[et_pb_section fb_built=\u201d1\u2033 admin_label=\u201dsection\u201d _builder_version=\u201d4.16\u2033 custom_padding=\u201d40px|||||\u201d da_disable_devices=\u201doff|off|off\u201d global_colors_info=\u201d{}\u201d da_is_popup=\u201doff\u201d da_exit_intent=\u201d off" da_has_close="on" da_alt_close=\u201doff\u201d da_dark_close=\u201doff\u201d da_not_modal=\u201don\u201d da_is_singular=\u201doff\u201d da_with_loader=\u201doff\u201d da_has_shadow=\u201don\u201d][et_pb_row admin_label=\u201drow\u201d _builder_version=\u201d4.16\u2033 background_size=\u201dinitial\u201d background_position= \u201ctop_left\u201d background_repeat=\u201crepeat\u201d custom_padding=\u201d3px||3px|||\u201d global_colors_info=\u201d{}\u201d][et_pb_column type=\u201d4_4\u2033 _builder_version=\u201d4.16\u2033 custom_padding=\u201d|||\u201d global_colors_info=\u201d{}\u201d custom_padding__hover=\u201d|||\u201d ][et_pb_text admin_label=\u201cText\u201d _builder_version=\u201c4.27.3\u2033 background_size=\u201cinitial\u201d background_position=\u201ctop_left\u201d background_repeat=\u201crepeat\u201d global_colors_info=\u201c{}\u201c]<\/p>\n
Structural complexity, or more precisely: low structural complexity, is a quality characteristic of a system or software design. If the system or software design is structurally complex, then it is<\/p>\n
So, controlling (preferrably: minimizing) structural complexity in the system and software design process is of utmost importance. Suppose we have two possible system or software architecture designs: how can we decide which design is better because its structural complexity is lower?<\/p>\n
How can the structural complexity of a system or software be quantified?<\/strong><\/p>\n A system architecture (and analogously a software architecture) is characterized by several aspects, namely<\/p>\n We seek a structural complexity measure by means of which<\/p>\n can be evaluated.<\/p>\n Moreover, we want to take into account an important feature of safety-critical systems: the higher the required safety integrity level of a system part, the less structurally complex that system part is allowed to be.<\/p>\n Section 2 describes how we propose to measure and reduce structural complexity. Section 3 summarizes the main ideas of this article. <\/p>\n The content of this Section 2.1 and the following Section 2.2 is largely based on [1].<\/p>\n As had already been mentioned in the introduction, structural complexity is about system- or software-internal dependencies: the more dependencies there are, the higher the structural complexity. Fig. 1 shows an example of a dependency tree<\/strong>\u00a0and the associated adjacency matrix<\/strong> A<\/em> (also known in the English literature as the \u201cDependency Structure Matrix\u201d, or \u201cDSM\u201d for short).<\/p>\n [\/et_pb_text][et_pb_image src=\u201dhttps:\/\/autoconformance.com\/wp-content\/uploads\/2025\/01\/Image1.png\u201d title_text=\u201dImage1\u2033 show_in_lightbox=\u201don\u201d _builder_version=\u201d4.27.3\u2033 _module_preset=\u201ddefault\u201d global_colors_info=\u201d{}\u201d][\/et_pb_image][et_pb_text _builder_version=\u201d4.27.3\u2033 _module_preset=\u201ddefault\u201d custom_padding=\u201d3px|||||\u201d global_colors_info="{}"]<\/p>\n Fig. 1:<\/strong> Dependency tree (left) and associated DSM (right), taken from [1], p. 34.<\/p>\n [\/et_pb_text][et_pb_text _builder_version=\u201c4.27.3\u2033 _module_preset=\u201cdefault\u201d custom_padding=\u201c||0px|||\u201c global_colors_info="{}"]<\/p>\n Note that<\/p>\n can all be represented by dependency trees and DSMs.<\/p>\n Affecting or being affected by other system elements need not be direct \u2013 it may occur through intermediate connections. To capture this, we propose a structural complexity measure that takes into account both direct and indirect connections in the system architecture.<\/p>\n In safety-critical systems, one cannot assume that all system elements have the same safety integrity. To take this fact into account, we weight the system elements according to their safety integrity. The corresponding weighting matrix is given by<\/p>\n [\/et_pb_text]Edit global_colors_info=\u201d{}\u201d][\/et_pb_image][et_pb_text _builder_version=\u201d4.27.3\u2033 _module_preset=\u201ddefault\u201d global_colors_info=\u201d{}\u201d]<\/p>\n As for the structural complexity measure<\/strong> c<\/em>\u00a0we propose to calculate it as the Frobenius norm of the column-weighted extended reachability matrix<\/strong> R<\/em>ext<\/small><\/sup>\u00a0where the columns are weighted according to the safety integrity of the system elements:<\/p>\n [\/et_pb_text]Edit global_colors_info=\u201d{}\u201d][\/et_pb_image][et_pb_text _builder_version=\u201d4.27.3\u2033 _module_preset=\u201ddefault\u201d global_colors_info=\u201d{}\u201d]<\/p>\n Which values should be chosen for the scalar weights?