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c9533d43656d96962487d3fd239720ac9b655194 | subsection | 14 | 16 | Demonstration | In this section, we present a program for calculating the Riemann curvature tensor , , , the fourth-order tensor that expresses the curvature of a manifold, to demonstrate our proposal.
Figure REF is the program for calculating the Riemann curvature tensor of S^2 using the formula of curvature form.In our system, when ... | {
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e5a18799b84c3d205518b90d3b63d9572be74356 | subsection | 15 | 16 | Discussion | We introduced several forms of syntax into a language to introduce the new concepts of scalar and tensor functions.
However, we think it is possible to introduce the concepts of scalar and tensor functions even using a static type system or the overloading feature of object-oriented programming.
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f0f997ee91a0f04c14cae2b655a849936ce7b731 | abstract | 0 | 122 | Abstract | Boolean nets are Petri nets that permit at most one token per place. Research
has approached this important subject in many ways which resulted in various
different classes of boolean nets. But yet, they are only distinguished by the
allowed interactions between places and transitions, that is, the possible
effects of ... | {
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e6bc2016cfee925b5a7ca02420682638d29f12be | subsection | 1 | 122 | Introduction | This paper contributes to the analysis of the computational complexity of boolean Petri net synthesis as a function of the specific net class.
While the efficient algorithms developed in this paper attack the synthesis problem itself, the proofs for intractable synthesis cases turn to feasibility, the corresponding dec... | {
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585b4f9ddb7e929beecc98d8eb5641766fc531b8 | subsection | 2 | 122 | Preliminary Notions | This section provides short formal definitions of all preliminary notions used in the paper.
For a detailed introduction into the field of Petri net synthesis, we propose the excellent monograph of Badouel, Bernardinello and Darondeau .
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1d3c61291b6d87f83d13063523b2098abdb04bad | subsection | 3 | 122 | Preliminary Notions | For every reachable marking M and transition t \in T with M [baseline=-1pt]{
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ace7d593cd92ea9b4f2aef1cd4e6332fd8adcfdc | subsection | 4 | 122 | Preliminary Notions | Aside from that, a TS A can be simple, which prohibits s [baseline=-1pt]{
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efa1b5c8db220a0680159238d5d37984ef71334f | subsection | 5 | 122 | Preliminary Notions | We also use sig^{-1}(\tau ^{\prime }) = \lbrace e \in E \mid sig(e) \in \tau ^{\prime }\rbrace for \tau ^{\prime } \subseteq \tau .For a TS A=(S, E, \delta , s_0) and a type of nets \tau , a pair of states s \ne s^{\prime } \in S is separable for \tau if there is a \tau -region (sup, sig) such that sup(s) \ne \sup (s^{... | {
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1d30c47a39e469fb01ef9cb9494b418f780063e2 | subsection | 6 | 122 | Preliminary Notions | In fact, having a region set \mathcal {R} for A that solves all its SSP and ESSP atoms with respect to some type of nets \tau , one can construct a \tau -net N(A, \mathcal {R}) = (\mathcal {R}, E(A), f, M_0) on place set \mathcal {R}, transition set E(A), flow function f(R, e) = sig(e) for all R = (sup, sig) \in \mathc... | {
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883a66a6df31da591108da5f88c9d830cf4494e0 | subsection | 7 | 122 | A Reduction Scheme yields the NP-completeness of Feasibility for 77 Boolean Petri Net Classes | This section presents our main result:
Let \tau _1 = \lbrace \textsf {nop}, \textsf {inp}, \textsf {out}\rbrace , \tau _2 = \lbrace \textsf {nop}, \textsf {inp}, \textsf {res}, \textsf {swap}\rbrace , \tilde{\tau }_2 = \lbrace \textsf {nop}, \textsf {out}, \textsf {set}, \textsf {swap}\rbrace , \tau _3 = \lbrace \text... | {
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088987415be611da0436074ab34c211d7ec3949c | subsection | 8 | 122 | A Reduction Scheme yields the NP-completeness of Feasibility for 77 Boolean Petri Net Classes | Although this demands for 77 NP-completeness proofs, executing them individually does not teach us a lot about the problem structure.
On the one hand, it is straight forward that \tau -feasibility is a member of NP for all considered type of nets \tau and we do not have to explicitly prove this here.
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fba8341a3d850557efc07292b0c9639865a79a89 | subsection | 9 | 122 | Unions of Transition Systems | If A_0=(S_0,E_0,\delta _0,s_0^0), \dots ,A_n=(S_n,E_n,\delta _n,s_0^n) are TSs with pairwise disjoint states (but not necessarily disjoint events) we say that U(A_0, \dots , A_n) is their union.
By S(U), we denote the entirety of all states in A_0, \dots , A_n and E(U) is the aggregation of all events.
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ba3da514b7a509d634b4af2d6e12bb52e4e1b05d | subsection | 10 | 122 | Unions of Transition Systems | For our NP-completeness scheme, we require one basic construction A(U) and an enhanced construction A^+(U).
If s^0_0, \dots , s^n_0 are the initial states of U's TSs then A(U) = (S(U) \cup \bot , E(U) \cup \odot \cup \ominus , \delta , \bot _0) and A^+(U) = (S(U) \cup \bot , E(U) \cup \odot \cup \ominus , \delta ^+, \b... | {
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b9af5f8160a1821fd7ddb2462c1cd7f803a1f1d9 | subsection | 11 | 122 | Unions of Transition Systems | Moreover, define the joiningA_\tau (U) = A(U) if \textsf {inp}\in \tau and \lbrace \textsf {out}, \textsf {set}, \textsf {swap}\rbrace \cap \tau \ne \emptyset and, otherwise,
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4bab23605199373a3ce5ac0e3081cdd028b634c8 | subsection | 12 | 122 | Unions of Transition Systems | A \tau -region R of U separating s and s^{\prime }, respectively inhibiting e at s, can be completed to become an equivalent \tau -region R^{\prime } of A(U) by settingsup_{R^{\prime }}(s^{\prime \prime }) &= {\left\lbrace \begin{array}{ll}
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9e58858c802e5c333de2160cac192cffbf0bd0d3 | subsection | 13 | 122 | Unions of Transition Systems | To separate the state \bot _i from all the other states of (S(U) \cup \bot )\setminus \lbrace \bot _i\rbrace we simply define the \tau -region R_i where only sup_{R_i}(\bot _i) = 1 and where the signature of all events is \textsf {nop} except for \odot _i, \ominus _i, \ominus _{i+1}.
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d8aa10ee6240b8ef0fe562b50d36b7f66ec96ebc | subsection | 14 | 122 | The General Reduction Scheme | Our general scheme can be set up to a specific reduction by the turn switch \sigma .
In each of its six positions, \sigma covers a whole collection of net classes.
Therefore, we simply understand the positions \sigma _1, \dots , \sigma _6 as the type sets managed by the respective reductions:\begin{array}{rcl@{\quad }r... | {
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bf8437eb0d3e5974a3c163bb2a086028667ae450 | subsection | 15 | 122 | The General Reduction Scheme | There is a key state s_{key} \in S(U^\sigma _\varphi ) and a key event k \in E(U^\sigma _\varphi ) with \lnot s_{key} [baseline=-1pt]{
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0.0377... | |
0880df72195fbe8b72c43323ebf266fc95082ca3 | subsection | 16 | 122 | The General Reduction Scheme | The support sup includes exactly the red emphasized states and the signature sig is assumed to be defined in accordance to Lemmas REF and REF from Sections REF and REF , respectively.
The example can be used to comprehend all the steps of the reduction scheme laid out in this section.
[Figure: The result TS A(U^{\sigma... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
-0.010515500791370869,
0.01066812127828598,
-0.0441071093082428,
-0.0029512846376746893,
0.004029161762446165,
-0.007028139661997557,
0.016330314800143242,
0.047495268285274506,
0.042733531445264816,
0.05646931007504463,
0.012125639244914055,
0.03797179460525513,
0.008020168170332909,
0.03... | |
9d69811736ae541613c9a9183eb6d8cb8c583d81 | subsection | 17 | 122 | The General Reduction Scheme | Then, every clause \zeta _i = \lbrace X_{i,0}, X_{i,1}, X_{i,2}\rbrace \subseteq V(\varphi ) is implemented as a translator T^\sigma _i=U(T^\sigma _{i,0},T^\sigma _{i,1},T^\sigma _{i,2}), a subunion of T^\sigma _\varphi .
