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1d2e432cebd769b7c3263d95a633829db27bbc51
subsection
17
39
A simulation study
We simulate an observational study where individuals may experience a terminating event D, so that the hazard for D depends additively on the treatment A and a covariate process L. A and L are counting processes that jump from 0 to 1 for an individual at the instant treatment is initiated or the covariate changes, resp...
{ "cite_spans": [] }
1802.01946
The additive hazard estimator is consistent for continuous-time marginal structural models
[ "Pål Christie Ryalen", "Mats Julius Stensrud", "Kjetil Røysland" ]
[ "stat.ME" ]
2,018
en
Statistics
[ -0.0220495592802763, 0.0177006833255291, -0.039033059030771255, -0.005932018160820007, 0.0005984471645206213, -0.027436058968305588, 0.019516529515385628, 0.05880899727344513, 0.008224714547395706, 0.012955069541931152, -0.026993542909622192, -0.00017226218187715858, 0.00320061924867332, -...
83901bd23f98eda9627da5957e4f507d95bc8b3a
subsection
18
39
Weight calculation using additive hazard models
We assume that the longitudinal data is organised such that each individual has multiple time-ordered rows; one row for each time either A, L or D changes.Our goal is to convert the data to a format suitable for weighted additive hazard regression. Heuristically, the additive hazard estimates are cumulative sums of lea...
{ "cite_spans": [] }
1802.01946
The additive hazard estimator is consistent for continuous-time marginal structural models
[ "Pål Christie Ryalen", "Mats Julius Stensrud", "Kjetil Røysland" ]
[ "stat.ME" ]
2,018
en
Statistics
[ -0.021459702402353287, 0.0008776193717494607, -0.03995839133858681, -0.0027702245861291885, -0.006452410481870174, 0.01347718108445406, 0.024298610165715218, 0.038096312433481216, -0.0012277131900191307, 0.001968919998034835, -0.03715001046657562, 0.038279466331005096, -0.0077421292662620544...
531c60fb5776ffc176ce1b2408b5422b2211f8bc
subsection
19
39
A marginal structural model
We now suppose the intervention that imposes a marginal treatment initiation rate is causally valid. This implies that the intensity for the event D has the same form under the randomised scenario \tilde{P}, i.e. that the hazard for D under \tilde{P} for the filtration \mathcal {F}_t^{A \cup D \cup L}, generated by A,D...
{ "cite_spans": [] }
1802.01946
The additive hazard estimator is consistent for continuous-time marginal structural models
[ "Pål Christie Ryalen", "Mats Julius Stensrud", "Kjetil Røysland" ]
[ "stat.ME" ]
2,018
en
Statistics
[ -0.044642943888902664, 0.02026173286139965, -0.017088208347558975, -0.008139784447848797, 0.007163315545767546, -0.018339309841394424, 0.01948360912501812, 0.04433779790997505, 0.04787749797105789, 0.0531260184943676, -0.05611645430326462, 0.010397869162261486, -0.007289188914000988, 0.045...
c8b24fef0cf1f80518f734d9b07b6ce3659ab853
subsection
20
39
Simulation details and results
We simulate subjects, none of which are treated at baseline. Initially, all the patients start with L = 0, and the hazards for transitioning from one state to another is constant. As described in Section REF , we fit additive hazard models for the time to treatment initiation, one for the observed treatment scenario, i...
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1802.01946
The additive hazard estimator is consistent for continuous-time marginal structural models
[ "Pål Christie Ryalen", "Mats Julius Stensrud", "Kjetil Røysland" ]
[ "stat.ME" ]
2,018
en
Statistics
[ -0.050504352897405624, -0.011023986153304577, -0.013823850080370903, -0.02674747072160244, -0.00905568990856409, 0.02563362941145897, 0.023909462615847588, 0.040617093443870544, 0.023863688111305237, 0.022551489993929863, -0.03939644619822502, 0.05251842364668846, -0.03509365767240524, 0.0...
0620e88f9069bc48c7538e90ca282228e75f3daa
subsection
21
39
Performance
In Figure REF we plot mean weight estimates based on aggregated simulations of the set-up in Section . The plot suggests that the discrete weights gradually approximate the continuous likelihood ratio as the time discretisation is refined. However, the continuous-time weights (REF ) are closer to the expected value of ...
{ "cite_spans": [ { "arxiv_id": "", "doi": "10.1093/biostatistics/kxy036", "end": 1888, "openalex_id": "https://openalex.org/W2886657410", "raw": "Pål Christie Ryalen, Mats Julius Stensrud, Sophie Fosså, and Kjetil Røysland. Causal inference in continuous time: an example on prostate...
1802.01946
The additive hazard estimator is consistent for continuous-time marginal structural models
[ "Pål Christie Ryalen", "Mats Julius Stensrud", "Kjetil Røysland" ]
[ "stat.ME" ]
2,018
en
Statistics
[ -0.03949648141860962, 0.009828336536884308, -0.04746292904019356, -0.011186599731445312, -0.003956515807658434, -0.03156055882573128, 0.04059530049562454, 0.044074900448322296, 0.0255780927836895, 0.03635263442993164, -0.03296460583806038, 0.027027923613786697, -0.03281199187040329, 0.0429...
58c05fbaf16d4e94b66fc2c8186078dfeebbbb0f
subsection
22
39
Performance
Utilising the fact that the true likelihood ratio R^i has a constant mean equal to 1, we can find precise estimates of the bias and variance of the additive hazard weight estimator (REF ) at time t_0.We plot the bias and variance of the weight estimator as a function of n under the strategies \kappa _n^1, \kappa _n^2, ...
{ "cite_spans": [] }
1802.01946
The additive hazard estimator is consistent for continuous-time marginal structural models
[ "Pål Christie Ryalen", "Mats Julius Stensrud", "Kjetil Røysland" ]
[ "stat.ME" ]
2,018
en
Statistics
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9e28ef6e09780928aaeb833bc0d8869cb835d77e
subsection
23
39
Censoring weights
Most standard martingale-based estimators in survival analysis are consistent when we have independent censoring, see . We have assumed independent censoring when conditioning on \mathcal {V}_0. A likely situation where this is violated is when we have independent censoring when conditioned on \mathcal {L} \cup \mathca...
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1802.01946
The additive hazard estimator is consistent for continuous-time marginal structural models
[ "Pål Christie Ryalen", "Mats Julius Stensrud", "Kjetil Røysland" ]
[ "stat.ME" ]
2,018
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Statistics
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47c96b7211672165b4f174591d522db2b80b336d
subsection
24
39
Censoring weights
If the model is causal with respect to this intervention, the corresponding likelihood ratio process is given byR_t^{i,c} = \prod _{s \le t} \Big ( \frac{\tilde{\lambda }_s^{i,c}}{\lambda _s^{i,c} }\Big )^{\Delta N_s^{i,c} } \exp \big ( -\int _0^t \tilde{\lambda }_s^{i,c} - \lambda _s^{i,c} ds \big ).However, as we onl...
{ "cite_spans": [] }
1802.01946
The additive hazard estimator is consistent for continuous-time marginal structural models
[ "Pål Christie Ryalen", "Mats Julius Stensrud", "Kjetil Røysland" ]
[ "stat.ME" ]
2,018
en
Statistics
[ -0.04357462748885155, 0.015325880609452724, -0.009192476980388165, 0.01393747329711914, -0.004416965879499912, -0.0007919452618807554, 0.0035930760204792023, 0.030850104987621307, 0.028408950194716454, 0.034328751266002655, -0.04674813151359558, -0.023496123030781746, -0.007327281404286623, ...
f293a325babc27448077d4c97c495fa538b29777
subsection
25
39
Discussion
Marginal structural modeling is an appealing concept for causal survival analysis. Here we have developed theory for continuous-time MSMs that may motivate the approach for practical research. Indeed, we show that the continuous-time MSMs yield consistent effect estimates, even if the treatment weights are estimated fr...
{ "cite_spans": [ { "arxiv_id": "", "doi": "", "end": 1169, "openalex_id": "", "raw": "Fred W. Huffer and Ian W. McKeague. Weighted least squares estimation for aalen's additive risk model. Journal of the American Statistical Association, 86(413):114–129, 1991.", "source_ref_id...
1802.01946
The additive hazard estimator is consistent for continuous-time marginal structural models
[ "Pål Christie Ryalen", "Mats Julius Stensrud", "Kjetil Røysland" ]
[ "stat.ME" ]
2,018
en
Statistics
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0ba08bce3f61178bea5c5e17b916e0933f134d04
subsection
26
39
Funding
The authors were all supported by the research grant NFR239956/F20 - Analyzing clinical health registries: Improved software and mathematics of identifiability.
{ "cite_spans": [] }
1802.01946
The additive hazard estimator is consistent for continuous-time marginal structural models
[ "Pål Christie Ryalen", "Mats Julius Stensrud", "Kjetil Røysland" ]
[ "stat.ME" ]
2,018
en
Statistics
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260f4d83fe7c19f933d7bcea9696407806dd2bd8
subsection
27
39
Appendix: proofs
We need some lemmas to prove Theorem REF .Lemma 1 Suppose that \lbrace V^i\rbrace _i are processes on [0, T] such that \sup _ i E\big [ \sup _{s} | V^i_s | \big ] < \infty , then\lim _{a \rightarrow \infty } \sup _n P \bigg ( \sup _s \big | \frac{1}{n} \sum _{i=1}^n V^{i}_{s} \big | \ge a \bigg ) = 0.By Markov's inequ...
