in the past. Similarly, (A2) states that if a follower a ∈ A sends a message μ to the leader, a will eventually appear in the inbox Iμ;and if a ∈ Iμ, a message μ has been sent by a to the leader at some point in the past. Conditions (A1) and (A2) describe an asynchronous network (i.e. a ...	{ "page_id": null, "source": 7332, "title": "from dpo" }
run for M over some universe A ={a1, . . . , a k } (k > 0). Remark 1 The behaviour of a synchronous distributed machine when the delivery of a mes-sage from a follower to the leader coincides with the leader deleting the content of the corresponding inbox or when the delivery of a message from the leader to a follower ...	{ "page_id": null, "source": 7332, "title": "from dpo" }
μ ∈ T L > out , a nullary predicate past μ; for each symbol μ ∈ T F > out , a unary predicate past μ (·); a nullary predicate tl τ for each τ ∈ T L ; and a unary predicate tf τ (·) for each τ ∈ T F . For each q ∈ Q L , sl q (short for state leader ) is to be viewed as stating that the leader is at state q. For each...	{ "page_id": null, "source": 7332, "title": "from dpo" }
Spec (M). 4.1 The leader’s execution We first write a collection of formulas describing the execution of the leader. The following two formulas (short for unique state leader and unique transition leader respectively) state that the leader’s states and transitions are unique: # 123 Formal Methods in System Design (2021...	{ "page_id": null, "source": 7332, "title": "from dpo" }
step-by-step execution of the leader. For each τ ∈ T L \ { idle }, let enbl τ := ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ , if τ ∈ T L > out ∪ T L > local ;∀x ¬μ( x), if τ =μ=0 ∈ T L > in ;∃ =1 xμ( x), if τ =μ=1 ∈ T L > in ;∃ ≥1 xμ( x), if τ =μ≥1 ∈ T L > in ;∃>1 xμ( x), if τ =μ>1 ∈ T L > in ;∀xμ( x), if τ =μall ∈ T L > in . Th...	{ "page_id": null, "source": 7332, "title": "from dpo" }
follower x, i.e. that follower x starts at an initial state with its inbox empty: initf (x) := ∨ > q∈QF > init sf q (x) ∧ ∧ > μ∈TF > in ¬μ( x) The next formula (short for persistence follower ) is the persistence condition for the inbox of follower x: the inbox does not lose any messages (although it can grow due to ...	{ "page_id": null, "source": 7332, "title": "from dpo" }
clear that ‖follower (x)‖ = O(‖F‖2). 4.3 The distributed machine’s execution Next, we write a collection of formulas describing the network of the leader and its followers, and, combining these formulas with the ones in the previous two subsections, we form Spec (M). # 123 Formal Methods in System Design (2021) 58:440–...	{ "page_id": null, "source": 7332, "title": "from dpo" }
short for broadcast leader asynchronous ) and when a follower sends a message to the leader ( sendfa , short for send follower asynchronous ). (Recall that the network is asynchronous.) They correspond to conditions (A1) and (A2) respectively from Sect. 3.4. bcastla := ∧ > μ∈TL > out (tl μ → ∀ x ♦μ( x)) ∧ ∀x ∧ > μ∈...	{ "page_id": null, "source": 7332, "title": "from dpo" }
sent then a message μ is received (by the intended recipient(s)) at some point in the future (possibly at different times by different recipients), we simply state that if tl μ or tf μ (x) (for any follower x) holds true at any moment then ♦μ( x) holds true next. That μ can be received multiple times is # 123 454 ...	{ "page_id": null, "source": 7332, "title": "from dpo" }
Lemma 3 Let M = 〈 L, F〉 be an asynchronous distributed machine and Spec (M) as above. If M is runnable, then Spec (M) is satisfiable over a finite domain. Proof Let A = { a1, . . . , a k } (k > 0) and let R = 〈 R0, R1, . . . , Rk 〉 be a run for M over A. Thus, R0 = {〈 c0 > n , τ 0 > n 〉} n∈N is a run for L over A and e...	{ "page_id": null, "source": 7332, "title": "from dpo" }
τ 0 > n′ = μ for some n′ μ = ⊥ ;(d) for each τ ∈ T L , tl An > τ = if τ 0 > n = τ , otherwise tl An > τ = ⊥ ;(e) for each q ∈ Q F , sf Anq = { ai ∈ A \| q in = q};(f) for each μ ∈ T F > in ,μAn = { ai ∈ A \|μ ∈ Iin };(g) for each μ ∈ T F > in , past An > μ = { ai ∈ A \| ∃ n′ τ = { ai ∈ A \| τ in = τ }.We now sh...	{ "page_id": null, "source": 7332, "title": "from dpo" }