The TS T^\sigma _{i,\alpha } that builds T^\sigma _i with its three copies for \alpha \in \lbrace ... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
0.01133755873888731,
-0.011200225912034512,
-0.03750702366232872,
0.08374229073524475,
0.019073953852057457,
0.007221399340778589,
0.005874778144061565,
0.02163749374449253,
0.024033183231949806,
0.019073953852057457,
-0.015503310598433018,
-0.004268751014024019,
-0.022919263690710068,
0.0... | |
c3c13b6ef917f3cfcd85afce0b03de664a196297 | subsection | 18 | 122 | The General Reduction Scheme | To effect this behavior, the TS T^\sigma _{i,\alpha } provides two pathsP_{i,\alpha } &= s^0_{i,\alpha } \dots s^1_{i,\alpha } [baseline=-1pt]{
[->,line width=0.3pt] (0,0) -- ++(0.6,0) node[anchor=base, yshift=2pt, pos=0.5] {\scalebox {0.75}{X_{i,\alpha }}};
} s^2_{i,\alpha } \dots s^3_{i,\alpha } [baseline=-1pt]{
[->,... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
-0.01916157454252243,
-0.011022482067346573,
-0.06737063080072403,
0.04311354085803032,
0.013226978480815887,
-0.002257892396301031,
-0.00429075863212347,
0.0495210736989975,
0.0347837470471859,
0.01836826093494892,
-0.009329062886536121,
0.002854784717783332,
-0.025675898417830467,
0.0054... | |
9114948f289b02a59fc3a521d72d738e6fdb90cb | subsection | 19 | 122 | The General Reduction Scheme | Notice, while cross checking with Figure REF , that states can be the same if they are linked by dots, like s^0_{i, \alpha } and s^1_{i, \alpha } both represent t_{i, \alpha , 2}, or have the same superscript, like s^0_{i, \alpha } and \dot{s}^0_{i, \alpha } both represent t_{i, \alpha , 2}, too.For an indicator region... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
0.006672436837106943,
0.011620493605732918,
-0.04126305133104324,
0.03494540974497795,
0.006790701765567064,
0.041232530027627945,
-0.030596308410167694,
0.06058221682906151,
0.00960617233067751,
0.034243449568748474,
-0.00894236285239458,
-0.007179832085967064,
0.021150365471839905,
0.007... | |
9cb0dc92f6f510df31cef7b590dc2e8e8044b60e | subsection | 20 | 122 | The General Reduction Scheme | Traversing the mapped paths in \tau , these interactions do not step from 0 to 1 or vice versa.
Since the mapped paths start and terminate at different states \tau , that is, sup(s^0_{i,\alpha }) = sup(\dot{s}^0_{i,\alpha }) \ne sup(s^7_{i,\alpha }) = sup(\dot{s}^7_{i,\alpha }), the remaining interactions sig(X_{i,0}),... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
-0.0017505839932709932,
-0.019161075353622437,
-0.040091805160045624,
0.02913825958967209,
0.00023324434005189687,
0.02915351651608944,
0.030968936160206795,
0.06602638214826584,
0.01059503760188818,
0.025919321924448013,
0.009504259563982487,
-0.043600600212812424,
0.013775837607681751,
-... | |
6490ba747dae16b1b44cfc038880aeae820140d5 | subsection | 21 | 122 | The General Reduction Scheme | Subsection REF does the same for K^\sigma _m.Before we start with our construction, we need some minor tools:
Firstly, we use so called generators G^{\eta ,\varrho }_j in K^\sigma _m and in T^\sigma _\varphi .
A template G^{\eta ,\varrho }_j serves as a blueprint for freezer gadget TSs as follows:
For j \in \mathbb {N}... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
-0.006466779392212629,
0.02400529570877552,
-0.026096031069755554,
0.007248897571116686,
0.017931483685970306,
-0.023516949266195297,
0.00929384957998991,
0.029163459315896034,
-0.004456164315342903,
0.05631248652935028,
0.0031380094587802887,
0.025683987885713577,
-0.016115443781018257,
0... | |
6cc4ca218d563342903ed4d154c843de969828ed | subsection | 22 | 122 | The General Reduction Scheme | Moreover, we use blanc events in s [baseline=-1pt]{
[<->,line width=0.3pt] (0,0) -- ++(0.6,0) node[anchor=base, yshift=2pt, pos=0.5] {\scalebox {0.75}{\_}};
}s^{\prime } to say that there is an anonymous event u that occurs exactly twice, namely at the transitions s [baseline=-1pt]{
[->,line width=0.3pt] (0,0) -- ++(0.... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
-0.021226782351732254,
0.001393436803482473,
-0.041721079498529434,
0.03235138580203056,
0.006298601161688566,
-0.02902468666434288,
0.013566218316555023,
0.029970813542604446,
0.030077632516622543,
0.052494700998067856,
0.007076865993440151,
0.02101314067840576,
0.031237399205565453,
0.00... | |
8027c583553dd1623b550f84457519efad043b95 | subsection | 23 | 122 | Details of the Translator Union | This section defines the translator union T^\sigma _\varphi for all cubic monotone boolean 3-CNF \varphi with m clauses and every \sigma \in \lbrace \sigma _1,\dots , \sigma _6\rbrace .
The union T^\sigma _\varphi =U(T^\sigma _0, \dots , T^\sigma _{m-1}, F^\sigma _T) consists of translator subunion T^\sigma _i for i \i... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
0.02449570968747139,
-0.0030066384933888912,
-0.051891226321458817,
0.01782616227865219,
0.004406938795000315,
0.013110165484249592,
-0.025747204199433327,
0.029623784124851227,
0.029852716252207756,
0.019474470987915993,
-0.025747204199433327,
0.01771932654082775,
-0.0070091309025883675,
... | |
c679c2485553175b59e4dc8567cdbfc1d54b3b79 | subsection | 24 | 122 | Details of the Translator Union | The red region fraction stands for X_{i,\alpha } \in M, the green for X_{i,\beta } \in M and blue for X_{i,\gamma } \in M.
In all three settings, the states taking part in the indicator support are exactly those within the colored area plus t_{i, \alpha , 0} for \sigma _1, \dots , \sigma _4 or plus t^{\prime }_{i, \alp... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
-0.03290665149688721,
-0.0024897484108805656,
-0.05668613314628601,
0.028785688802599907,
0.022054782137274742,
-0.016087017953395844,
-0.005960133392363787,
0.07088056206703186,
0.02290950156748295,
0.022558456286787987,
-0.04664319381117821,
-0.011096074245870113,
0.024588411673903465,
0... | |
ef6604d694049b0a71982206e223605abdbaeb22 | subsection | 25 | 122 | Details of the Translator Union | Aside from k, the interface consist of V=\lbrace v_0, \dots , v_{3m-1}\rbrace , W=\lbrace w_0,\dots , w_{3m-1}\rbrace and, for \sigma _5 and \sigma _6, Acc=\lbrace a_0, \dots , a_{18m-1}\rbrace .