{ "cite_spans": [] }
1802.01946
The additive hazard estimator is consistent for continuous-time marginal structural models
[ "Pål Christie Ryalen", "Mats Julius Stensrud", "Kjetil Røysland" ]
[ "stat.ME" ]
2,018
en
Statistics
[ -0.0385708250105381, 0.026532700285315514, -0.03054540790617466, -0.00433692941442132, -0.009177354164421558, 0.002267637988552451, -0.011725347489118576, 0.011687204241752625, -0.014464059844613075, 0.03133879601955414, -0.05453011393547058, -0.0017403025412932038, -0.00836870912462473, 0...
13ca8f11ccf9c4968fc672aaa91452ead016a0d8
subsection
28
39
Appendix: proofs
Finally, the weak law of large numbers and the triangle inequality yields& \lim \limits _{n\longrightarrow \infty } E\bigg [ \bigg | \frac{1}{n} \sum _{i = 1} ^n S_{{(i,n)}} V_{{i}} -E_{ P} [S_{{1}} V_{{1}}] \bigg | \bigg ] \\\le & \lim \limits _{n\longrightarrow \infty } E \bigg [\bigg | \frac{1}{n} \sum _{i = 1}^n S_...
{ "cite_spans": [] }
1802.01946
The additive hazard estimator is consistent for continuous-time marginal structural models
[ "Pål Christie Ryalen", "Mats Julius Stensrud", "Kjetil Røysland" ]
[ "stat.ME" ]
2,018
en
Statistics
[ -0.0346614271402359, 0.0199089627712965, -0.05479922890663147, -0.001962291542440653, -0.029184555634856224, 0.0018678954802453518, 0.005492127500474453, 0.026118118315935135, -0.030359260737895966, 0.01987845078110695, -0.055134858936071396, 0.013806598260998726, -0.0025973187293857336, -...
a6a7bf6d66b6dffbb0662d218ebd3c270114c6a2
subsection
29
39
Appendix: proofs
If the assumptions of Theorem REF are satisfied, then\lim \limits _{n\longrightarrow \infty } P \bigg ( \sup _t \bigg | \int _0^t \Gamma ^{(n)-1} \frac{1}{n} \sum _{i=1}^n R^{(i,n)}_{s-}X_{s-}^{i \intercal } ( \lambda ^{i,D}_s - \gamma ^i_s ) ds \bigg | \ge \delta \bigg )= 0for every \delta > 0.Assumption \ref {eq:boun...
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1802.01946
The additive hazard estimator is consistent for continuous-time marginal structural models
[ "Pål Christie Ryalen", "Mats Julius Stensrud", "Kjetil Røysland" ]
[ "stat.ME" ]
2,018
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Statistics
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539fb0cb1904585db50f8f53fd431fa6f1ce3065
subsection
30
39
Appendix: proofs
Then\Xi ^{(n)}_t := \frac{1}{n} \int _0^t \Gamma ^{(n)-1}_s X_{s-}^{(n)\intercal } Y^{(n),D}_{s} d M_s^{(n)}defines a square integrable local martingale with respect to the filtration \mathcal {F}_{s}^{1,\mathcal {V}_0 \cup \mathcal {L}} \otimes \dots \otimes \mathcal {F}_{s}^{n,\mathcal {V}_0 \cup \mathcal {L}} and\li...
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1802.01946
The additive hazard estimator is consistent for continuous-time marginal structural models
[ "Pål Christie Ryalen", "Mats Julius Stensrud", "Kjetil Røysland" ]
[ "stat.ME" ]
2,018
en
Statistics
[ -0.0432564839720726, 0.039776429533958435, -0.007551569025963545, -0.022528784349560738, 0.015515249222517014, -0.009463310241699219, 0.030160484835505486, -0.008081971667706966, 0.0236430112272501, 0.05439877137541771, -0.013477585278451443, 0.019888214766979218, -0.01185966469347477, 0.0...
476870fb295ec30fedd460169d6e73990373ddec
subsection
31
39
Appendix: proofs
Moreover, Lemma REF implies that \int _0^\cdot \Gamma ^{(n)-1} \frac{1}{n} \sum _{i=1}^n R^{(i,n)}_{s-}X_{s-}^{i \intercal } ( \lambda ^{i,D}_s - \gamma ^i_s ) ds converges in same sense to 0, which proves the consistency.To see that B^{(n)} is P-UT, note that it coincides with the sum of B_t, \Xi ^{(n)} and \int _0^...
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1802.01946
The additive hazard estimator is consistent for continuous-time marginal structural models
[ "Pål Christie Ryalen", "Mats Julius Stensrud", "Kjetil Røysland" ]
[ "stat.ME" ]
2,018
en
Statistics
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012e1bbce77c9bea22b6f82c0e3a6f4163996238
subsection
32
39
Proof of Theorem
Lemma 6 Suppose that REF . and REF . from Theorem REF are satisfied, and that\lim _{a \rightarrow \infty } \sup _n P \bigg ( \sup _t \big | \theta ^{(i,n)}_t \big | \ge a \bigg ) = 0, \theta _{t-}^{(i,n)} converges to \theta ^i_t in probability for each i and t.Then we have that K^{(i,n)} is predictably uniformly t...
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1802.01946
The additive hazard estimator is consistent for continuous-time marginal structural models
[ "Pål Christie Ryalen", "Mats Julius Stensrud", "Kjetil Røysland" ]
[ "stat.ME" ]
2,018
en
Statistics
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495e679b48656dcc642cdc77d0b748b218e8ebc7
subsection
33
39
Proof of Theorem
The latter property holds due to \ref {item:tightthetan}), \ref {item:thetaconvergence}) and .Since \lbrace \int _0^t Y_s^i Z_{s-}^{i\intercal } dW^{(n)}_s \rbrace _n converges in the skorokhod topology, we have that \lbrace \sup _{t \le T}|\int _0^t Y_s^i Z_{s-}^{i\intercal } dW^{(n)}_s | \rbrace _n is tight . Therefo...
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1802.01946
The additive hazard estimator is consistent for continuous-time marginal structural models
[ "Pål Christie Ryalen", "Mats Julius Stensrud", "Kjetil Røysland" ]
[ "stat.ME" ]
2,018
en
Statistics
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be32b6c7fc2cc6df45b2da7d7a9e63ef6e0936d1
subsection
34
39
Proof of Theorem
For the same reason we also have\lim \limits _{n\longrightarrow \infty } P\bigg ( \sup _{t \le T} | n ^{-1/2} \int _0^t Y_s^i \tilde{Z}_{s-}^{i \intercal } d\tilde{W}^{(n)}_s |\ge \epsilon \bigg ) = 0.By combining (REF ),(REF ) and (REF ), we obtain that\lim \limits _{n\longrightarrow \infty } P\bigg ( \sup _{t \le T} ...
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1802.01946
The additive hazard estimator is consistent for continuous-time marginal structural models
[ "Pål Christie Ryalen", "Mats Julius Stensrud", "Kjetil Røysland" ]
[ "stat.ME" ]
2,018
en
Statistics
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a17448641d3c59b975dcf41e7791a71bc290298b
subsection
35
39
Proof of Theorem
This means that K^{(i,n)} is a sum of three processes that are P-UT, and must therefore be P-UT itself.Lemma 7 Suppose that\lbrace \kappa _n\rbrace _n increasing sequence of positive numbers such that \lim \limits _{n\longrightarrow \infty } \kappa _n = \infty \text{ and } \sup _n \frac{\kappa _n }{\sqrt{n}} < \infty ...
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1802.01946
The additive hazard estimator is consistent for continuous-time marginal structural models
[ "Pål Christie Ryalen", "Mats Julius Stensrud", "Kjetil Røysland" ]
[ "stat.ME" ]
2,018
en
Statistics
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c0d05ccfaf5c68fa2377c7f552ead5087ee0339b
subsection
36
39
Proof of Theorem
Therefore implies that \int _0 ^\cdot Y_s^{i,A} Z_{s-}^{i\intercal } d W^{(n)}_s converges in law to a continuous process, so it is C-tight. Moreover, from we have that\lim _{n \longrightarrow \infty } P \bigg ( \sup _{1/\kappa _n \le t \le T} \Big | \int _0 ^t Y_s^{i,A} Z_{s-}^{i\intercal } d W^{(n)}_s - \int _0 ^{...
{ "cite_spans": [ { "arxiv_id": "", "doi": "10.2307/2290171", "end": 143, "openalex_id": "https://openalex.org/W2081877251", "raw": "Jean Jacod and Albert N. Shiryaev. Limit theorems for stochastic processes, volume 288 of Grundlehren der Mathematischen Wissenschaften [Fundamental Pr...
1802.01946
The additive hazard estimator is consistent for continuous-time marginal structural models
[ "Pål Christie Ryalen", "Mats Julius Stensrud", "Kjetil Røysland" ]
[ "stat.ME" ]
2,018
en
Statistics
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58ebc934e80e48650b8ca91e09e41bb62e3230c0
subsection
37
39
Proof of Theorem
Combining (REF ) and (REF ) yields the claim.[Proof of Theorem REF ]Combining (REF ) and the decomposition in the proof of Lemma REF , we see that\lim \limits _{n\longrightarrow \infty } P \bigg ( \sup _{ 1/ \kappa _n \le t \le T} \bigg | \kappa _n \int _{t- 1/\kappa _n} ^t Y_s^{i,A} \tilde{Z}_{s-}^{i\intercal } d \til...
{ "cite_spans": [ { "arxiv_id": "", "doi": "10.2307/2290171", "end": 1604, "openalex_id": "https://openalex.org/W2081877251", "raw": "Jean Jacod and Albert N. Shiryaev. Limit theorems for stochastic processes, volume 288 of Grundlehren der Mathematischen Wissenschaften [Fundamental P...