¬ tl μ=1 , ¬tl μ≥1 , ¬tl μ >1 , and ¬tl μall , then, from (d), # 123 Formal Methods in System Design (2021) 58:440–468 455 τ 0 > n =μ=1 ,μ≥1 ,μ >1 ,μall , which by Lemma 1 implies that, for all a ∈ A, if a ∈ I > μ > n then a ∈ I > μ > n+1 , which, in turn, implies due to (b) that An \| ∀ x(μ( x) →μ( x)); ...	{ "page_id": null, "source": 7332, "title": "from dpo" }
. Further, it is straightforward to establish from the definition of enbl τ (τ ∈ T L \{ idle }) and (b) that An \| ∧ > 〈q0 > n,τ, q′〉∈ δL ¬enbl τ . (The latter still holds when the conjunction is empty, in which case it evaluates to by convention.) Consequently, An \| sl q0 > n ∧∧ > 〈q0 > n,τ, q′〉∈ δL ¬enbl τ ∧tl idl...	{ "page_id": null, "source": 7332, "title": "from dpo" }
persf (ai ) is then evident. To show that, for all n ∈ N, for all ai ∈ A, An \| stepf (ai ),we consider two cases. If enb (cin ) = ∅ , we have τ in ∈ enb (cin ). Thus, from (e) and (h) above, An \| sf q in (ai ), An \| tf τ in (ai ), and An \| sf q in+1 (ai ). Further, it is straightforward to establish from the defi...	{ "page_id": null, "source": 7332, "title": "from dpo" }
the truth of ∀x follower (x) in M.Moving to neta , we first show that pastm is true in M. For the first conjunct of pastm ,let μ ∈ T L > out . If τ 0 > n = μ for all n ∈ N, then, from (d), An \| ¬ tl μ for all n ∈ N, and, from (c), An \| ¬ past μ for all n ∈ N. It follows that A0 \| ¬past μ, thus A0 \| ¬ past ...	{ "page_id": null, "source": 7332, "title": "from dpo" }
# 123 456 Formal Methods in System Design (2021) 58:440–468 pastm is true in M. The argument for the second conjunct is completely analogous. Thus, pastm is true in M. It is straightforward to show that bcastla and sendfa are true in M.As a result, neta is true in M. For the converse of Lemma 3, we require the fol...	{ "page_id": null, "source": 7332, "title": "from dpo" }
An \| past μ implies, then, that n′ ≤ n. It also follows that An′ −1 \| ¬ past μ ∧ ¬ tl μ ∧ tl μ. Since Am \| (¬past μ ∧ ¬ tl μ ∧ tl μ ) → ¬past μ for all m ∈ N, we must have An′ \| ¬ past μ, thus n′ 0. From A0 \| ¬ past μ W tl μ, we have that either A0 \| ¬past μ or A0 \| ¬ past μ U tl μ. In the former...	{ "page_id": null, "source": 7332, "title": "from dpo" }
be the unary predicate in M corresponding to the symbol μ ∈ T Fout . Let a ∈ A. For all n ∈ N, (a) if An \| past μ (a), there exists n ′ 0) and let M = { An }n∈N be a sequence, where each An (n ∈ N) is a M -structure ( M as in the beginning of this section) with universe A, such that Spec (M) is true in M. We constr...	{ "page_id": null, "source": 7332, "title": "from dpo" }
μrn 〉, such that An \| sl q0 > n and I > μin =μAni (1 ≤ i ≤ r); # 123 Formal Methods in System Design (2021) 58:440–468 457 (b) set τ 0 > n = τ ∈ T L such that An \| tl τ ;(c) for each i (1 ≤ i ≤ k), set cin = 〈 q in , Iin 〉, such that An \| sf q in (ai ) and Iin = { μ ∈ T F > in \| An \| μ( ai )};(d) for each i (1...	{ "page_id": null, "source": 7332, "title": "from dpo" }
,μall for all μ ∈ T L > in , which, due to (b), implies that An \| ¬ tl μ=1,μ≥1,μ>1,μall for all μ ∈ T L > in ,then, because An \| persl , An \| ∀ x(μ( x) →μ( x)) for all μ ∈ T L > in , which, due to (a), implies that I > μjn ⊆ I > μjn+1 (for all 1 ≤ j ≤ r); and if τ 0 > n =μ=1 ,μ≥1 ,μ>1 ,μall for some ...	{ "page_id": null, "source": 7332, "title": "from dpo" }
for the latter case, it follows from (a) that q0 > n = q0 > n+1 = q and from (b) that τ 0 > n = idle . That is, 〈σ ( c0 > n ), τ 0 > n , σ ( c0 > n+1 ) = 〈 q0 > n , τ 0 > n , q0 > n+1 〉 = 〈 q, τ, q′ 〉 ∈ δL or τ 0 > n = idle and σ ( c0 > n ) = q0 > n = q0 > n+1 = σ ( c0 > n+1 ). It is straightforward to show from An \| ...	{ "page_id": null, "source": 7332, "title": "from dpo" }
Ii > 0 = ∅ (1 ≤ i ≤ k). It is a consequence of the assumption that Spec (M) is true in M that An \| persf (ai ) and An \| stepf (ai ), for all n ∈ N. To show that Iin τ in Iin+1 , for all n ∈ N, we consider two possibilities: if τ in /∈ T F > in ,which, due to (d), implies that A \| ¬ tf μ (ai ) for all μ ∈ T F > in...	{ "page_id": null, "source": 7332, "title": "from dpo" }