If \varphi is a cubic monotone boolean 3-CNF with m clauses, \sigma \in \lbrace \sigma _1, \dots , \sigma _6\rbrace a turn ... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
-0.0017302827909588814,
0.009393508546054363,
-0.022739769890904427,
-0.020786285400390625,
0.011453823186457157,
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0.04056530445814133,
0.02519688569009304,
-0.024098049849271774,
-0.0020011758897453547,
0.005829927511513233,
... | |
c355473a0fa19aa185765ca2494b0c56d931a68c | subsection | 26 | 122 | Details of the Translator Union | It is important that, on the interface, (sup, sig) is compatible with a key region (sup_K,sig_K) such that both of them can be combined to a region of U^\sigma _\varphi that inhibits k at the key state.For given \varphi with m-clauses and one-in-three model M, our approach is as follows:
We first define for every claus... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
-0.0029453663155436516,
-0.006115831900388002,
-0.028431179001927376,
0.009187101386487484,
0.0036798003129661083,
-0.0070391204208135605,
0.02070913091301918,
0.03241429105401039,
0.025546856224536896,
0.04398210346698761,
0.0020850293803960085,
-0.00025657497462816536,
0.02223522774875164,... | |
9e99e3b156bbd11270d32601755acf711b97ccc4 | subsection | 27 | 122 | Details of the Translator Union | Consider the following state sets for our objective:S^0_{\sigma _1,i}=S^0_{\sigma _2, i}=S^0_{\sigma _3, i}=S^0_{\sigma _4, i}=\lbrace t_{i,0,0},t_{i,1,0},t_{i,2,0}\rbrace ,
S^0_{\sigma _5, i}=\emptyset and S^0_{\sigma _6, i}=\lbrace t^{\prime }_{i,j,0}, t^{\prime }_{i,j,1} \mid 0\le j\le 2 \rbrace ,
S^1_{\sigma _1,i... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
-0.028340116143226624,
0.016665270552039146,
-0.05637500435113907,
-0.011316206306219101,
0.007241456303745508,
-0.009614578448235989,
-0.003672234946861863,
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0.0249368604272604,
-0.01690945029258728,
0.0005308048566803336,
-0.03461248427629471,
-... | |
ae4d42e38d7660d79c8433b1c21f83b3c8f4893f | subsection | 28 | 122 | Details of the Translator Union | For a transition s [baseline=-1pt]{
[->,line width=0.3pt] (0,0) -- ++(0.6,0) node[anchor=base, yshift=2pt, pos=0.5] {\scalebox {0.75}{e}};
} s^{\prime } of T^\sigma _\varphi ,we can then set sig(e) = \textsf {nop} if and only if e is not in \lbrace k, q_2, q_3\rbrace \cup \lbrace X_{i,\alpha _i}, x_{i,\alpha _i}, v_{3i... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
-0.014683057554066181,
-0.012135451659560204,
-0.018687527626752853,
0.034446071833372116,
0.017482372000813484,
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0.020518142729997635,
0.025064170360565186,
0.0033351515885442495,
-0.022714881226420403,
... | |
c77c42e7b4a1caeb9fe3be94ca47f114df5b60c6 | subsection | 29 | 122 | Details of the Translator Union | The following Lemma REF realizes and justifies this idea:
[Without proof]
For every cubic monotone boolean 3-CNF \varphi with one-in-three model M, every \sigma \in \lbrace \sigma _1, \dots , \sigma _6\rbrace , every \tau \in \sigma and V = \lbrace v_0, \dots , v_{3m-1}\rbrace , we get an indicator \tau -region (sup^\s... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
-0.015397057868540287,
-0.005188304930925369,
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-0.01074284315109253,
0.004535952117294073,
... | |
888f72d560f954cc7f5fce137f5b7f4dd95ff1c3 | subsection | 30 | 122 | Details of the Key Union | This subsection defines the key union K^\sigma _m for all numbers m of clauses and every \sigma \in \lbrace \sigma _1,\dots , \sigma _6\rbrace .
In particular, K^\sigma _m = U(H^\sigma , D^\sigma , G^\sigma , F^\sigma _K) consists of the head H^\sigma , the duplicator D^\sigma , the generator G^\sigma , and the freezer... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
0.038491856306791306,
0.010561603121459484,
-0.06025608256459236,
-0.010523446835577488,
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0.03583618998527527,
-0.005162518005818129,
-0.0035065438132733107,
-0.00455965148285031... | |
a3420acca6dfb6958d88e6f11337feb1284aa2a7 | subsection | 31 | 122 | Details of the Key Union | But since there are differences in the constructed regions as defined in Figure REF , we keep the two switch positions distinguished to make our argumentation simpler.For \sigma \in \lbrace \sigma _5, \sigma _6\rbrace we create different key union ingredients as follows:Here H^\sigma = U(H^{\prime }_0, \dots , H^{\prim... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
0.01778777688741684,
0.012989958748221397,
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0.0038653325755149126,
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0.0531497523188591,
0.013363715261220932,
0.00031583409872837365,
0.006334797944873571,
0.0... | |
d73f32eeed69cef91cb4b709d3adc3acb7295368 | subsection | 32 | 122 | Details of the Key Union | If (sup_K,sig_K) is a \tau -key region of K^\sigma _m, that is, where k is inhibited at the key state, theneither sig_K(k)=\textsf {inp}, V \subseteq sig^{-1}_K(\textsf {enter}) and W \subseteq sig^{-1}_K(\textsf {keep}^-) or sig_K(k)=\textsf {out}, V \subseteq sig^{-1}_K(\textsf {exit}) and W \subseteq sig^{-1}_K(\tex... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
-0.009701738134026527,
0.03264421224594116,
-0.011646661907434464,
-0.009694110602140427,
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0.025566214695572853,
0.009213600307703018,
-0.013057684525847435,
0.004488578997552395,
... | |
135b01304d1f2ae7fbd5c8f47e7f63babba27da8 | subsection | 33 | 122 | Details of the Key Union | If: If \varphi has a one-in-three model, then using the corresponding region (sup^\sigma ,sig^\sigma ) of T^\sigma _\varphi defined in Lemma REF and the key region (sup^\sigma _K, sig^\sigma _K) of K^\sigma _\varphi introduced in Lemma REF yields a combined region R of U^\sigma _\varphi that inhibits the key event k at... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
0.0010088906856253743,
0.026242602616548538,
-0.002698639640584588,
-0.0011509745381772518,
-0.014738577418029308,
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-0.0012291683815419674,
0.045375291258096695,
0.008383900858461857,
0.050898440182209015,
0.01563875935971737... | |
e02cf285bab799db94a4ab3f8e530f9cad67f847 | subsection | 34 | 122 | NP-completeness of Feasibility for seven more Petri Net Classes | This section presents and proves the following theorem:
Deciding \tau -feasibility as well as \tau -language viability is NP-complete formodest TSs and \tau = \lbrace \textsf {nop}, \textsf {inp}, \textsf {free}\rbrace or \tau = \lbrace \textsf {nop}, \textsf {inp}, \textsf {used}, \textsf {free}\rbrace ,
modest TSs... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/s00454-001-0047-6",
"end": 1938,
"openalex_id": "https://openalex.org/W2007385106",
"raw": "Cristopher Moore, J. M. Robson, Hard Tiling Problems with Simple Tiles, Discrete & Computational Geometry 26(4): 573-590 (2001)",
"sou... | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
0.024212215095758438,
0.02085576392710209,
0.005942443385720253,
0.0029750356916338205,
0.00868100207298994,
-0.015912627801299095,
0.04784467816352844,
0.04830237478017807,
0.027721231803297997,
0.06682387739419937,
-0.02338835783302784,
-0.0018603509524837136,
0.013746190816164017,
0.018... | |
e94bb33fdd82380c03af1a3dc519677b8be12ebb | subsection | 35 | 122 | NP-completeness of Feasibility for seven more Petri Net Classes | Like before, the variables V(\varphi ) are used as events in A^{\tau }_\varphi and their key signature sig tells us how to find M and vice versa.This idea is put into practice by creating six directed labeled paths per clause C_i=\lbrace X_{i,0},X_{i,1},X_{i,2}\rbrace that commonly start at state t_{i,0}, terminate at ... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
-0.005413990002125502,
-0.010064908303320408,
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0.0449296310544014,
0.013391898944973946,
-0.005753556732088327,
-0.002563919173553586,
0... | |
67193260bde003420b459dcfbc9534a7777a123b | subsection | 36 | 122 | NP-completeness of Feasibility for seven more Petri Net Classes | See Figure REF for a visualization of the following concepts.
Firstly, we call A_\varphi the basic TS with states S = \lbrace s_0, s_1, q\rbrace \cup \lbrace t_{i,0},\dots , t_{i,8} \mid 0\le i \le m-1\rbrace and events E = \lbrace k, h\rbrace \cup \lbrace h_i, r_i \mid 0 \le i \le m-1\rbrace \cup V(\varphi ).
To omit ... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
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765275d7fc78b7c96aee6bdd3e7901932fb4c3cf | subsection | 37 | 122 | NP-completeness of Feasibility for seven more Petri Net Classes | Aside from this, the brown arcs present the remaining transition function \delta (A^{+}_\varphi )(s,e) for all s \in \lbrace s_0, t_{i,0}, \dots , t_{i,8}, m_0, \dots , m_4, p_{i,0}, \dots , p_{i,3}\rbrace and all e \in \lbrace k, h, h_i, a, c, u, v, a_i, b_i, x_{i,0}, x_{i,1}, x_{i,2}\rbrace .While still being depicta... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
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92d4204e73fe9c0fce6a8fae75f21ee54c237a53 | subsection | 38 | 122 | NP-completeness of Feasibility for seven more Petri Net Classes | The next lemma shows the equivalence between the one-in-three satisfiability of \varphi and the inhibitability of k at q:
If \tau is \lbrace \textsf {nop}, \textsf {inp}, \textsf {free}\rbrace or \lbrace \textsf {nop}, \textsf {inp}, \textsf {used}, \textsf {free}\rbrace or \lbrace \textsf {nop}, \textsf {set}, \text... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
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ca9afab0b0778133caf41a344ead1e7ecf211a61 | subsection | 39 | 122 | NP-completeness of Feasibility for seven more Petri Net Classes | In A^\tau _\varphi , there is a path t_{i,0} [baseline=-1pt]{
[->,line width=0.3pt] (0,0) -- ++(0.6,0) node[anchor=base, yshift=2pt, pos=0.5] {\scalebox {0.75}{X_{i,\alpha }}};
} t_1 [baseline=-1pt]{
[->,line width=0.3pt] (0,0) -- ++(0.6,0) node[anchor=base, yshift=2pt, pos=0.5] {\scalebox {0.75}{X_{i,\beta }}};
} t_2 ... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
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add244477669f1dcecf4c5d24403557d0c54522e | subsection | 40 | 122 | NP-completeness of Feasibility for seven more Petri Net Classes | For every j\in \lbrace 0,1,2\rbrace , we have sig(X_{i,j}) \in \lbrace \textsf {nop}, \textsf {res}, \textsf {free}\rbrace by [baseline=-1pt]{
[->,line width=0.3pt] (0,0) -- ++(0.6,0) node[anchor=base, yshift=2pt, pos=0.5] {\scalebox {0.75}{X_{i,j}}};
} t_{i,5}.