1802.01946
The additive hazard estimator is consistent for continuous-time marginal structural models
[ "Pål Christie Ryalen", "Mats Julius Stensrud", "Kjetil Røysland" ]
[ "stat.ME" ]
2,018
en
Statistics
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2c4d746d1595e62d14d472251a83da5e3cf042df
subsection
38
39
Proof of Theorem
Since K^{(i,n)} is P-UT,R_t^{(i,n)} = 1 + \int _0^t R_{s-}^{(i,n)} dK_{s}^{(i,n)}andR_t^{i} = 1 + \int _0^t R_{s-}^{i} dK_{s}^{i} implies that R^{(i,n)} converges to R^{i} in probability.
{ "cite_spans": [ { "arxiv_id": "", "doi": "10.2307/2290171", "end": 187, "openalex_id": "https://openalex.org/W2081877251", "raw": "Jean Jacod and Albert N. Shiryaev. Limit theorems for stochastic processes, volume 288 of Grundlehren der Mathematischen Wissenschaften [Fundamental Pr...
1802.01946
The additive hazard estimator is consistent for continuous-time marginal structural models
[ "Pål Christie Ryalen", "Mats Julius Stensrud", "Kjetil Røysland" ]
[ "stat.ME" ]
2,018
en
Statistics
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907ce9527fc5fff25884030cd445e04604f5edfc
abstract
0
10
Abstract
We propose a flow-insensitive analysis that prunes out portions of code which are irrelevant to a specified set of data-flow paths. Our approach is fast and scalable, in addition to being able to generate a certificate as an audit for the computed result. We have implemented our technique in a tool called DSlicer and a...
{ "cite_spans": [] }
1808.01232
Data-Flow Guided Slicing
[ "Mohamed Nassim Seghir" ]
[ "cs.PL", "cs.SE" ]
2,018
en
Computer Science
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5c18d8d3f50e809ca15674afc9c7eeb75c0e179b
subsection
1
10
Introduction
Applying static analysis naively may result in exploring parts of code which are irrelevant to a property of interest. We propose a lightweight approach for slicing programs that can be invoked by other tools as a preprocessing phase. As slicing criterion, our technique accepts a set of data-flow paths and returns, as ...
{ "cite_spans": [] }
1808.01232
Data-Flow Guided Slicing
[ "Mohamed Nassim Seghir" ]
[ "cs.PL", "cs.SE" ]
2,018
en
Computer Science
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92c9c9f26e792e04a924f95d991ffd26b0d8a3d4
subsection
2
10
Body
To illustrate our approach, let us consider the simple Java example in Figure REF (a). In the function main, an instance of the class C, defined in Figure REF (b), is created and its methods m1, m3, m4 and m5 are consecutively invoked. The function source returns some private (integer) information, it could be for exam...
{ "cite_spans": [] }
1808.01232
Data-Flow Guided Slicing
[ "Mohamed Nassim Seghir" ]
[ "cs.PL", "cs.SE" ]
2,018
en
Computer Science
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842db8d3a640724636cbef1ee8bd2bea7954e1b0
subsection
3
10
Body
They appear (are used) in the methods m3, m4 and m5, hence they are the relevant methods. Methods m1 and m2 can then be safely discarded. In the next section, we cover the translation from a program to an assignment graph in more details.A key question that may arise is: how can we trust the result (soundness) of the a...
{ "cite_spans": [ { "arxiv_id": "", "doi": "", "end": 1682, "openalex_id": "", "raw": "S. Arzt, S. Rasthofer, C. Fritz, E. Bodden, A. Bartel, J. Klein, Y. L. Traon, D. Octeau, and P. McDaniel. Flowdroid: precise context, flow, field, object-sensitive and lifecycle-aware taint analysi...
1808.01232
Data-Flow Guided Slicing
[ "Mohamed Nassim Seghir" ]
[ "cs.PL", "cs.SE" ]
2,018
en
Computer Science
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52d5021c69e980c84bb7933a8d401d4a33784bb5
subsection
4
10
Assignment Graph: a Compact Representation
Our analysis is applied to object oriented programs, therefore we need to account for the main features distinguishing them from simple imperative programs. We use a Jimple-like intermediary representation which allows to encode most of the language constructs using a minimal set of three-address-based instructions. It...
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1808.01232
Data-Flow Guided Slicing
[ "Mohamed Nassim Seghir" ]
[ "cs.PL", "cs.SE" ]
2,018
en
Computer Science
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803ebb3e6ea7d3d98285e18b303e7339c367816e
subsection
5
10
Slice Computation
In this section we describe our approach for computing a slice with respect to a set of sources SR and sinks SK. The slicing criterion is all data-flow paths r \leadsto k such that r \in SR and k \in SK.This is achieved via algorithm ComputeSlice (Algorithm ) which takes as parameters an assignment graph G, a set of so...
{ "cite_spans": [] }
1808.01232
Data-Flow Guided Slicing
[ "Mohamed Nassim Seghir" ]
[ "cs.PL", "cs.SE" ]
2,018
en
Computer Science
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b7f66794abbf6454a166e9d8d90d6ae3642eb225
subsection
6
10
Slice Computation
\mathsf {marked}(id^{\prime },G,+) \wedge \lnot \mathsf {marked}(id^{\prime },G,-) \mathsf {mark}(id^{\prime },G,-) L := L \cup \lbrace id^{\prime }\rbraceids := \lbrace id \;|\; id \in \mathsf {nodes}(G) \wedge \mathsf {marked}(id,G,+) \wedge \mathsf {marked}(id,G,-)\rbracevia\_locals := \lbrace m \;|\; \exists c \;\...
{ "cite_spans": [] }
1808.01232
Data-Flow Guided Slicing
[ "Mohamed Nassim Seghir" ]
[ "cs.PL", "cs.SE" ]
2,018
en
Computer Science
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f1d02e1d4e5cb15ce4d5320263cefc87f772d742
subsection
7
10
DSlicer: Implementation and Experiments
We implemented our approach in a tool called DSlicer, which is written in Python and uses Androguardhttps://github.com/androguard as front-end for parsing and decompiling Android applications. It takes as input an Android application in bytecode format (APK) and a configuration file containing the sources and sinks to ...
{ "cite_spans": [] }
1808.01232
Data-Flow Guided Slicing
[ "Mohamed Nassim Seghir" ]
[ "cs.PL", "cs.SE" ]
2,018
en
Computer Science
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e2785b31020498cc2f085f9fb4e65eaacd6d7851
subsection
8
10
Related work
Our work is at the intersection of several topics: program slicing, data-flow analysis, and program certification.Program slicing was originally proposed by Weiser . While the initial proposal defined a slice as a set of statements that might influence a program point of interest (criterion), a dynamic variant was exam...
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1808.01232
Data-Flow Guided Slicing
[ "Mohamed Nassim Seghir" ]
[ "cs.PL", "cs.SE" ]
2,018
en
Computer Science
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bd7b10b6c873e4c3a60e431541763c1efc14054f
subsection
9
10
Conclusion and Further Work
We presented a lightweight slicing approach for object oriented programs that can detect and discard parts of code that are irrelevant to a specified set of data-flow paths. A distinguishing feature of our approach lies in the compact representation it uses, allowing it to efficiently encode programs without sacrificin...
{ "cite_spans": [] }
1808.01232
Data-Flow Guided Slicing
[ "Mohamed Nassim Seghir" ]
[ "cs.PL", "cs.SE" ]
2,018
en
Computer Science
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987dcba2258758f7c71ed1cd3cb530433dd06995
abstract
0
22
Abstract
Proof-carrying-code was proposed as a solution to ensure a trust relationship between two parties: a (heavyweight) analyzer and a (lightweight) checker. The analyzer verifies the conformance of a given application to a specified property and generates a certificate attesting the validity of the analysis result. It suff...
{ "cite_spans": [] }
1808.01246
Certificate Enhanced Data-Flow Analysis
[ "Mohamed Nassim Seghir" ]
[ "cs.PL", "cs.CR" ]
2,018
en
Computer Science
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84564422ee6de20ae50a1c2bbd8735058c034f4d
subsection
1
22
Introduction
Static data-flow analysis has proven its effectiveness in assessing the security of Android applications by identifying data leaks , , , , , , . Once we want to install an application on a mobile device, how can we trust the outcome of these tools? The analysis might be broken, or a (malicious) tool can provide a false...
{ "cite_spans": [ { "arxiv_id": "", "doi": "", "end": 144, "openalex_id": "", "raw": "A. P. Fuchs, A. Chaudhuri, and J. S. Foster. SCanDroid: Automated Security Certification of Android Applications. Technical Report CS-TR-4991, Department of Computer Science, University of Maryland,...
1808.01246
Certificate Enhanced Data-Flow Analysis
[ "Mohamed Nassim Seghir" ]
[ "cs.PL", "cs.CR" ]
2,018
en
Computer Science
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17e5925f07326386107013bf3b8fd2ccd9ee3f3d
subsection
2
22
Example
We start by illustrating our idea through an example. Consider the simple code in Figure REF as part of an Android application. To ease the presentation, we omit irrelevant details. We have the root procedure foo which makes call to function bar which, in turn, calls procedures getId, Send and getNumber. Function getId...
{ "cite_spans": [] }
1808.01246
Certificate Enhanced Data-Flow Analysis
[ "Mohamed Nassim Seghir" ]
[ "cs.PL", "cs.CR" ]
2,018
en
Computer Science
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7db04ff71b192dfd6fc0cc1c688792b9bf6fb8da
subsection
3
22
Body
Our analysis attempts to find all data leaks, i.e., paths leading from sources to sinks. In addition, it outputs a certificate corroborating its outcome.Let us use id and num to respectively refer to the sources getDeviceId and getLine1Number. We also write sms to refer to the sink sendTextMessage. Our analysis compute...