q in = q and q in+1 = q′, and from (d) that τ in = τ ; and, for the latter case, it follows from (c) that q in = q in+1 = q and from (d) that τ in = idle . It is straightforward to show from An \| enbf τ (ai ) (a consequence of the # 123 458 Formal Methods in System Design (2021) 58:440–468 assumption that An \| sf q (...	{ "page_id": null, "source": 7332, "title": "from dpo" }
\| tl τ → ∀ x ♦μ( x), and, as a result, An \| ∀ x ♦μ( x). It is then straightforward to show from the semantics of MFOTL and (c) above that, for each i (1 ≤ i ≤ k), there exists n′ > n, such that μ ∈ Iin′ . Now, for each i (1 ≤ i ≤ k), supposing μ ∈ Iin (n ∈ N), we have from (c) above that An \| μ( ai ), thus, fro...	{ "page_id": null, "source": 7332, "title": "from dpo" }
ϕP is the formula (over M ) expressing P in MFOTL , is unsatisfiable over finite domains. 4.5 Execution in a synchronous setting If N = 〈 L, F〉 is a synchronous distributed machine, we can (automatically) construct an MFOTL -sentence Spec ′(N ) (with ‖Spec ′(N )‖ quadratic in ‖N ‖) such that N is runnable if and only i...	{ "page_id": null, "source": 7332, "title": "from dpo" }
∀ xμ( x)) ∧ ∀ x(( ¬μ( x) ∧μ( x)) → tl μ )), sendfs := ∀ x ∧ > μ∈TF > out ((tf μ (x) →μ( x)) ∧ (( ¬μ( x) ∧μ( x)) → tf μ (x)) ). As in the definition of bcastla and sendfa , the symbol μ ∈ T F > in (resp. μ ∈ T L > in ), and, thus, the predicate μ( ·) appearing in bcastls (resp. sendfs ) exists because T L >...	{ "page_id": null, "source": 7332, "title": "from dpo" }
runs of a runnable synchronous machine N have a property P (expressible in MFOTL ) then Spec ′(N ) ∧ ¬ ϕP ,where ϕP is the formula (over N ) expressing P in MFOTL , is unsatisfiable over finite domains. Relevant to Rem. 1, it should also be clear that for an arbitrary (not necessarily runnable) synchronous distributed ...	{ "page_id": null, "source": 7332, "title": "from dpo" }
and only if tl μ holds true # 123 460 Formal Methods in System Design (2021) 58:440–468 at time n (n ∈ N). One then enforces the condition that μ( x) holds true whenever t μ > m holds true and if ¬μ( x), μ( x) hold true at times n, n + 1 respectively then t μ > m also holds true at time n + 1 ( n ∈ N), for each f...	{ "page_id": null, "source": 7332, "title": "from dpo" }
• At the first round of computations, every process broadcasts its input bit. • At every round the (tentative) output bit is set to the minimum value ever seen so far. The correctness criterion for this protocol is that, eventually, the output bits of all processes will be the same. We now give a formal specification o...	{ "page_id": null, "source": 7332, "title": "from dpo" }
> in = { 0,1}, T F > out = { 0, 1}, and T F > local = ∅ . Let T F = T F > in ∪ T F > out ∪ T F > local ∪ { idle }. Let Q F = { i0, i1, o0, o1}, Q F > init = { i0, i1},and δF = {〈 i0, 0, o0〉, 〈i1, 1, o1〉, 〈o1,0, o0〉, 〈o1,1, o1〉, 〈o0,0, o0〉, 〈o0,1, o0〉} . # 123 Formal Methods in System Design (2021) 58:440–468 ...	{ "page_id": null, "source": 7332, "title": "from dpo" }
described in , with minor simplifications. In this setting, k followers (“resource managers”) coordinated by a leader (“transaction manager”) collectively perform a transaction. The transaction is aborted if one or more followers abort, otherwise it is committed. Initially, the leader and each follower are in a working...	{ "page_id": null, "source": 7332, "title": "from dpo" }
followers) it enters a committing state. When in an aborting state, the leader broadcasts an abort message to the followers and enters an aborted state; and while in a committing state, it broadcasts a commit message to the followers and enters a committed state. (Similar to , we could allow the leader (“transaction ma...	{ "page_id": null, "source": 7332, "title": "from dpo" }
in = { ←−−−−prepare , ←−−−− commit , ←−− abort }, T F > out = { −−−−−→ prepared , −−−−→ aborted }, and T F > local = ∅ . Let T F = T F > in ∪ T F > out ∪ T F > local ∪{idle }. Let Q F = { working , preparing , prepared , committed , aborted }, Q F > init = { working },and δF = {〈 working , ←−−−−prepare , preparing 〉, 〈...	{ "page_id": null, "source": 7332, "title": "from dpo" }