Accordingly, [baseline=-1pt]{
[->,line width=0.3pt] (0,0)... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
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fd4e027e94b970447d8ed00efa0a99f4ae844d3d | subsection | 41 | 122 | NP-completeness of Feasibility for seven more Petri Net Classes | We define a \tau -region (sup, sig) of A^{\tau }_\varphi that inhibts k at q by sup = \lbrace s_0\rbrace \cup \lbrace s \mid s [baseline=-1pt]{
[->,line width=0.3pt] (0,0) -- ++(0.6,0) node[anchor=base, yshift=2pt, pos=0.5] {\scalebox {0.75}{X}};
}: X \in M\rbrace and sig(k) = \textsf {inp}, sig(X) = \textsf {inp} for ... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
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67eee603e5b49c39131a2a41f9d9a7006df5b2d9 | subsection | 42 | 122 | NP-completeness of Feasibility for seven more Petri Net Classes | Using the same lemma, we can finish our proof for Theorem by the following lemma:
If \tau is \lbrace \textsf {nop}, \textsf {inp}, \textsf {free}\rbrace or \lbrace \textsf {nop}, \textsf {inp}, \textsf {used}, \textsf {free}\rbrace or \lbrace \textsf {nop}, \textsf {set}, \textsf {res}\rbrace \cup \omega with non-emp... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
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59e3adb592ad483ebf8ae5b5e88f8d3ca540fd5c | subsection | 43 | 122 | NP-completeness of Feasibility for seven more Petri Net Classes | For X \in V(\varphi ) let sup_X = \lbrace s \mid s [baseline=-1pt]{
[->,line width=0.3pt] (0,0) -- ++(0.6,0) node[anchor=base, yshift=2pt, pos=0.5] {\scalebox {0.75}{k}};
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[->,line width=0.3pt] (0,0) -- ++(0.6,0) node[anchor=base, yshift=2pt, pos=0.5] {\scalebox {0.75}{X}};... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
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e12c82bbbc3238f25ebc1e96954fffef9823ceeb | subsection | 44 | 122 | NP-completeness of Feasibility for seven more Petri Net Classes | For the \tau -ESSP of A^{\times }_\varphi , it is sufficient to prove the inhibition of e \in E(A^{+}_\varphi ) at states s\in S(A^{+}_\varphi ) where \lnot s [baseline=-1pt]{
[->,line width=0.3pt] (0,0) -- ++(0.6,0) node[anchor=base, yshift=2pt, pos=0.5] {\scalebox {0.75}{e}};
} and \lnot [baseline=-1pt]{
[->,line wid... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
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1391aa5051026b6364ac92188ee61546a9caa3eb | subsection | 45 | 122 | NP-completeness of Feasibility for seven more Petri Net Classes | For X \in V(\varphi ) let sup_X = \lbrace s, s^{\prime } \mid s [baseline=-1pt]{
[->,line width=0.3pt] (0,0) -- ++(0.6,0) node[anchor=base, yshift=2pt, pos=0.5] {\scalebox {0.75}{k}};
}s^{\prime }\rbrace \cup \lbrace s\mid s [baseline=-1pt]{
[->,line width=0.3pt] (0,0) -- ++(0.6,0) node[anchor=base, yshift=2pt, pos=0.5... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
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b48b4b1302b399ee664e099d636d7661b5d83675 | subsection | 46 | 122 | NP-completeness of Feasibility for seven more Petri Net Classes | Hence, for X\in V(\varphi ) the regions sup^1_{X,x} = \lbrace t_{n,0},\dots , t_{n,7}\mid 0\le n < m, X \in C_n\rbrace \cup \lbrace p_{n,0}, \dots , p_{n,3} \mid [baseline=-1pt]{
[->,line width=0.3pt] (0,0) -- ++(0.6,0) node[anchor=base, yshift=2pt, pos=0.5] {\scalebox {0.75}{X}};
}s [baseline=-1pt]{
[<-,line width=0.3... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
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... | |
61d6f270c59f07c1e4fac7949bab0be1b40da593 | subsection | 47 | 122 | NP-completeness of Feasibility for seven more Petri Net Classes | The regions sup^1_v=\lbrace m_1,m_2\rbrace \cup \lbrace s \mid [baseline=-1pt]{
[->,line width=0.3pt] (0,0) -- ++(0.6,0) node[anchor=base, yshift=2pt, pos=0.5] {\scalebox {0.75}{k}};
}s \rbrace \cup \lbrace p_{i,0},\dots , p_{n,3}\mid 0 \le n< m\rbrace and sup^2_v=\lbrace m_0,\dots , m_4\rbrace \cup \lbrace p_{n,0},\do... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
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f8db02fc2f32de2dc53f416513e3834fdd3ab377 | subsection | 48 | 122 | Polynomial Time Net Synthesis for 36 Types of Nets | This section proves the tractability of synthesis for the 36 types of nets given in the following theorem:
There is a polynomial time algorithm, that, on input TS A, synthesizes a \tau -net N with state graph isomorphic to A or rejects A if N does not exist, for every\tau = \lbrace \textsf {nop}, \textsf {set}\rbrace ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/3-540-60922-9_42",
"end": 1452,
"openalex_id": "https://openalex.org/W1500168942",
"raw": "Vincent Schmitt, Flip-Flop Nets, STACS 1996: 517-528",
"source_ref_id": "5066c44173d4449d30a4b200fb9cb47069bd2844",
"start": 1347... | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
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d92e1200fddea7b26d6c1b028575ff210c14ff2d | subsection | 49 | 122 | Polynomial Time Net Synthesis for 36 Types of Nets | However, sig(e) \in \tau implies that sup(s_0) = sup(s_1).
Hence, s_0 and s_1 are not \tau -separable.
As all states have to be reachable, this implies that input TSs with more than one state cannot be \tau -feasible and, thus, are discarded after a constant time check.Being reduced, A = (\lbrace s\rbrace , E, \lbrace ... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
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95e501dc0f25dd1993b3ee292ee2d10ff8a03191 | subsection | 50 | 122 | Net Synthesis by Incremental Region Growing | The result of this section is the following contribution to Theorem :
If \tau =\lbrace \textsf {nop}, \textsf {res}\rbrace \cup \omega is a type of nets with \omega \subseteq \lbrace \textsf {inp}, \textsf {used}, \textsf {free}\rbrace or \tau =\lbrace \textsf {nop}, \textsf {set}\rbrace \cup \omega ^{\prime } with \o... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
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c10218fccb9c3485c1eca5b0ab937925ecc44d3c | subsection | 51 | 122 | Net Synthesis by Incremental Region Growing | TS A and set of states Q \subseteq S(A)
A support sup \supseteq Q for a region of A.\exists \ s \in Q, s^{\prime } \notin Q, e \in E(A): (s^{\prime } [baseline=-1pt]{
[->,line width=0.3pt] (0,0) -- ++(0.6,0) node[anchor=base, yshift=2pt, pos=0.5] {\scalebox {0.75}{e}};
} s) \vee (s [baseline=-1pt]{
[->,line width=0.3pt... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
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6e8b70320ea1410985f975ff83c7aa98d247ea0c | subsection | 52 | 122 | Net Synthesis by Incremental Region Growing | If \tau =\lbrace \textsf {nop}, \textsf {res}\rbrace \cup \omega is a type of nets with \omega \subseteq \lbrace \textsf {inp}, \textsf {used}, \textsf {free}\rbrace and A is a TS and Q \subseteq S(A) then the result sup of Algorithm REF started on Q forms a \tau -region (sup, sig) of A withsig(e) =
{\left\lbrace \begi... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
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... | |
b73b526d26d63cdf33c26fdc5587c8b6a98de804 | subsection | 53 | 122 | Net Synthesis by Incremental Region Growing | Algorithm REF terminates after \mathcal {O}(\vert E(A)\vert \vert S(A)\vert ^5) time.