{ "cite_spans": [] }
1808.01246
Certificate Enhanced Data-Flow Analysis
[ "Mohamed Nassim Seghir" ]
[ "cs.PL", "cs.CR" ]
2,018
en
Computer Science
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90af14e566f3b02346c91f2e9b613db16c304b51
subsection
4
22
Body
For example, assuming the summary for \mathsf {bar} is \lbrace \mathsf {(sms, id)}, \mathsf {(ret, num)}\rbrace , the summary for \mathsf {foo} must be \lbrace \mathsf {(sms, id)}, \mathsf {(ret, num)}\rbrace , which is the case. This is performed by the checker, which takes as input a certificate (computed map) and an...
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1808.01246
Certificate Enhanced Data-Flow Analysis
[ "Mohamed Nassim Seghir" ]
[ "cs.PL", "cs.CR" ]
2,018
en
Computer Science
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96f8fca6cad7b4adaef8f3277136809b2bfd8cec
subsection
5
22
Preliminaries
In this section we provide some ingredients required for the presentation. We start by introducing a language that we consider along our study.
{ "cite_spans": [] }
1808.01246
Certificate Enhanced Data-Flow Analysis
[ "Mohamed Nassim Seghir" ]
[ "cs.PL", "cs.CR" ]
2,018
en
Computer Science
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51daee1ce8593b7fee52b26705def03bbd660e98
subsection
6
22
Intermediary Representation
Without loss of generality, we consider an object oriented language, accounting for the features distinguishing it from simple imperative languages. We use a Jimple-like intermediary representation which allows us to encode most of the language constructs using a minimal set of instructions. It has been shown that the ...
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1808.01246
Certificate Enhanced Data-Flow Analysis
[ "Mohamed Nassim Seghir" ]
[ "cs.PL", "cs.CR" ]
2,018
en
Computer Science
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043b2c44be30755ee21443ffcb393994ef6942e3
subsection
7
22
Term Representative
A key challenge faced when designing any program analysis is aliasing. As we want our analysis to scale to large applications, we need a light but sound solution for the aliasing problem. Therefore, we use the approach proposed by Vallée et al . Their idea is based on the observation that only objects of compatible typ...
{ "cite_spans": [ { "arxiv_id": "", "doi": "10.1145/353171.353189", "end": 245, "openalex_id": "https://openalex.org/W1974608873", "raw": "V. Sundaresan, L. J. Hendren, C. Razafimahefa, R. Vallée-Rai, P. Lam, E. Gagnon, and C. Godin. Practical virtual method call resolution for java....
1808.01246
Certificate Enhanced Data-Flow Analysis
[ "Mohamed Nassim Seghir" ]
[ "cs.PL", "cs.CR" ]
2,018
en
Computer Science
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0da841e5cafe4169b297661cde70ea7aa8f61326
subsection
8
22
Field Access
Two terms o_1.f and o_2.f are aliases if o_1 and o_2 points to the same object. Hence, all potential field access that are aliases must map to the same representative. Observe that in type safe languages, like Java, o_1.f and o_2.f can be aliases only if o1 and o2 belong to the same type hierarchy.Given an object o who...
{ "cite_spans": [] }
1808.01246
Certificate Enhanced Data-Flow Analysis
[ "Mohamed Nassim Seghir" ]
[ "cs.PL", "cs.CR" ]
2,018
en
Computer Science
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0c4e580757ffbf10397717e8d826ce9e358fa537
subsection
9
22
Array Access
An l-value representing an array access also leads to aliasing. Here the aliasing is due to two causes: reference and index. Two terms a[i] and a[j] refer to the same memory location if i = j. In another scenario, terms a[i] and b[i] also refer to the same memory location if a and b points to the same array, i.e, store...
{ "cite_spans": [] }
1808.01246
Certificate Enhanced Data-Flow Analysis
[ "Mohamed Nassim Seghir" ]
[ "cs.PL", "cs.CR" ]
2,018
en
Computer Science
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1432f7536788a9b1f44ead6ae7ead60470644354
subsection
10
22
Control Flow Graph and Call Graph
Our analysis proceeds at the control flow graph level. Each function is associated with a corresponding control flow graph \mathit {CFG}. A \mathit {CFG} is a directed graph (N,E) where N is the set of pairs (\ell ,st), such that st is a program basic statement and \ell is the corresponding program location. The relati...
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1808.01246
Certificate Enhanced Data-Flow Analysis
[ "Mohamed Nassim Seghir" ]
[ "cs.PL", "cs.CR" ]
2,018
en
Computer Science
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fddd1a71057317cf137bbd8142b4ce3be8bbe39e
subsection
11
22
Data-Flow Certification
As seen in section , our approach has two main components: an analyzer and a checker. The analyzer takes as input an application and produces a certificate. The checker takes as input an application and a certificate, and answers whether the certificate is valid with respect to the input application. As we are in the c...
{ "cite_spans": [] }
1808.01246
Certificate Enhanced Data-Flow Analysis
[ "Mohamed Nassim Seghir" ]
[ "cs.PL", "cs.CR" ]
2,018
en
Computer Science
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24ff9a98bdb15a7217acebc4dfdfe41e57fa33a7
subsection
12
22
Abstract Domain
A natural way for encoding inter-variable flows is through an abstract domain D representing the powerset of pairs of l-value representatives (section ). Hence, aliases are taken into account in D. We also need to include symbols associated with sources and sinks. Let SR be the set of method identifiers representing so...
{ "cite_spans": [] }
1808.01246
Certificate Enhanced Data-Flow Analysis
[ "Mohamed Nassim Seghir" ]
[ "cs.PL", "cs.CR" ]
2,018
en
Computer Science
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1f31073d6ca64ba48cc812e6cdb788cde164c784
subsection
13
22
Transfer Function
To capture variable dependency relations induced by the program, we model the effect of program basic statements, belonging to our language L, on elements of the abstract domain D. This is achieved through function F that we define below.Let L_b denotes the set of basic statements in L. Given a set of facts d \in D and...
{ "cite_spans": [] }
1808.01246
Certificate Enhanced Data-Flow Analysis
[ "Mohamed Nassim Seghir" ]
[ "cs.PL", "cs.CR" ]
2,018
en
Computer Science
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8a2985edbe7c835f69f22b9d931398bb99813237
subsection
14
22
Transfer Function
Note that summaries for functions are computed iteratively, on-the-fly, during the analysis as we will see later.Finally, a conditional cond does not have any effect on the input facts, making the transfer function F behave as an identity function. [Table: Definition of functions \mathsf {flow}and \mathsf {kill} for th...
{ "cite_spans": [] }
1808.01246
Certificate Enhanced Data-Flow Analysis
[ "Mohamed Nassim Seghir" ]
[ "cs.PL", "cs.CR" ]
2,018
en
Computer Science
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869d5454bf87bb8f843e01494d98c7edc7d4b050
subsection
15
22
The analyser
In what follows, we describe the algorithm that performs the data-flow analysis. A key feature of our algorithm is that it produces a certificate.To compute all possible flows from sources to sinks, we propose a bottom-up inter-procedural data-flow analysis that computes method summaries that are context-independent. A...
{ "cite_spans": [] }
1808.01246
Certificate Enhanced Data-Flow Analysis
[ "Mohamed Nassim Seghir" ]
[ "cs.PL", "cs.CR" ]
2,018
en
Computer Science
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52a2fbb21e4562d3d2a2a675e426bac884d7fb39
subsection
16
22
The analyser
When the set of facts associated with a location is updated, all successor locations need to be considered (lines 17-18). Once a fix-point is reached, the algorithm returns the set of facts accumulated at the final location (line 19).In the next section we describe the algorithm implementing the checker. [h!] method M ...
{ "cite_spans": [] }
1808.01246
Certificate Enhanced Data-Flow Analysis
[ "Mohamed Nassim Seghir" ]
[ "cs.PL", "cs.CR" ]
2,018
en
Computer Science
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d8ae26702ac046a1ae20b0ea2fdec0ca937be5f5
subsection
17
22
The checker
The checker component takes as input a program and certificate, and answers whether the certificate is valid with respect to the input program. This is implemented via procedure Checker (Algorithm REF ). First, it extracts all methods in the program (line 2). For each method it checks its presence in the certificate, w...
{ "cite_spans": [] }
1808.01246
Certificate Enhanced Data-Flow Analysis
[ "Mohamed Nassim Seghir" ]
[ "cs.PL", "cs.CR" ]
2,018
en
Computer Science
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feab6de1fcc61afe2f5738e0415f1bcf3e5d9708
subsection
18
22
Discussion
We discuss some aspects related to our analysis without deep diving into detail.s
{ "cite_spans": [] }
1808.01246
Certificate Enhanced Data-Flow Analysis
[ "Mohamed Nassim Seghir" ]
[ "cs.PL", "cs.CR" ]
2,018
en
Computer Science
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e00d01c914f066d05e5fe14686be4ca18bfe7b81
subsection
19
22
Implementation and Experiments
We have implemented our approach in a tool called DCert, which is written in Python and uses Androguardhttps://github.com/androguard as front-end for parsing and decompiling Android applications. It accepts Android applications in bytecode format (APK), so no source code is required. One can simply download an app from...
{ "cite_spans": [] }
1808.01246
Certificate Enhanced Data-Flow Analysis
[ "Mohamed Nassim Seghir" ]
[ "cs.PL", "cs.CR" ]
2,018
en
Computer Science
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b2a5fc9c5fc82cd95525f4c68e4f8401899fb5b9
subsection
20
22
Related Work
Our work is related to many topics: certification, taint analysis and Android security. Along these axes we report on related work.The idea of associating proofs with code was initially proposed by Necula under the moniker Proof-Carrying Code (PCC) , . It was then used to support resource policies for mobile code , . F...