The original protocol involves a degree of randomness, which we are unable to model in our framework and first-order temporal logic in general. The simplifications made here address this limitation while staying faithful to the core ideas behind the protocol. # 123 464 Formal Methods in System Design (2021) 58:440–468 ...	{ "page_id": null, "source": 7332, "title": "from dpo" }
waits for a random number of rounds before it bids again. This random number depends on the contention in the system (i.e. on how many sensors have bid up to that point) and its specific properties are of no concern to us. What is important to us is that it is calculated in such a way that if two sensors bid for the tr...	{ "page_id": null, "source": 7332, "title": "from dpo" }
q0〉, 〈q3, −→go , q0〉, 〈q4, −−→nogo , q0〉} . Let L = 〈 Q L , Q L > init , T L , δ L 〉. L is depicted in Fig. 7. For the followers, let T F > in = { ←−go , ←−−nogo }, T F > out = { −→ bid }, and T F > local = { sending , wait , ε }. Let T F = T F > in ∪ T F > out ∪ T F > local ∪ { idle }. Let > Fig. 7 Control-MAC leader ...	{ "page_id": null, "source": 7332, "title": "from dpo" }
finite domains: Spec (N MAC ) ∧ ∀x(( tf −→ > bid (x) ∧ ∃ y(x = y ∧ tf −→ > bid (y))) → ♦(tf −→ > bid (x) ∧ ¬∃ y(x = y ∧ tf −→ > bid (y)))) ∧ ¬ ∀x(tf −→ > bid (x) → ♦tf sending (x)) ## 6 Related work presents an approach in which the states and transitions of a parameterized or infinite-state system are represent...	{ "page_id": null, "source": 7332, "title": "from dpo" }
LTL(MSO) can be straightforwardly trans-lated to monadic MFOTL ) and approximates to some extent the expressiveness of approaches based on numerical abstractions (the latter are more flexible in expressing properties involving complex numerical constraints, but MFOTL can express simple numerical constraints, pos-sibly ...	{ "page_id": null, "source": 7332, "title": "from dpo" }
provers for said logic can then be used to check whether that translation (and thus the original protocol) satisfies a property of interest. Thus, our framework constitutes a flexible, high-level approach to parameterized verification. To demonstrate the applicability of our framework, we have specified in it three pro...	{ "page_id": null, "source": 7332, "title": "from dpo" }
2(4):604–643 4. Benghabrit W, Grall H, Royer JC, Sellami M (2015) Abstract accountability language: translation, com-pliance and application. In: Asia-Pacific Software Engineering Conference, pp. 214–221. IEEE 5. Benghabrit W, Grall H, Royer JC, Sellami M (2015) Checking accountability with a prover. In: Computer Softw...	{ "page_id": null, "source": 7332, "title": "from dpo" }
A (2009) Temporal verification of fault-tolerant protocols. In: Methods, Models and Tools for Fault Tolerance, pp. 44–56. Springer 18. Fisher M, Lisitsa A (2003) Deductive verification of cache coherence protocols. In: Proceedings of the 3rd Workshop on Automated Verification of Critical Systems AVoCS, 3: 177–186 19. G...	{ "page_id": null, "source": 7332, "title": "from dpo" }
Sci 57(2–3):317–325 30. Wolter F, Zakharyaschev M (2001) Decidable fragments of first-order modal logics. J Symb Logic 66(3):1415–1438 31. Wolter F, Zakharyaschev M (2002) Axiomatizing the monodic fragment of first-order temporal logic. Annals Pure Appl logic 118(1–2):133–145 Publisher’s Note Springer Nature remains ne...	{ "page_id": null, "source": 7332, "title": "from dpo" }
Title: A Brief Introduction to Chemical Reaction Optimization URL Source: Markdown Content: This website utilizes technologies such as cookies to enable essential site functionality, as well as for analytics, personalization, and targeted advertising. To learn more, view the following link: Privacy Policy Manage Prefe...	{ "page_id": null, "source": 7334, "title": "from dpo" }