That the algorithm terminates is trivial as every iteration extends Q, which is possible for at most \vert S(A)\vert times.
After termination, sup obviously contains input Q.
Moreover, there are no events e \in E(A) participating in... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
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0.0... | |
ae202640fbc5efaf83659527d52f7a38517500ea | subsection | 54 | 122 | Net Synthesis by Incremental Region Growing | As s^{\prime } is added to Q_{i+1}, there are s \in Q_i \subseteq sup^{\prime } and e \in E(A) such that either s^{\prime } [baseline=-1pt]{
[->,line width=0.3pt] (0,0) -- ++(0.6,0) node[anchor=base, yshift=2pt, pos=0.5] {\scalebox {0.75}{e}};
} s or s [baseline=-1pt]{
[->,line width=0.3pt] (0,0) -- ++(0.6,0) node[anch... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
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0.008721952326595783,
-0.01053807232528925,... | |
604a12884eb5d395fd57b8053864df849671122f | subsection | 55 | 122 | Net Synthesis by Incremental Region Growing | If \tau =\lbrace \textsf {nop}, \textsf {res}\rbrace \cup \omega with \omega \subseteq \lbrace \textsf {inp}, \textsf {used}, \textsf {free}\rbrace and A is a TS then e \in E(A) is \tau -inhibitable at s \in S(A) where \lnot (s [baseline=-1pt]{
[->,line width=0.3pt] (0,0) -- ++(0.6,0) node[anchor=base, yshift=2pt, pos=... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
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24662f4b7f0ae04e571142e2f0bf4c5518b7d888 | subsection | 56 | 122 | Net Synthesis by Incremental Region Growing | Let X = \lbrace x \mid x [baseline=-1pt]{
[->,line width=0.3pt] (0,0) -- ++(0.6,0) node[anchor=base, yshift=2pt, pos=0.5] {\scalebox {0.75}{e}};
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[->,line width=0.3pt] (0,0) -- ++(0.6,0) node[anchor=base, yshift=2pt, pos=0.5] {\scalebox {0.75}{e}};
} y\rbrace , and Z = \lbr... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
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6a4e56fff6785196c04bf50f66d7c261aebc8475 | subsection | 57 | 122 | Net Synthesis by Incremental Region Growing | Depending on the availability of \textsf {inp}, \textsf {used}, \textsf {free} in \tau , we have to test the inhibitability of e at s by up to three calls of Algorithm REF with inputs Q_\textsf {inp}= \lbrace z \mid z [baseline=-1pt]{
[->,line width=0.3pt] (0,0) -- ++(0.6,0) node[anchor=base, yshift=2pt, pos=0.5] {\sca... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
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8bca6a690bde0aa856cd406b3fb9c2740c47529a | subsection | 58 | 122 | Net Synthesis by Incremental Region Growing | Computing N(A, \mathcal {R}) consumes \mathcal {O}(\vert \mathcal {R}\vert \vert E\vert ) = \mathcal {O}(\vert E\vert \vert S\vert \max \lbrace \vert E\vert , \vert S\vert \rbrace ) time, which is dominated by the previous costs.If \tau =\lbrace \textsf {nop}, \textsf {set}\rbrace \cup \omega ^{\prime } with \omega ^{\... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
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0264fb3148b1fb39614d20e132cfac760aac3a71 | subsection | 59 | 122 | Net Synthesis for Relatives of Flip-Flop-Nets | Last step in proving Theorem is to cover item REF , the relatives of flip-flop nets:
If \tau =\lbrace \textsf {nop}, \textsf {swap}\rbrace \cup \omega with \omega \subseteq \lbrace \textsf {inp}, \textsf {out}, \textsf {used}, \textsf {free}\rbrace then a given TS A can be synthesized into a \tau -net N with A(N) iso... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/3-540-60922-9_42",
"end": 549,
"openalex_id": "https://openalex.org/W1500168942",
"raw": "Vincent Schmitt, Flip-Flop Nets, STACS 1996: 517-528",
"source_ref_id": "5066c44173d4449d30a4b200fb9cb47069bd2844",
"start": 399
... | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
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20ed7d295ac982c475450270124a9fe21953062f | subsection | 60 | 122 | Net Synthesis for Relatives of Flip-Flop-Nets | In A^{\prime }, however, every node s \in S(A) is now reached by exactly one directed path \pi _s = s_0 [baseline=-1pt]{
[->,line width=0.3pt] (0,0) -- ++(0.6,0) node[anchor=base, yshift=2pt, pos=0.5] {\scalebox {0.75}{e_1}};
} \dots [baseline=-1pt]{
[->,line width=0.3pt] (0,0) -- ++(0.6,0) node[anchor=base, yshift=2pt... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
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4c826da6405d0f94301743bc979afb0cf8b49e20 | subsection | 61 | 122 | Net Synthesis for Relatives of Flip-Flop-Nets | Another possibility would be to select the complementary support sup(s) = (1 + \sum _{e \in E} \psi _s(e) \cdot \rho (e)) \mod {2}.Based on the specific type of nets \tau and the atom (A,x,y), we augment \Psi with additional equations to obtain M_{x,y}.
If, beside \textsf {nop} and \textsf {swap}, \tau does not contain... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
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493671201d01f7ff5a283998b6bc4ae48e283d6c | subsection | 62 | 122 | Net Synthesis for Relatives of Flip-Flop-Nets | As solving M_{x,y} is in polynomial time, solving (E)SSP atoms is tractable for flip-flop nets.
Having the polynomial size set \mathcal {R} of regions for all these atoms after polynomial time, we can synthesize the net N(A, \mathcal {R}) in polynomial time, too.
Keeping this approach in mind, we are ready to prove the... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
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055c968c1cd0496296d85c0a528213820bb10cf4 | subsection | 63 | 122 | Net Synthesis for Relatives of Flip-Flop-Nets | If \rho is a solution to M^\textsf {used}_{e,s}, we again define support and signature like in all previous cases except for e.
In fact, we let sig(e) = \textsf {used} if sup(z) = 1 for any (that also means all) z [baseline=-1pt]{
[->,line width=0.3pt] (0,0) -- ++(0.6,0) node[anchor=base, yshift=2pt, pos=0.5] {\scalebo... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
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0a6732c50a0af537609b3f4c55b78fb6d3f9a7fe | subsection | 64 | 122 | Conclusion | In this paper we investigate the complexity of boolean net synthesis for the 128 practically more relevant \textsf {nop}-afflicted classes.
In total, we prove 84 cases NP-hard and provide polynomial time algorithms for 36 classes.
As a side product, this paper introduces a very general reduction scheme that serves well... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
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975038084caa9156a493a63f97c9c854ba450a55 | subsection | 65 | 122 | Technical Proofs from Section | [Proof of Lemma REF ]
(REF -REF ):
Firstly, if T^\sigma _\varphi installs a non empty freezer then, by Lemma REF , we have sig(x_j)\ne \textsf {swap}.Let i\in \lbrace 0,\dots , m-1\rbrace .
For abbreviation we define S_0=\lbrace t_{i,\alpha ,2}\mid 0\le i \le m-1, 0\le \alpha \le 2\rbrace and S_1=\lbrace t_{i,\alpha ,5... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
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0.0... | |
a535819482590dd41410bca5b478625d8c2c0033 | subsection | 66 | 122 | Technical Proofs from Section | By a similar argument and sig(x_j)\ne \textsf {swap} for all j\in \lbrace 0,\dots , m-1\rbrace , we obtain that S_0\subseteq sup^{-1}(1), S_1\subseteq sup^{-1}(0) (S_0\subseteq sup^{-1}(0), S_1\subseteq sup^{-1}(1)) implies that at least one element of \lbrace x_{i,0},x_{i,1},x_{i,2}\rbrace has a signature from \lbrace... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
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4e8b0fca8a51fbb95cb5754724405f56995b7e5e | subsection | 67 | 122 | Technical Proofs from Section | Moreover, by sig(x_{i,\beta }),sig(x_{i,\gamma })\notin \lbrace \textsf {res},\textsf {inp}\rbrace (sig(x_{i,\beta }),sig(x_{i,\gamma })\notin \lbrace \textsf {set},\textsf {out}\rbrace ), we have sup(t_{i,\beta ,2})=sup(t_{i,\beta ,3})=1 (sup(t_{i,\beta ,2})=sup(t_{i,\beta ,3})=0).
The inclusion (exclusion) of t_{i,\b... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
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041c4d571c0cfae2deac96136a68efae59963254 | subsection | 68 | 122 | Technical Proofs from Section | Consequently, Acc\cap sig_K^{-1}(\textsf {swap})=\emptyset assures state synchronization for a-labeled transitions: sup(s)=sup(s^{\prime }).