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1808.01246
Certificate Enhanced Data-Flow Analysis
[ "Mohamed Nassim Seghir" ]
[ "cs.PL", "cs.CR" ]
2,018
en
Computer Science
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fe57ab34afd1d8e9a49ada193b4514a5e979b9d0
subsection
21
22
Conclusion and Further Work
We have presented a data-flow certification approach inspired by the proof-carrying-code idea. It consists of splitting the verification process between a heavyweight analyzer and a lightweight checker and using a certificate to ensure trust between the two elements. We implemented our technique in a tool called DCert ...
{ "cite_spans": [] }
1808.01246
Certificate Enhanced Data-Flow Analysis
[ "Mohamed Nassim Seghir" ]
[ "cs.PL", "cs.CR" ]
2,018
en
Computer Science
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45ec73ff827620946d2e17c6ee9c9df1c51b6fe6
abstract
0
12
Abstract
The inherent stochasticity in many nano-scale devices makes them prospective candidates for low-power computations. Such devices have been demonstrated to exhibit probabilistic switching between two stable states to achieve stochastic behavior. Recently, superparamagnetic nanomagnets (having low energy barrier EB $\sim...
{ "cite_spans": [] }
10.1109/LMAG.2018.2839097
1803.01431
Design of a Low Voltage Analog-to-Digital Converter using Voltage Controlled Stochastic Switching of Low Barrier Nanomagnets
[ "Indranil Chakraborty", "Amogh Agrawal", "Kaushik Roy" ]
[ "cs.ET" ]
2,018
en
Computer Science
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23f1269dce337c8d2dd0e3e2b9c189fdf4ec282e
subsection
1
12
Introduction
Magnetization dynamics of nanomagnets is a function of temperature. In memory applications, reasonably high current is applied to devices based on nanomagnets to achieve deterministic switching across a range of temperature. Recently, however, the stochasticity introduced by the temperature dependence of nanomagnetic d...
{ "cite_spans": [ { "arxiv_id": "", "doi": "10.7567/apex.7.083001", "end": 553, "openalex_id": "https://openalex.org/W1999802284", "raw": "A. Fukushima, T. Seki, K. Yakushiji, H. Kubota, H. Imamura, S. Yuasa, and K. Ando, “Spin dice: A scalable truly random number generator based on ...
10.1109/LMAG.2018.2839097
1803.01431
Design of a Low Voltage Analog-to-Digital Converter using Voltage Controlled Stochastic Switching of Low Barrier Nanomagnets
[ "Indranil Chakraborty", "Amogh Agrawal", "Kaushik Roy" ]
[ "cs.ET" ]
2,018
en
Computer Science
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b9ff2d1c5f5aa07e22c5ca6858acb632b91322ff
subsection
2
12
Lifetime of a Magnetic State
The energy profile of a nanomagnet shown in Fig. REF (b) gives us insight on its stability. The two low-energy states are separated by an energy barrier (E_B) which is essential for the nanomagnet to retain its alignment along the easy axis (x-axis). On receiving an input stimulus, \vec{m} proceeds towards the opposite...
{ "cite_spans": [ { "arxiv_id": "", "doi": "10.1147/rd.491.0079", "end": 812, "openalex_id": "https://openalex.org/W1973893439", "raw": "L. Sun, Y. Hao, C.-L. Chien, and P. C. Searson, “Tuning the properties of magnetic nanowires,” ChemInform, vol. 37, no. 18, may 2006. [Online]. Ava...
10.1109/LMAG.2018.2839097
1803.01431
Design of a Low Voltage Analog-to-Digital Converter using Voltage Controlled Stochastic Switching of Low Barrier Nanomagnets
[ "Indranil Chakraborty", "Amogh Agrawal", "Kaushik Roy" ]
[ "cs.ET" ]
2,018
en
Computer Science
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0c8d90a6ef09bd3e4e73850bbd563f0815930d2f
subsection
3
12
A Voltage Controlled Stochastic Device
Ferromagnets are traditionally switched using the current induced spin transfer torque (STT) phenomenon . However, the STT phenomenon leads to sub optimal device performance due to various demerits, such as high write current requirement, switching asymmetry and shared read-write paths . Some of these issues are mitiga...
{ "cite_spans": [ { "arxiv_id": "", "doi": "10.1016/0304-8853(96)00062-5", "end": 105, "openalex_id": "https://openalex.org/W1964504828", "raw": "J. Slonczewski, “Current-driven excitation of magnetic multilayers,” Journal of Magnetism and Magnetic Materials, vol. 159, no. 1-2, pp. L...
10.1109/LMAG.2018.2839097
1803.01431
Design of a Low Voltage Analog-to-Digital Converter using Voltage Controlled Stochastic Switching of Low Barrier Nanomagnets
[ "Indranil Chakraborty", "Amogh Agrawal", "Kaushik Roy" ]
[ "cs.ET" ]
2,018
en
Computer Science
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6ef7d29305fc356ca765c1cf6de8948a9419e39d
subsection
4
12
Device Modeling
The device described in the previous section is modeled using the Landau-Lifshitz-Gilbert (LLG) equation for magnetization dynamics, which in its implicit form is expressed as :\vspace{-2.84526pt} \frac{\partial \hat{m}}{\partial t}=-|\gamma | \hat{m}\times H_{eff}+\alpha \hat{m}\times \frac{\partial \hat{m}}{\partial ...
{ "cite_spans": [ { "arxiv_id": "", "doi": "10.1109/tmag.2004.836740", "end": 489, "openalex_id": "https://openalex.org/W2107971449", "raw": "T. Gilbert, “Classics in magnetics a phenomenological theory of damping in ferromagnetic materials,” IEEE Transactions on Magnetics, vol. 40, ...
10.1109/LMAG.2018.2839097
1803.01431
Design of a Low Voltage Analog-to-Digital Converter using Voltage Controlled Stochastic Switching of Low Barrier Nanomagnets
[ "Indranil Chakraborty", "Amogh Agrawal", "Kaushik Roy" ]
[ "cs.ET" ]
2,018
en
Computer Science
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6c2045d83b7e472ca39aca2b2942485f32b56788
subsection
5
12
Device Modeling
Our aim here is to demonstrate the ADC operation, by sampling the telegraphic switching of the low-barrier magnet over a long duration. A non-linear \alpha _{ME} would only affect the switching characteristics of the magnet, and not the stochastic switching behavior averaged over a long duration. Thus, a linear approxi...
{ "cite_spans": [ { "arxiv_id": "", "doi": "10.48550/arxiv.1512.05428", "end": 476, "openalex_id": "https://openalex.org/W2329766060", "raw": "S. Manipatruni, D. E. Nikonov, and I. A. Young, “Spin-orbit logic with magnetoelectric nodes: A scalable charge mediated nonvolatile spintron...
10.1109/LMAG.2018.2839097
1803.01431
Design of a Low Voltage Analog-to-Digital Converter using Voltage Controlled Stochastic Switching of Low Barrier Nanomagnets
[ "Indranil Chakraborty", "Amogh Agrawal", "Kaushik Roy" ]
[ "cs.ET" ]
2,018
en
Computer Science
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f24c43d2eb7cec8410bb5f581645507b41c5eb14
subsection
6
12
Influence of
The role of thermal noise and external stimulus on the switching behavior fundamentally depends on E_B. For high E_B devices, the main driving force is the magnetoelectric interaction and thermal noise acts as an instigator of stochasticity. The variation of magnetization switching dynamics for different runs, confirms...
{ "cite_spans": [ { "arxiv_id": "", "doi": "10.1109/ted.2016.2568762", "end": 2290, "openalex_id": "https://openalex.org/W2404427863", "raw": "A. Sengupta, M. Parsa, B. Han, and K. Roy, “Probabilistic deep spiking neural systems enabled by magnetic tunnel junction,” IEEE Transactions...
10.1109/LMAG.2018.2839097
1803.01431
Design of a Low Voltage Analog-to-Digital Converter using Voltage Controlled Stochastic Switching of Low Barrier Nanomagnets
[ "Indranil Chakraborty", "Amogh Agrawal", "Kaushik Roy" ]
[ "cs.ET" ]
2,018
en
Computer Science
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dde2eb4398c8033ed3bbad41cf8709e2188d034c
subsection
7
12
Influence of
Analog input is fed at the ME oxide terminal of the MTJ and digital output is obtained at the counter output.]
{ "cite_spans": [] }
10.1109/LMAG.2018.2839097
1803.01431
Design of a Low Voltage Analog-to-Digital Converter using Voltage Controlled Stochastic Switching of Low Barrier Nanomagnets
[ "Indranil Chakraborty", "Amogh Agrawal", "Kaushik Roy" ]
[ "cs.ET" ]
2,018
en
Computer Science
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2aeb71439dcd9e15cc2f574ae6995804af7d6f60
subsection
8
12
Device Readout
The sensing circuit, shown in Fig. REF , consists of a reference MTJ (R_{Ref}) forms a voltage divider with the resistance of the MTJ (R_{MTJ}). The resistance in the P state (R_P) is lower than the resistance in the AP state (R_{AP}). Considering the magnetization of PL (m_{x,p}) pinned along x-axis, R_{MTJ} = R_{AP} ...