American Chemical Society Subjects what are subjects Algorithms Catalysts Chemical Reactions Kinetic Modeling Optimization 1. Introduction Chemical reaction optimization is a term that has a variety of meanings depending on the chemist defining it, with a large corresponding variance in expectations of optimization cap...	{ "page_id": null, "source": 7334, "title": "from dpo" }
of these methodologies borrow concepts from related fields, such as statistics, computer science, process chemistry, and engineering, this review will help to diversify the chemist’s toolkit and serve as a comprehensible reference for optimization campaigns. Although reaction optimization is often related to reaction y...	{ "page_id": null, "source": 7334, "title": "from dpo" }
another factor until each factor is optimized and the scientist believes that they have arrived at the optimum reaction conditions. (20) These factors can be any number of experimental conditions (such as temperature, stoichiometry, reaction time, etc.) which, when combined, constitute a multidimensional space with man...	{ "page_id": null, "source": 7334, "title": "from dpo" }
Figure 1 Figure 1. An example of an OFAT experimental procedure in varying temperature and reagent equivalents, where ○ represents a numbered experimental data point and the blue region indicates the true optimum area of parameter space. Response surface is contoured from red (low response) to blue (high response). Gen...	{ "page_id": null, "source": 7334, "title": "from dpo" }
perform as a rudimentary technique to achieve improved reaction yields. However, as research laboratories are beginning to diversify their equipment by incorporating advanced technologies such as automated retrosynthesis software and experimentation, (30,31) it is also important for chemists to evolve at the same pace ...	{ "page_id": null, "source": 7334, "title": "from dpo" }
response to small changes in the experimental factors; this is important on a process scale to understand how possible deficiencies in reactors may lead to suboptimal outputs. The practical manner of running DoE campaigns focuses on performing predefined experiments from a structured experimental design. These designs ...	{ "page_id": null, "source": 7334, "title": "from dpo" }
value for each factor, e.g., a reaction temperature of 50 °C in this example, and are conducted throughout the course of the 17 experiment campaign. The outputs from each experiment were then inputted into the DoE software, MODDE, to identify the optimum reaction conditions that afforded the highest yield of 7. After s...	{ "page_id": null, "source": 7334, "title": "from dpo" }
experimental factors and to estimate the main factor effects and interactions, hence giving an accurate statistical model for the process and thereby optimizing the product output. Scheme 3 Scheme 3. One Reaction of Interest, Optimizing the Yield and Selectivity of the Desired 3,4-Dihydroxymandelic Acid Intermediate, 1...	{ "page_id": null, "source": 7334, "title": "from dpo" }
by Minisci and co-workers to refit the model and plot the response surface using MODDE Pro. (40) There are many advantages to running optimization campaigns using DoE. The use of predefined, space-filling experimental designs removes the necessity for chemical-intuition-guided optimization, and it has been shown numero...	{ "page_id": null, "source": 7334, "title": "from dpo" }
However, software options have undoubtedly helped to facilitate the employment of DoE overall due to the high expertise barrier for typical bench scientists to use the statistical methods unaided. Another major disadvantage is the difficulty in exploring categorical variables in DoE studies, as these experimental desig...	{ "page_id": null, "source": 7334, "title": "from dpo" }
(63−68) and delivery, (69−71) analytical method development, (72−75) and more. (76−79) This is because there are numerous and undisputed benefits to the use of DoE for experimental parameter screening and optimization, especially when compared with traditional human intuition-guided experimentation. With the rise of us...	{ "page_id": null, "source": 7334, "title": "from dpo" }

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