For abbreviation we define S_0=\lbrace t^{\prime }_{i,\alpha ,2}\mid 0\le i \le m-1, 0\le \alpha \le 2\rbrace and S_1=\lbrace t^{\prime }_{i,\alpha ,11}\mid 0\le i \le m-1, 0\le \... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
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0.... | |
4c9ab4d21b3f6a792cde6e391fa525896d304c4c | subsection | 69 | 122 | Technical Proofs from Section | For Y\in \zeta _i\setminus \lbrace X\rbrace there are transitions s [baseline=-1pt]{
[<->,line width=0.3pt] (0,0) -- ++(0.6,0) node[anchor=base, yshift=2pt, pos=0.5] {\scalebox {0.75}{Y}};
}s^{\prime } and s^{\prime \prime } [baseline=-1pt]{
[<->,line width=0.3pt] (0,0) -- ++(0.6,0) node[anchor=base, yshift=2pt, pos=0.... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
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bdc6e4be6dbe5d4489788e4deb8aa0ccad07b7c6 | subsection | 70 | 122 | Technical Proofs from Section | Hence, similar to the case \sigma =\sigma _5, that results in M=\lbrace X\in V(\varphi )\mid sig(X)\ne \textsf {nop}\rbrace being a one-in-three model of \varphi .[Proof of Lemma REF ]
Let C=\lbrace c_0,\dots , c_{6m-2}\rbrace and Z=\lbrace z_0,\dots , z_{3m-1}\rbrace .(REF ): Firstly, we show that sig_K(k)\in \lbrace ... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
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da923a9cd6c220eb403ca28f29774de983b7925e | subsection | 71 | 122 | Technical Proofs from Section | Thus, S(F^\sigma _K)\subseteq sup_K^{-1}(1) (S(F^\sigma _K)\subseteq sup_K^{-1}(0)) which clearly implies sup_K(h_{0,4})=sup_K(h_{0,6})=1 (sup_K(h_{0,4})=sup_K(h_{0,6})=0).
Consequently, if sig_K(k)\in \lbrace \textsf {used},\textsf {free}\rbrace then k is not inhibited at h_{0,6}, a contradiction.
Hence, sig_K(k)\in \... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
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a5be05db5e6bbde10cfc84ed85cd08be4b14ae27 | subsection | 72 | 122 | Technical Proofs from Section | Symmetrically, if sig_K(k)=\textsf {out} we obtain that V\subseteq sig_K^{-1}(\textsf {exit}) and W\subseteq sig_K^{-1}(\textsf {keep}^+).To prove the existence of an announced key region of K^\sigma _\varphi for \sigma \in \lbrace \sigma _1,\dots , \sigma _4\rbrace we, firstly, define the following subsets and operati... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
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0.00531142670661211... | |
714130c7f2bb8b7833d6758e13a5126027243e74 | subsection | 73 | 122 | Technical Proofs from Section | Figure REF .
Hence, the first claim is proven.(REF ):
Let j\in \lbrace 0,\dots , 3m-1\rbrace .
By definition of \sigma clearly we have either sig_K(k)=\textsf {used}, sup_K(h^{\prime }_{j,2})=0 and sup_K(h^{\prime }_{j,1})=sup_K(h^{\prime }_{j,3})=1 or sig_K(k)=\textsf {free}, sup_K(h^{\prime }_{j,2})=1 and sup_K(h^{\p... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
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-0.02742810919880867,
... | |
2294225deb6b5d34339cf2253276683d93897e71 | subsection | 74 | 122 | Technical Proofs from Section | Hence, we have (W\cup Acc)\cap sig^{-1}_K(\textsf {swap})=\emptyset .To prove the existence of an announced key region of K^\sigma _\varphi for \sigma \in \lbrace \sigma _5, \sigma _6\rbrace we, firstly, define the following subsets and, secondly, show how they are to composed to a corresponding region:S^{\sigma _5}_0=... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
0.007164732553064823,
0.030291788280010223,
-0.015962321311235428,
-0.01411582063883543,
-0.017503617331385612,
-0.043095216155052185,
0.002971952548250556,
0.019075432792305946,
0.016298050060868263,
0.005989686120301485,
0.010888257063925266,
0.0015355723444372416,
-0.013909805566072464,
... | |
ccff07dcdb556a354034714a6d522bc8fdf5251d | subsection | 75 | 122 | Technical Proofs from Section | Figure REF .
Hence, the lemma is proven. | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
-0.03123416006565094,
0.02886909246444702,
-0.029097968712449074,
-0.014716828241944313,
-0.028289269655942917,
-0.02455093525350094,
0.0002732225984800607,
0.02368120104074478,
-0.005645647179335356,
0.00040744978468865156,
0.006835810374468565,
0.012763739563524723,
-0.027984099462628365,
... | |
7a08be77f3e34212c4fc2d06ef55956013806e25 | subsection | 76 | 122 | Concluding the ESSP and the SSP from a Key Region | In this section we show for \sigma \in \lbrace \sigma _1,\dots , \sigma _6\rbrace and \tau \in \sigma that the inhibition of the key event at the key state in U^\sigma _\varphi by a \tau -region implies the ESSP and the SSP for U^\sigma _\varphi with respect to \tau .
In our reduction, events of the same kind are numbe... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
0.024764470756053925,
-0.002374596893787384,
-0.00874309428036213,
-0.0167232658714056,
0.003854667069390416,
-0.006748051382601261,
-0.014136957935988903,
0.04946791008114815,
0.016982659697532654,
0.0445546880364418,
0.013053608126938343,
0.021895881742239,
-0.04498192295432091,
-0.00389... | |
7e1539a1ff53ee403ae451174dae33a1d1503d75 | subsection | 77 | 122 | Concluding the ESSP and the SSP for | In this section, we show for \sigma \in \lbrace \sigma _1,\dots , \sigma _4\rbrace that U^\sigma _\varphi has the (E)SSP if k is inhibitable at s_{key} in U^\sigma _\varphi .
Our approach for the ESSP is as follows:
Let s\in \bigcup _{i=1}^{4} S(U^{\sigma _i}_\varphi ) be a state and e\in \bigcup _{i=1}^{4} E(U^{\sigma... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
0.03580741211771965,
0.011737382039427757,
-0.008150535635650158,
-0.01385896373540163,
-0.004857505671679974,
-0.043438997119665146,
0.0022246078588068485,
0.04169899597764015,
0.012286856770515442,
0.06349480897188187,
0.011821329593658447,
0.043957944959402084,
0.004983427003026009,
-0.... | |
15b113fd87901cd2c74e8aad78882d5154037559 | subsection | 78 | 122 | Concluding the ESSP and the SSP for | More exactly, given a set S defined by a certain row of the table the implied support sup allows a \tau -signature sig such that for each \sigma \in \lbrace \sigma _1,\dots ,\sigma _4\rbrace , \tau \in \sigma and e\in E(U^\sigma _\varphi ) it holds:sig(e)=
{\left\lbrace \begin{array}{ll}
\textsf {inp}, & \text{if } e\i... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
0.022312909364700317,
0.02216029167175293,
-0.015902908518910408,
-0.007581353187561035,
-0.011309074237942696,
-0.04465634375810623,
0.013689931482076645,
0.040810342878103256,
0.01741383783519268,
0.05250096321105957,
0.01434619352221489,
0.022007672116160393,
-0.00403295923024416,
0.032... | |
03130e9e68f167aaaf254fa98959a24245b0ce35 | subsection | 79 | 122 | Concluding the ESSP and the SSP for | That sig(k)=\textsf {inp} implies for all generators G^{\eta ,\varrho }_j installed by the respective union U^\sigma _\varphi that exactly the source states of the k-labeled transition has to be included by the support.