{ "cite_spans": [] }
10.1109/LMAG.2018.2839097
1803.01431
Design of a Low Voltage Analog-to-Digital Converter using Voltage Controlled Stochastic Switching of Low Barrier Nanomagnets
[ "Indranil Chakraborty", "Amogh Agrawal", "Kaushik Roy" ]
[ "cs.ET" ]
2,018
en
Computer Science
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8867f059d6a8f5504e763e14d057b723f0082c42
subsection
9
12
ADC Implementation
The influence of an analog voltage on the stochastic switching of the spintronic device, described earlier, was used to implement a low-voltage 8-bit ADC. An increase in the input voltage decreases \langle m_x \rangle , thus increasing the probability of the magnet being in AP state. When R_{MTJ} = R_{AP} (R_P) or m_x ...
{ "cite_spans": [ { "arxiv_id": "", "doi": "10.1109/vlsit.2015.7223662", "end": 756, "openalex_id": "https://openalex.org/W1515011322", "raw": "W. H. Choi, Y. Lv, H. Kim, J.-P. Wang, and C. H. Kim, “An 8-bit analog-to-digital converter based on the voltage-dependent switching probabi...
10.1109/LMAG.2018.2839097
1803.01431
Design of a Low Voltage Analog-to-Digital Converter using Voltage Controlled Stochastic Switching of Low Barrier Nanomagnets
[ "Indranil Chakraborty", "Amogh Agrawal", "Kaushik Roy" ]
[ "cs.ET" ]
2,018
en
Computer Science
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22b893c5ade87402526ad81f71592ea4be1b8ea9
subsection
10
12
ADC Implementation
Moreover, since E_B = K_{u2}V, low E_B magnets are area-eficient. As a result, circuits that utilize low E_B devices promise to achieve lower power and compact area.
{ "cite_spans": [] }
10.1109/LMAG.2018.2839097
1803.01431
Design of a Low Voltage Analog-to-Digital Converter using Voltage Controlled Stochastic Switching of Low Barrier Nanomagnets
[ "Indranil Chakraborty", "Amogh Agrawal", "Kaushik Roy" ]
[ "cs.ET" ]
2,018
en
Computer Science
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72a4375a7686673cb3846ba27059e7ef0aa53015
subsection
11
12
Conclusion
We explore a voltage controlled stochastic spintronic device based on superparamagnetic nanomagnets as a possible candidate to perform approximate low voltage Analog-to-Digital conversions. The simulated device achieves voltage controllable telegraphic switching at a ns scale which enables digital encoding of analog in...
{ "cite_spans": [] }
10.1109/LMAG.2018.2839097
1803.01431
Design of a Low Voltage Analog-to-Digital Converter using Voltage Controlled Stochastic Switching of Low Barrier Nanomagnets
[ "Indranil Chakraborty", "Amogh Agrawal", "Kaushik Roy" ]
[ "cs.ET" ]
2,018
en
Computer Science
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9f23b589ed8ef5e774639805242e16d480b8caa4
abstract
0
31
Abstract
Education in the practical applications of logic and proving such as the formal specification and verification of computer programs is substantially hampered by the fact that most time and effort that is invested in proving is actually wasted in vain: because of errors in the specifications respectively algorithms that...
{ "cite_spans": [] }
10.4204/EPTCS.267.8
1803.01472
Teaching the Formalization of Mathematical Theories and Algorithms via the Automatic Checking of Finite Models
[ "Wolfgang Schreiner", "Alexander Brunhuemer", "Christoph Fürst" ]
[ "cs.LO" ]
2,018
en
Computer Science
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0c351df6681abcb89008f29f3c473bbc69289003
subsection
1
31
Introduction
From a student's perspective, education in the practical applications of logic and proving (such as the formal specification of a computational problem, the development of an algorithm that is supposed to solve this problem, and the formal verification that the algorithm indeed implements the specification) is often a ...
{ "cite_spans": [ { "arxiv_id": "", "doi": "10.1007/s00165-008-0069-4", "end": 3338, "openalex_id": "https://openalex.org/W2007452988", "raw": "Wolfgang Schreiner (2009): The RISC ProofNavigator: A Proving Assistant for Program Verification in the Classroom. Formal Aspects of Computi...
10.4204/EPTCS.267.8
1803.01472
Teaching the Formalization of Mathematical Theories and Algorithms via the Automatic Checking of Finite Models
[ "Wolfgang Schreiner", "Alexander Brunhuemer", "Christoph Fürst" ]
[ "cs.LO" ]
2,018
en
Computer Science
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37c359530575212d73124369889a78cefed172b2
subsection
2
31
Introduction
RISCAL has been designed in such a way thatevery type has (arbitrarily many but) only finitely many values, and thus every language construct is executable, and thus every constant, function, predicate, theorem, procedure can be evaluated.While every RISCAL type such as \mathbb {N}[n] is finite, it may depend on a co...
{ "cite_spans": [ { "arxiv_id": "", "doi": "", "end": 2505, "openalex_id": "", "raw": "Jon Barwise Dave Barker-Plummer, John Etchemendy & Albert Liu (2008): Tarski's World: Revised and Expanded. CSLI Publications, Stanford, CA, USA.", "source_ref_id": "f010cd13b5ebdd524994bd953...
10.4204/EPTCS.267.8
1803.01472
Teaching the Formalization of Mathematical Theories and Algorithms via the Automatic Checking of Finite Models
[ "Wolfgang Schreiner", "Alexander Brunhuemer", "Christoph Fürst" ]
[ "cs.LO" ]
2,018
en
Computer Science
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6ab2aa6bb5d7f95e9525e7057374c21535232590
subsection
3
31
Introduction
As for real programming languages with support for formal specification, the programming language WhyML of the verification environment Why3  and Microsoft's programming language Dafny  provide a rich variety of built-in specification constructs; however, both WhyML and Dafny do not support model checking. Also for ind...
{ "cite_spans": [ { "arxiv_id": "", "doi": "", "end": 307, "openalex_id": "", "raw": "Jean-Christophe Filliâtre & Andrei Paskevich (2013): Why3 — Where Programs Meet Provers. In M. Felleisen & P. Gardner, editors: ESOP 2013, Rome, Italy, March 16-24, 2013, Lecture Notes in Computer S...
10.4204/EPTCS.267.8
1803.01472
Teaching the Formalization of Mathematical Theories and Algorithms via the Automatic Checking of Finite Models
[ "Wolfgang Schreiner", "Alexander Brunhuemer", "Christoph Fürst" ]
[ "cs.LO" ]
2,018
en
Computer Science
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920f8c7169a0d9ba3c0422e780a62abfedfb1cb0
subsection
4
31
The RISCAL Language and System
In this section, we give a short account of the RISCAL language and software system; for more details, see the tutorial and reference manual .
{ "cite_spans": [ { "arxiv_id": "", "doi": "", "end": 142, "openalex_id": "", "raw": "Wolfgang Schreiner (2017): The RISC Algorithm Language (RISCAL) — Tutorial and Reference Manual (Version 1.0). Technical Report, RISC, Johannes Kepler University, Linz, Austria. Download from .", ...
10.4204/EPTCS.267.8
1803.01472
Teaching the Formalization of Mathematical Theories and Algorithms via the Automatic Checking of Finite Models
[ "Wolfgang Schreiner", "Alexander Brunhuemer", "Christoph Fürst" ]
[ "cs.LO" ]
2,018
en
Computer Science
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6e62419f3c425e16aa3c520726f40e4d4ef12b5c
subsection
5
31
System
The user interface of the RISCAL software system is depicted in Figure REF ; it contains an editor frame for RISCAL specifications on the left and the control widgets and output frame of the checker on the right. The RISCAL specification language is based on a statically typed variant of first order predicate logic. On...
{ "cite_spans": [] }
10.4204/EPTCS.267.8
1803.01472
Teaching the Formalization of Mathematical Theories and Algorithms via the Automatic Checking of Finite Models
[ "Wolfgang Schreiner", "Alexander Brunhuemer", "Christoph Fürst" ]
[ "cs.LO" ]
2,018
en
Computer Science
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d62f3760d4ed021b201e3199ce68f20c8cc1aab2
subsection
6
31
Language
RISCAL specifications consist of declarations of the following kinds of entities:Types type I = T introduces a named type I defined by the type expression T; types include booleans, integers, sets, tuples, records, arrays, and maps (partial functions). All types are finite, e.g., for integer constants N and M with N\le...
{ "cite_spans": [] }
10.4204/EPTCS.267.8
1803.01472
Teaching the Formalization of Mathematical Theories and Algorithms via the Automatic Checking of Finite Models
[ "Wolfgang Schreiner", "Alexander Brunhuemer", "Christoph Fürst" ]
[ "cs.LO" ]
2,018
en
Computer Science
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83498cf09a8e806d619992294c2cde708a53d210
subsection
7
31
Language
These entities may be also defined recursively; by an annotation decreases E a termination measure E is stated, i.e., an expression E that evaluates to a natural number which is decreased in every recursive invocation.The algorithmic language RISCAL also supports non-deterministic expression evaluations respectively co...
{ "cite_spans": [] }
10.4204/EPTCS.267.8
1803.01472
Teaching the Formalization of Mathematical Theories and Algorithms via the Automatic Checking of Finite Models
[ "Wolfgang Schreiner", "Alexander Brunhuemer", "Christoph Fürst" ]
[ "cs.LO" ]
2,018
en
Computer Science
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37d561b6c18a2766e34ada37748a6ede88761fe8
subsection
8
31
Example
The specification listed in Figure REF introduces the mathematical theory of the greatest common divisor and its computation by the Euclidean algorithm; this theory is restricted to the domain of all natural numbers less than equal N:The theory first introduces the undefined constant N which is then used as the domain ...