For readability, the table does not enumerate these states explicitly, but there are assumed to be i... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
-0.014013431966304779,
0.01655369997024536,
-0.01804887317121029,
-0.011961383745074272,
-0.009215147234499454,
-0.03826422244310379,
0.015623031184077263,
0.006442212034016848,
-0.023693913593888283,
0.05208694189786911,
-0.01614176481962204,
0.02441098727285862,
-0.014204142615199089,
0.... | |
4f0afe8860e39f7cf659f3abac31d4fd801b4a0e | subsection | 80 | 122 | Concluding the ESSP and the SSP for | 1.2
p0.5cm p7cm p4.5cmE Support Target States\lbrace q_j\rbrace
\bigcup _{i=0}^{6m-1}\lbrace h_{i,1},h_{i,4}\rbrace , \bigcup _{i=0}^{3m-1}\lbrace h_{i,2}\rbrace , \bigcup _{i=3m}^{6m-1}\lbrace h_{i,5}\rbrace ,
\bigcup _{i=0}^{6m-2}\lbrace g^{c,c}_{i,2}, g^{c,c}_{i,3}\rbrace , \bigcup _{i=0}^{3m-1}\lbrace g^{\_,q}_... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
-0.012295813299715519,
-0.00585805531591177,
-0.007475122343748808,
-0.02668161131441593,
0.01188391912728548,
-0.040243618190288544,
-0.0027116388082504272,
0.01998451165854931,
-0.03481271490454674,
0.035941608250141144,
-0.04668137803673744,
0.017269058153033257,
-0.011205055750906467,
... | |
4135fa13fe44caad70f858d8353297abcaa5a607 | subsection | 81 | 122 | Concluding the ESSP and the SSP for | The first row of the following table is dedicated to the inhibition of Z at certain states of H and the sources/sinks of k in F_2, G^{n,\_}_0 and G^{\_,q}_0,\dots ,G^{\_,q}_{3m-1}.
Hence, for F_2 and each generator G^{\eta ,\varrho }_j installed by U^\sigma _\varphi we, firstly, assume the sinks g^{\eta ,\varrho }_{j,2... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
0.016574041917920113,
-0.004589910618960857,
-0.014071146957576275,
-0.00957662146538496,
0.0036818485241383314,
-0.04474685713648796,
0.022190291434526443,
0.022297121584415436,
-0.017749181017279625,
0.026417739689350128,
-0.009775021113455296,
0.0036551407538354397,
-0.01620776392519474,
... | |
311f1afef7ef8800b91bc7f7eceebaf52007dd42 | subsection | 82 | 122 | Concluding the ESSP and the SSP for | 1.4
p0.5cm p9cm p3cmE Support Target StatesZ
\bigcup _{i=0}^{6m-1} \lbrace h_{i,1}, h_{i,4}\rbrace , \bigcup _{i=3m}^{6m-1} \lbrace h_{i,2}\rbrace , \bigcup _{i=0}^{3m-1} \lbrace h_{i,5}\rbrace ,
\lbrace f_{0,1},f_{0,2}, f_{1,2}\rbrace
\bigcup _{i=0}^{6m-1}\lbrace h_{i,0}\rbrace , \lbrace f_{0,0},f_{0,3}\rbrace , ... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
0.004634125158190727,
-0.014440163038671017,
-0.011846578679978848,
-0.017910989001393318,
0.02131316252052784,
-0.04897297918796539,
0.011221067048609257,
0.009695429354906082,
-0.023830464109778404,
0.045769136399030685,
-0.05315322428941727,
0.010893055237829685,
-0.012601769529283047,
... | |
c91438a3034dbafef03b0e3da9b30c8400c2bbae | subsection | 83 | 122 | Concluding the ESSP and the SSP for | If j\in \lbrace 0,\dots , 3m-1\rbrace then the following table proves y_j to be inhibitable.1.4
p0.5cm p9cm p2.5cmE Support Target StatesY
\bigcup _{i=0}^{6m-1}\lbrace h_{i,1}, h_{i,4}\rbrace , \bigcup _{i=3m}^{6m-1}\lbrace h_{i,2}\rbrace , \bigcup _{i=0}^{3m-1}\lbrace h_{i,5}\rbrace \lbrace f_{0,1}, f_{0,2}, f_{0,4}... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
0.0053437561728060246,
0.006072277203202248,
-0.014097447507083416,
-0.028088096529245377,
0.005904450546950102,
-0.014867925085127354,
-0.03167348727583885,
0.024990929290652275,
-0.019422132521867752,
0.02772192843258381,
-0.044947054237127304,
0.005603124853223562,
-0.016599591821432114,
... | |
1f66ae108fd14f137e20cc0fade2f4bcb8efb6b3 | subsection | 84 | 122 | Concluding the ESSP and the SSP for | For simplicity, we refrain from presenting these states explicitly.The third row, inhibits the events c_j,r_j (p_j) for j\in \lbrace 0,\dots , 6m-2\rbrace (j\in \lbrace 0,\dots , 3m-1\rbrace ) at the remaining states of U^\sigma _\varphi .1.4
p1.6cm p8.3cm p2.4cmE Support Target States\lbrace c_j,r_j, c_{j+3m}, r_{j+3... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
0.006022326648235321,
0.003119862638413906,
-0.004782772157341242,
-0.026774371042847633,
-0.008871394209563732,
-0.023402784019708633,
-0.025599654763936996,
0.03859781101346016,
-0.022594213485717773,
0.03231231868267059,
-0.02537081390619278,
0.01769701950252056,
-0.015988342463970184,
... | |
26b3d4da3a76d1bac3e8a47b17e781095bf8d851 | subsection | 85 | 122 | Concluding the ESSP and the SSP for | The following table proves n_0 to be inhibitable in U^\sigma _\varphi :
1.4
p0.5cm p8cm p3cm
E Support Target States\lbrace n_0\rbrace
\bigcup _{i=0}^{6m-1}\lbrace h_{i,1},h_{i,4}\rbrace , \bigcup _{i=0}^{3m-1}\lbrace h_{i,2}\rbrace , \bigcup _{i=3m}^{6m-1}\lbrace h_{i,5}\rbrace ,\lbrace t_{i,\alpha ,1}\mid 0\le i\l... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
0.03542094677686691,
0.005743319168686867,
-0.0017351992428302765,
-0.014918900094926357,
-0.003182469867169857,
-0.028297096490859985,
-0.011250193230807781,
0.04021086171269417,
0.015422298572957516,
0.038898974657058716,
-0.027061481028795242,
0.027412334457039833,
-0.007478518411517143,
... | |
b18c900ff9fa908ab04ce473691fe0eb575cfa4b | subsection | 86 | 122 | Concluding the ESSP and the SSP for | The third (fourth) row deals with the inhibition of x_{3i},x_{3i+1},x_{3i+2} (w_{3i},w_{3i+1},w_{3i+2}) at the remaining states, that is, the rest of \bigcup _{j=3m}^{6m-1}\lbrace h_{j,0,\dots , h_{j,6}}\rbrace (\bigcup _{j=0}^{3m-1}\lbrace h_{j,0,\dots , h_{j,6}}\rbrace ), h_{3i+\alpha ,1} and G^{\_,y}_0,\dots , G^{\_... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
-0.00641339085996151,
0.002333009149879217,
-0.02951456792652607,
-0.016054460778832436,
-0.0038133158814162016,
-0.007187881041318178,
-0.014566523022949696,
0.029590873047709465,
-0.0030617169104516506,
0.010323994792997837,
-0.01741267926990986,
-0.011109930463135242,
-0.03131535276770592... | |
8072ef09874e5f0d17de2911641abca9cde3b5ef | subsection | 87 | 122 | Concluding the ESSP and the SSP for | Having this insight, we now can define the following subsets of S(U^\sigma _\varphi ) to, finally, combine them to a fitting support of U^\sigma _\varphi :S_0=\lbrace t_{n,\alpha _n,0},t_{n,\alpha _n,1}, t_{n,\beta _n,0},t_{n,\beta _n,1}, t_{n,\gamma _n,0},t_{n,\gamma _n,1}\mid n\in \lbrace i,j,\ell \rbrace \rbrace ,
... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
-0.016344770789146423,
0.018038766458630562,
-0.05207129195332527,
0.010522613301873207,
-0.003632171079516411,
-0.029057368636131287,
0.002922524232417345,
0.03900768607854843,
0.003597833449020982,
0.0425482913851738,
0.00030641673947684467,
0.019488582387566566,
-0.0022262309212237597,
... | |
c0bf859b3bc4d0ce35be93842c90558f498762b1 | subsection | 88 | 122 | Concluding the ESSP and the SSP for | 1.4
p2cm p7cm p3cm
E Support Target StatesV, W
\bigcup _{i=0}^{6m-1}\lbrace h_{i,0},h_{i,4}\rbrace , \bigcup _{i=0}^{3m-1}\lbrace h_{i,2}\rbrace , \bigcup _{i=3m}^{6m-1}\lbrace h_{i,5}\rbrace ,
\lbrace f_{0,1},f_{0,4},f_{1,0},f_{1,2},f_{2,1},f_{2,2}\rbrace ,
\lbrace g^{n,\_}_{0,1}, g^{n,\_}_{0,2}\rbrace , \bigcup _{... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
0.014099770225584507,
-0.006367022637277842,
-0.018662357702851295,
-0.0368364118039608,
0.013611466623842716,
-0.030396906659007072,
0.00243579619564116,
0.033906590193510056,
-0.011406470090150833,
0.021286990493535995,
-0.0366838164627552,
0.015015339478850365,
-0.01207025721669197,
-0.... | |
6416f5f7283d4a7821200c75371d7a2e1009343b | subsection | 89 | 122 | Concluding the ESSP and the SSP for | For arbitrary i \in \lbrace 0,\dots , m-1\rbrace , \alpha \in \lbrace 0,1,2\rbrace we present regions of U^\sigma _\varphi that inhibits the event X_{i,\alpha } and x_{i,\alpha } at all states of S(U^\sigma _\varphi )\setminus \lbrace s\in S(T_j)\mid j\ne i, X_{i,\alpha }, x_{i,\alpha } \in E(T_j)\rbrace .