{ "cite_spans": [] }
10.4204/EPTCS.267.8
1803.01472
Teaching the Formalization of Mathematical Theories and Algorithms via the Automatic Checking of Finite Models
[ "Wolfgang Schreiner", "Alexander Brunhuemer", "Christoph Fürst" ]
[ "cs.LO" ]
2,018
en
Computer Science
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693552c43e6054ddfd4147eb187b234ce334a498
subsection
9
31
Example
Its contract specified by the clauses requires and ensures states that the procedure behaves exactly as the implicitly defined function; the loop annotations invariant and decreases describe essential knowledge for proving the total correctness of the procedure (here the automatically introduced constants old_a and old...
{ "cite_spans": [] }
10.4204/EPTCS.267.8
1803.01472
Teaching the Formalization of Mathematical Theories and Algorithms via the Automatic Checking of Finite Models
[ "Wolfgang Schreiner", "Alexander Brunhuemer", "Christoph Fürst" ]
[ "cs.LO" ]
2,018
en
Computer Science
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1cc219fd05d6cf076940e73747bb6674d810220c
subsection
10
31
Evaluation, Execution, and Checking
Whenever the user saves a specification, it is automatically syntactically and semantically processed, i.e., parsed, type-checked, and translated into an executable internal representation (see below for more details on the implementation). Errors are reported by graphical markers in the editor frame and by textual mes...
{ "cite_spans": [] }
10.4204/EPTCS.267.8
1803.01472
Teaching the Formalization of Mathematical Theories and Algorithms via the Automatic Checking of Finite Models
[ "Wolfgang Schreiner", "Alexander Brunhuemer", "Christoph Fürst" ]
[ "cs.LO" ]
2,018
en
Computer Science
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492af45fe1515e07f7358a8305e6e4df0c78192a
subsection
11
31
Implementation
RISCAL has been implemented in Java (using the Eclipse Standard Widget Toolkit SWT for its graphical user interface). The executable internal representation of a specification is essentially a Java version of a denotational semantics of the specification, implemented on top of the lambda expressions introduced in Java ...
{ "cite_spans": [] }
10.4204/EPTCS.267.8
1803.01472
Teaching the Formalization of Mathematical Theories and Algorithms via the Automatic Checking of Finite Models
[ "Wolfgang Schreiner", "Alexander Brunhuemer", "Christoph Fürst" ]
[ "cs.LO" ]
2,018
en
Computer Science
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b1aaa99fb3da1772efbba3515faa3d0f001d5bd5
subsection
12
31
Implementation
The multi-threaded version can be selected by the check box Multi-Threaded where by the input box Threads the number of worker threads can be selected. These threads run on the local computer and iteratively request from the main thread new inputs to which the chosen operation is to be applied; thus the domain of input...
{ "cite_spans": [ { "arxiv_id": "", "doi": "", "end": 969, "openalex_id": "", "raw": "Wolfgang Schreiner (2017): The RISC Algorithm Language (RISCAL) — Tutorial and Reference Manual (Version 1.0). Technical Report, RISC, Johannes Kepler University, Linz, Austria. Download from .", ...
10.4204/EPTCS.267.8
1803.01472
Teaching the Formalization of Mathematical Theories and Algorithms via the Automatic Checking of Finite Models
[ "Wolfgang Schreiner", "Alexander Brunhuemer", "Christoph Fürst" ]
[ "cs.LO" ]
2,018
en
Computer Science
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ae8f4e4149ab67cd320187a3bb807526f134b742
subsection
13
31
Proof-Based Verification
Since RISCAL is based on checking rather than deriving and proving verification conditions, there is not any direct connection of RISCAL to a verification calculus like Hoare's axiomatic system or Dijkstra's predicate transformers. However, as will be sketched in the following section, we plan as future work to integra...
{ "cite_spans": [] }
10.4204/EPTCS.267.8
1803.01472
Teaching the Formalization of Mathematical Theories and Algorithms via the Automatic Checking of Finite Models
[ "Wolfgang Schreiner", "Alexander Brunhuemer", "Christoph Fürst" ]
[ "cs.LO" ]
2,018
en
Computer Science
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974eed9e321b2b09705350a8e8c0d2dadbf32ad5
subsection
14
31
Using RISCAL in Education
RISCAL shall support the education in formal logic with emphasis on the formal specification and verification of programs respectively algorithms (abstract programs). The particular goal of RISCAL is to give students immediate feedback about the interpretation and adequacy of formal definitions and specifications befor...
{ "cite_spans": [] }
10.4204/EPTCS.267.8
1803.01472
Teaching the Formalization of Mathematical Theories and Algorithms via the Automatic Checking of Finite Models
[ "Wolfgang Schreiner", "Alexander Brunhuemer", "Christoph Fürst" ]
[ "cs.LO" ]
2,018
en
Computer Science
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a100389e66eaef8c0e0f344a2d126a933ccb5287
subsection
15
31
Using RISCAL in Education
If none of these theorems hold, the specification is trivial: a proof of correctness of an algorithm with respect to the specification typically quickly succeeds but is pointless (indeed small logical errors may lead to the definitions of postconditions that are equivalent to true). Check the validity of \forall x. P(...
{ "cite_spans": [] }
10.4204/EPTCS.267.8
1803.01472
Teaching the Formalization of Mathematical Theories and Algorithms via the Automatic Checking of Finite Models
[ "Wolfgang Schreiner", "Alexander Brunhuemer", "Christoph Fürst" ]
[ "cs.LO" ]
2,018
en
Computer Science
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4a70fe64afc32ec77042215de8f7f7b88264c3a1
subsection
16
31
Using RISCAL in Education
The corresponding RISCAL specification is based on the following domains:val N:ℕ; val M:ℕ;type index = ℤ[-N,N]; type elem  = ℤ[-M,M]; type array = Array[N,elem];Rather than basing our specifications on a single integer type, we use two constants N and M to bound the types index of array indices/lengths and the type ele...
{ "cite_spans": [] }
10.4204/EPTCS.267.8
1803.01472
Teaching the Formalization of Mathematical Theories and Algorithms via the Automatic Checking of Finite Models
[ "Wolfgang Schreiner", "Alexander Brunhuemer", "Christoph Fürst" ]
[ "cs.LO" ]
2,018
en
Computer Science
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f1916d80ff82536d477ca836c000f7869e6d7439
subsection
17
31
Using RISCAL in Education
Theorems postNotValid (the postcondition is not generally valid), postSat (the postcondition is satisfiable), and resultUnique (the postcondition determines the result uniquely) are individually verified by corresponding calls of the checker (in non-deterministic mode with silent execution):Executing postNotValid(Array...
{ "cite_spans": [] }
10.4204/EPTCS.267.8
1803.01472
Teaching the Formalization of Mathematical Theories and Algorithms via the Automatic Checking of Finite Models
[ "Wolfgang Schreiner", "Alexander Brunhuemer", "Christoph Fürst" ]
[ "cs.LO" ]
2,018
en
Computer Science
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2ae08f4b07f7ad67348a417be8f73d7f7e0c4efe
subsection
18
31
Using RISCAL in Education
0 ≤ k ∧ k < i ⇒ m ≥ a[k]);fun Termination(a:array, n:index, m:elem, i:index):index = n-i;By checking the procedure (activity 3a), we verify its correctness with respect to the specification, and (partially) validate the adequacy of invariant and termination term:Executing maxProc(Array[ℤ],ℤ) with all 875 inputs.Executi...
{ "cite_spans": [ { "arxiv_id": "", "doi": "10.1007/s00165-008-0069-4", "end": 1968, "openalex_id": "https://openalex.org/W2007452988", "raw": "Wolfgang Schreiner (2009): The RISC ProofNavigator: A Proving Assistant for Program Verification in the Classroom. Formal Aspects of Computi...
10.4204/EPTCS.267.8
1803.01472
Teaching the Formalization of Mathematical Theories and Algorithms via the Automatic Checking of Finite Models
[ "Wolfgang Schreiner", "Alexander Brunhuemer", "Christoph Fürst" ]
[ "cs.LO" ]
2,018
en
Computer Science
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d301d6ef393d55fe100b6963dbeb0e7515283818
subsection
19
31
Sample Specifications
Since the release of the first version of RISCAL in March 2017, we have started to develop first prototypes of formally checked specifications. They are intended to serve as the nucleus of a future comprehensive library and accompanying lecture materials to be used in the class room and for self-instructed learning in ...
{ "cite_spans": [] }
10.4204/EPTCS.267.8
1803.01472
Teaching the Formalization of Mathematical Theories and Algorithms via the Automatic Checking of Finite Models
[ "Wolfgang Schreiner", "Alexander Brunhuemer", "Christoph Fürst" ]
[ "cs.LO" ]
2,018
en
Computer Science
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5557e1ea6e77dbd325eb1e5496a08347e0c0655c
subsection
20
31
Number Theory
In , we describe the application of RISCAL to number-theoretic algorithms that arise in, e.g., cryptography. In such algorithms, prime numbers play an important role; thus as our first example we pick the problem of generating, for a given bound n\in \mathbb {N}, all prime numbers less than equal n: formally, we wish t...
{ "cite_spans": [ { "arxiv_id": "", "doi": "", "end": 108, "openalex_id": "", "raw": "Christoph Fürst & Wolfgang Schreiner (2017): Formalization of Two Algorithms Arising in Number Theory. Technical Report, RISC, Johannes Kepler University, Linz, Austria. To appear.", "source_r...
10.4204/EPTCS.267.8
1803.01472
Teaching the Formalization of Mathematical Theories and Algorithms via the Automatic Checking of Finite Models
[ "Wolfgang Schreiner", "Alexander Brunhuemer", "Christoph Fürst" ]
[ "cs.LO" ]
2,018
en
Computer Science
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5ccb624197a02c773d8f36ed0b3e515a6d2b5f72
subsection
21
31
Number Theory
But then o<m is also a divisor of n, i.e., m is not the least divisor, which contradicts our assumption.While this argument can be easily understood by students, we can give the theorem further credibility by formulating it in RISCAL:theorem leastProperDivisor(n:nat,m:nat) ⇔n ≥ 2 ∧ m > 1 ∧ divides(m,n) ∧ (∀m0:nat. m0 >...
{ "cite_spans": [] }
10.4204/EPTCS.267.8
1803.01472
Teaching the Formalization of Mathematical Theories and Algorithms via the Automatic Checking of Finite Models
[ "Wolfgang Schreiner", "Alexander Brunhuemer", "Christoph Fürst" ]
[ "cs.LO" ]
2,018
en
Computer Science
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895202ade11be4fb66f3fb75352970e3a23728b5
subsection
22
31
Sieve of Eratosthenes (Set-Based)
The following algorithm computes for given n\in \mathbb {N} the set P of all values p\le n with \mathit {isPrime}(p,n):[H] [1] P = \lbrace p\ |\ p\in \mathbb {N} \wedge p\le n \wedge \mathit {isPrime}(p) \rbrace P \leftarrow \emptyset C \leftarrow \lbrace 2,\ldots ,n\rbrace C \ne \emptyset p \leftarrow \min (C) P\l...
{ "cite_spans": [] }
10.4204/EPTCS.267.8
1803.01472
Teaching the Formalization of Mathematical Theories and Algorithms via the Automatic Checking of Finite Models
[ "Wolfgang Schreiner", "Alexander Brunhuemer", "Christoph Fürst" ]
[ "cs.LO" ]
2,018
en
Computer Science
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c979578150cde58e7b5c71e3346bc6bd92729b3d
subsection
23
31
Sieve of Eratosthenes (Array-Based)
The following algorithm computes for given n\in \mathbb {N} the Boolean array P of length n such that P[p] has value true if and only if the property \mathit {isPrime}(p) holds:[H] [1] \forall p\in \mathbb {N}.\ p\le n \Rightarrow (P[p] = \textrm {T} \Leftrightarrow \mathit {isPrime}(p)) P \leftarrow ({\rm T},{\rm T},\...
{ "cite_spans": [] }
10.4204/EPTCS.267.8
1803.01472
Teaching the Formalization of Mathematical Theories and Algorithms via the Automatic Checking of Finite Models
[ "Wolfgang Schreiner", "Alexander Brunhuemer", "Christoph Fürst" ]
[ "cs.LO" ]
2,018
en
Computer Science
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5da3c224848e8ecec0f62ba17253f3acea029f37
subsection
24
31
Sieve of Eratosthenes (Array-Based)
Checking also this algorithm with N=30Executing SieveOfEratosthenesArray(ℤ) with all 31 inputs.Execution completed for ALL inputs (171 ms, 31 checked, 0 inadmissible).again validates both that the algorithm satisfies its contract and that the loop annotations are correct (if they should not be sufficient for a proof, t...
{ "cite_spans": [] }
10.4204/EPTCS.267.8
1803.01472
Teaching the Formalization of Mathematical Theories and Algorithms via the Automatic Checking of Finite Models
[ "Wolfgang Schreiner", "Alexander Brunhuemer", "Christoph Fürst" ]
[ "cs.LO" ]
2,018
en
Computer Science
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15abde23f2314ec2c7acb43f565d98a1a966852a
subsection
25
31
Discrete Mathematics
In , chosen theories from discrete mathematics have been formalized in RISCAL. This includes the development of the mathematical theories, the specification of algorithms and their annotation with meta-information. The concepts have been validated on small finite domains which should work as a ground layer for further ...
{ "cite_spans": [ { "arxiv_id": "", "doi": "", "end": 78, "openalex_id": "", "raw": "Alexander Brunhuemer (2017): Validating the Formalization of Theories and Algorithms of Discrete Mathematics by the Computer-Supported Checking of Finite Models. Bachelor thesis, Research Institute f...
10.4204/EPTCS.267.8
1803.01472
Teaching the Formalization of Mathematical Theories and Algorithms via the Automatic Checking of Finite Models
[ "Wolfgang Schreiner", "Alexander Brunhuemer", "Christoph Fürst" ]
[ "cs.LO" ]
2,018
en
Computer Science
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5bfe3dd7b32f0ba2746f4d8673f43b924448f67f
subsection
26
31
Problem Specification and Implicit Definition
Our problem is to compute the transitive closure of a given binary relation based on the following definition.Definition 3 (Transitive Closure) S is the transitive closure of relation R, if S is the smallest transitive relation S that contains R, i.e., S is a subset of every transitive relation that contains R.The corr...
{ "cite_spans": [] }
10.4204/EPTCS.267.8
1803.01472
Teaching the Formalization of Mathematical Theories and Algorithms via the Automatic Checking of Finite Models
[ "Wolfgang Schreiner", "Alexander Brunhuemer", "Christoph Fürst" ]
[ "cs.LO" ]
2,018
en
Computer Science
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d7a564bc12f36f1d0d31f25441bddb845ae092ee
subsection
27
31
Explicit (Recursive) Definition
A natural approach to compute the transitive closure of a relation r as a recursive function is based on the following fundamental steps:If r is transitive, then we are finished and r is our result. Otherwise r contains some pairs \langle x,z \rangle and \langle z,y \rangle that violate the transitivity of r in the se...
{ "cite_spans": [] }
10.4204/EPTCS.267.8
1803.01472
Teaching the Formalization of Mathematical Theories and Algorithms via the Automatic Checking of Finite Models
[ "Wolfgang Schreiner", "Alexander Brunhuemer", "Christoph Fürst" ]
[ "cs.LO" ]
2,018
en
Computer Science
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8d565a4f2b614f12e931e446e79281aa77f15222
subsection
28
31
Procedural Definition
Another algorithm which may be easiest expressed as a procedure is based on the following main steps:We initialize the result variable \mathit {res} with the empty set and an auxiliary variable \mathit {new} with r. We choose some pair x\in \mathit {new} and check for every pair y\in \mathit {res}, if the combination ...
{ "cite_spans": [] }
10.4204/EPTCS.267.8
1803.01472
Teaching the Formalization of Mathematical Theories and Algorithms via the Automatic Checking of Finite Models
[ "Wolfgang Schreiner", "Alexander Brunhuemer", "Christoph Fürst" ]
[ "cs.LO" ]
2,018
en
Computer Science
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fda5bffdeec275cf5c57e228bbb3152b32bc5b5b
subsection
29
31
Procedural Definition
Since the loop terminates when \mathit {new} is the empty set, the third invariant implies that then \mathit {res} is transitive and thus itself the transitive closure.The correctness of the invariant of the outer proof has to be verified with the help of the invariant of the inner loop:invariant new = old_new∪ { ⟨y0.1...
{ "cite_spans": [] }
10.4204/EPTCS.267.8
1803.01472
Teaching the Formalization of Mathematical Theories and Algorithms via the Automatic Checking of Finite Models
[ "Wolfgang Schreiner", "Alexander Brunhuemer", "Christoph Fürst" ]
[ "cs.LO" ]
2,018
en
Computer Science
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747f909a0ba12c577b30015ab436ce8e1573d554
subsection
30
31
Conclusions and Further Work
RISCAL is still in its infancy with first contents having been developed and first classroom experience having been gained; subsequent experience with the use of RISCAL will shape the further development of language and system. This will proceed along several possible strands:We will automatize the generation of formul...
{ "cite_spans": [] }
10.4204/EPTCS.267.8
1803.01472
Teaching the Formalization of Mathematical Theories and Algorithms via the Automatic Checking of Finite Models
[ "Wolfgang Schreiner", "Alexander Brunhuemer", "Christoph Fürst" ]
[ "cs.LO" ]
2,018
en
Computer Science
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285d60ace1fa3c274fe31d40a3ead7afb7cdb9fb
abstract
0
224
Abstract
A fundamental tool in noncommutative geometry is Connes' character formula. This formula is used in an essential way in the applications of noncommutative geometry to index theory and to the spectral characterisation of manifolds. A non-compact space is modelled in noncommutative geometry by a non-unital spectral trip...
{ "cite_spans": [] }
1803.01551
The Connes character formula for locally compact spectral triples
[ "Fedor Sukochev", "Dmitriy Zanin" ]
[ "math.OA" ]
2,018
en
Mathematics
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26dd9f931966c4032eb550ac18e77b14bc224cc7
subsection
1
224
Introduction
One of the fundamental tools in noncommutative geometry is the Chern character. The Connes Character Formula (also known as the Hochschild character theorem) provides an expression for the class of the Chern character in Hochschild cohomology, and it is an important tool in the computation of the Chern character. The f...
{ "cite_spans": [ { "arxiv_id": "", "doi": "10.1007/978-3-0348-9102-8_4", "end": 526, "openalex_id": "https://openalex.org/W2112440022", "raw": "Connes A., Moscovici H., The local index formula in noncommutative geometry. Geom. Funct. Anal. 5 (1995), no. 2, 174–243.", "source_r...
1803.01551
The Connes character formula for locally compact spectral triples
[ "Fedor Sukochev", "Dmitriy Zanin" ]
[ "math.OA" ]
2,018
en
Mathematics
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8b0bdef553ee9e4c579dbbbca9496323310dde99
subsection
2
224
Introduction
The conventional definition of a non-compact spectral triple is to replace the condition that (1+D^2)^{-1/2} be compact with the assumption that for all a \in \mathcal {A} the operator a(1+D^2)^{-1/2} is compact.This raises an important question: is the Connes Character Formula true for locally compact spectral triples...
{ "cite_spans": [] }
1803.01551
The Connes character formula for locally compact spectral triples
[ "Fedor Sukochev", "Dmitriy Zanin" ]
[ "math.OA" ]
2,018
en
Mathematics
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