Having this,... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
0.02028208039700985,
-0.0012914756080135703,
-0.028919896110892296,
-0.022876476868987083,
-0.017245110124349594,
-0.010202084667980671,
-0.018832270056009293,
0.04846943914890289,
0.018160779029130936,
0.009469549171626568,
-0.020800959318876266,
-0.008775166235864162,
-0.04254810884594917,... | |
b80629a831b2cefea928e7d930e5616a02f1f26a | subsection | 90 | 122 | Concluding the ESSP and the SSP for | The states t_{n,\alpha _n,3}, t_{n,\beta _n,5}, t_{n,\gamma _n,4} are the sinks of X_{i,\alpha _i} and the sources of x_{i,\alpha _i} in T_n.We now define subsets of S(U^\sigma _\varphi ) which will be used to combine supports of regions of U^\sigma _\varphiS_0=\lbrace t_{n,0,0}, t_{n,1,0},t_{n,2,0}, t_{n,0,1}, t_{n,1,... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
0.013369584456086159,
0.01005008164793253,
-0.030051041394472122,
-0.022435262799263,
0.0009314640192314982,
-0.03220299631357193,
0.011034485884010792,
0.05005963146686554,
0.011057378724217415,
0.025609774515032768,
-0.019840708002448082,
-0.0005914057255722582,
-0.03134831786155701,
-0.... | |
43488f4c406bffb771e780f4b674ea2eedae4626 | subsection | 91 | 122 | Concluding the ESSP and the SSP for | Observe that, if \tau \in \sigma _3 then \textsf {swap}\in \tau .
1.4
p1cm p8cm p3cm
E Support Target StatesX_{i,\alpha _i}
S_1,S_2,S_4,S_5 and for \sigma _1 if 0\in \lbrace i,j,\ell \rbrace :f_{1,0} and for \sigma _3:S_0
S(U^\sigma _\varphi )\setminus supX_{i,\alpha _i}
S_1,S_3,S_6 and for \sigma _1,\sigma _2,\si... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
-0.003941373433917761,
0.008042995817959309,
-0.02849387750029564,
-0.004738804884254932,
0.02086295187473297,
-0.03574325889348984,
-0.011225092224776745,
0.06248202174901962,
0.026815073564648628,
0.04273318499326706,
-0.018970482051372528,
0.012774170376360416,
-0.02231282740831375,
-0.... | |
851af71f90319a2eb29f3a2b1ebff3460f956d29 | subsection | 92 | 122 | Concluding the ESSP and the SSP for | If j\in \lbrace 0,\dots , m-1\rbrace \setminus \lbrace i\rbrace such that X_{i,\gamma _i}\in E(T^\sigma _j) and X_{i,\alpha _i}\notin E(T^\sigma _j) then if for \delta \in \lbrace 0,1,2\rbrace and \varepsilon \in \lbrace 3,4,5\rbrace the state t_{j,\delta ,\varepsilon } is a sink of X_{i,\gamma _i} in T^\sigma _j then ... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
-0.023771053180098534,
-0.012160160578787327,
-0.04174428805708885,
0.01827838458120823,
0.014708149246871471,
0.015944000333547592,
-0.024274548515677452,
0.05401125177741051,
-0.006053379736840725,
0.05175315588712692,
-0.011526977643370628,
0.009177335537970066,
-0.020338134840130806,
0... | |
df20660ef7171bd9d96b92cd4e24979f8f652561 | subsection | 93 | 122 | Concluding the ESSP and the SSP for | Finally, for x_{i,\alpha _i} and x_{i,\gamma _i} choose the support with their respective generator-sources which, by the presence of \textsf {swap} for the relevant cases is always fitting.
[Figure: All cases of how two events X_{i,\alpha _i},X_{i,\gamma _i} from T_i can occur as events X_{j,0},X_{j,1},X_{j,2} of T_j... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
0.022805996239185333,
0.01630743034183979,
-0.04253051057457924,
0.009213927201926708,
-0.02213478274643421,
-0.037252336740493774,
-0.008969849906861782,
0.05732771381735802,
0.04814429581165314,
0.031074121594429016,
-0.0031329640187323093,
-0.009641063399612904,
-0.01456837821751833,
-0... | |
187474d1b4703ff4c7c053c08d1c64c0408ac015 | subsection | 94 | 122 | Concluding the ESSP and the SSP for | With the definitions above, we define the following sets which will be used to yield a support that inhibits x_{i,\alpha _i} at g^{\_,x}_{j,0}, respectively g^{x,\_}_{j,2} where x_j=x_{i,\alpha _i}:S_{7}=\bigcup _{n=0}^{3m-1}\lbrace h_{n,1},h_{n,2},h_{n,3}\rbrace ,
S_{8}=\lbrace h_{3n+3m,n^{\prime }}, h_{3n+3m+1,n^{\p... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
0.0023161505814641714,
0.00514742499217391,
-0.047192998230457306,
-0.006322671193629503,
0.004548355005681515,
-0.05280975624918938,
-0.008676978759467602,
0.03189953416585922,
0.02381017990410328,
-0.00993617158383131,
-0.04270569235086441,
0.005212292540818453,
-0.043987780809402466,
0.... | |
311c7c147fff2a65ede6bc897c0816aa64b967e4 | subsection | 95 | 122 | Concluding the ESSP and the SSP for | S_{17}=\lbrace g^{\_,q}_{3n,0},g^{\_,q}_{3n,3}, g^{\_,q}_{3n+1,0},g^{\_,q}_{3n+2,3}, g^{\_,q}_{3n+2,0},g^{\_,q}_{3n+3,3} \mid n\in \lbrace i,j,\ell \rbrace \rbrace ,Now, to enrich the set S_7\cup \dots \cup S_{13} to a fitting support of U^\sigma _\varphi if \sigma = \sigma _4, respectively the set S_{10}\cup \dots \cu... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
0.04870045930147171,
0.050683874636888504,
-0.003940129652619362,
-0.02105470933020115,
-0.018506783992052078,
-0.04891405627131462,
-0.02085636742413044,
0.017408277839422226,
0.023221207782626152,
0.04201787710189819,
-0.004756380803883076,
0.039698805660009384,
-0.01980363205075264,
0.0... | |
927df61ea4a8af01779585f7204167704598701e | subsection | 96 | 122 | Concluding the ESSP and the SSP for | Thirdly, for i\in \lbrace 0,\dots , m-1\rbrace and \ell \in \lbrace 0,\dots , 2\rbrace it is true, that each event of T_{i,\ell } is unique in T_{i,\ell }.
Hence, the states of T_{i,\ell } are separable.
If i\in \lbrace 0,\dots , 6m-1\rbrace then the set of states \lbrace h_{i,0},\dots , h_{i,6}\rbrace can be extended ... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
0.00021277765335980803,
0.013777978718280792,
0.00825458113104105,
-0.0038507389836013317,
0.02220039814710617,
-0.029570013284683228,
-0.057736292481422424,
0.02021685615181923,
-0.0093913022428751,
0.032285939902067184,
-0.04748291149735451,
-0.007480237167328596,
-0.021544303745031357,
... | |
133a9c16394dfbb833d73693e4ae7de24eab3130 | subsection | 97 | 122 | Concluding the ESSP and the SSP for | For each applied generator G^{\eta ,\varrho }_j, an input key region separates \lbrace g^{\eta ,\varrho }_{j,0}, g^{\eta ,\varrho }_{j,1}\rbrace from \lbrace g^{\eta ,\varrho }_{j,2}, g^{\eta ,\varrho }_{j,3}\rbrace .
Finally, we note that for each applied generator, there is at least one region presented such that \lb... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
0.060012008994817734,
0.007940073497593403,
-0.006601473316550255,
0.01076219417154789,
-0.0004111619491595775,
-0.07133100181818008,
-0.02459058351814747,
0.04170636087656021,
0.007482432760298252,
0.05384910851716995,
0.0003091461258009076,
0.016795430332422256,
-0.012882597744464874,
0.... |
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