<\/strong><\/p>\n In Tables 1 and 2 below we provide recommendations depending on whether<\/p>\n are used to classify the system elements.<\/p>\n [\/et_pb_text][et_pb_text _builder_version=\u201c4.27.3\u2033 _module_preset=\u201cdefault\u201c global_colors_info=\u201c{}\u201c]<\/p>\n Table 1:<\/strong> Recommended weighting of system elements with SIL classification<\/p>\n [\/et_pb_text][et_pb_text _builder_version=\u201c4.27.3\u2033 _module_preset=\u201cdefault\u201d custom_padding=\u201c||0px|||\u201c global_colors_info="{}"]<\/p>\n Weight <\/strong>w<\/em>j<\/sub> <\/p>\n [\/et_pb_text][et_pb_text _builder_version=\u201c4.27.3\u2033 _module_preset=\u201cdefault\u201c global_colors_info=\u201c{}\u201c]<\/p>\n Table 2:<\/strong> Recommended weighting of system elements with ASIL classification<\/p>\n [\/et_pb_text][et_pb_text _builder_version=\u201c4.27.3\u2033 _module_preset=\u201cdefault\u201c global_colors_info=\u201c{}\u201c]<\/p>\n [\/et_pb_text][et_pb_text _builder_version=\u201c4.27.3\u2033 _module_preset=\u201cdefault\u201d custom_padding=\u201c||0px|||\u201c global_colors_info="{}"]<\/p>\n ISO 26262:2018, Part 9, allows for a so-called \"ASIL decomposition\" of safety requirements. In practice, this idea is extended to ASIL decompositions of system elements and system functions that meet the decomposed safety requirements. The weights in Table 2 were chosen such that any kind of ASIL decomposition is \"complexity-neutral\". An example of such an ASIL decomposition is shown in Fig. 2 a) and b). Note that the structural complexity in a) and b) is the same.<\/p>\n [\/et_pb_text][et_pb_image src=\u201chttps:\/\/autoconformance.com\/wp-content\/uploads\/2025\/01\/Beitrag10_Abb2a.png\u201d title_text=\u201cBeitrag10_Abb2a\u201d show_in_lightbox=\u201con\u201d _builder_version=\u201c4.27.3\u2033 _module_preset=\u201cdefault\u201c global_colors_info=\u201d{}\u201d][\/et_pb_image][et_pb_image src=\u201dhttps:\/\/autoconformance.com\/wp-content\/uploads\/2025\/01\/Beitrag10_Abb2b.png\u201d title_text=\u201dBeitrag10_Abb2b\u201d show_in_lightbox=\u201don\u201d _builder_version=\u201d4.27.3\u2033 _module_preset="default" global_colors_info=\u201d{}\u201d][\/et_pb_image][et_pb_text _builder_version=\u201d4.27.3\u2033 _module_preset=\u201ddefault\u201d global_colors_info=\u201d{}\u201d]<\/p>\n Fig. 2:<\/strong> Dependency trees (a) without and (b) with ASIL decomposition.<\/p>\n [\/et_pb_text][et_pb_text _builder_version=\u201c4.27.3\u2033 _module_preset=\u201cdefault\u201c global_colors_info=\u201c{}\u201c]<\/p>\n Assume we have a candidate of the static system architecture. For this candidate system architecture we determine its c<\/em>-value. Reducing the structural complexity of the system design means looking for alternative architectural designs with lower c<\/em>-values (by \u201ctrial and error\u201d).<\/p>\n Low c<\/em>-values are typically attained by<\/p>\n [\/et_pb_text][et_pb_text admin_label=\u201dText\u201d _builder_version=\u201d4.27.3\u2033 background_size=\u201dinitial\u201d background_position=\u201dtop_left\u201d background_repeat=\u201drepeat\u201d custom_padding=\u201d||0px|||\u201d global_colors_info="{}"]<\/p>\n In this article, a structural complexity measure c<\/em> was taken from network theory and adapted to safety-critical systems. This measure c<\/em> for structural complexity is based on a column-weighted extended reachability matrix. Such a reachability matrix can be determined for<\/p>\n The following properties expected of a structural complexity measure<\/strong> are met by c<\/em>:<\/p>\n <\/p>\n [1] Sturtevant, DJ: \u201cSystem design and the cost of architectural complexity\u201d, 2013, https:\/\/dspace.mit.edu\/handle\/1721.1\/79551<\/a><\/p>\n\n
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<\/strong><\/p>\n2. What is the most appropriate measure of structural complexity?<\/strong><\/h2>\n
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2.1. Structural Complexity of What?<\/strong><\/h3>\n
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2.2. Recommended Structural Complexity Measure<\/strong><\/h3>\n
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\n Safety Integrity Level (SIL)
<\/strong>of the system element s<\/em>j<\/sub><\/td>\n\n
<\/strong>of the system element s<\/em>j<\/sub><\/p>\n<\/td>\n<\/tr>\n\n None \/ QM<\/td>\n 1<\/td>\n<\/tr>\n \n SIL 1<\/td>\n 2<\/td>\n<\/tr>\n \n SIL 2<\/td>\n 4<\/td>\n<\/tr>\n \n SIL 3<\/td>\n 8<\/td>\n<\/tr>\n \n SIL 4<\/td>\n 16<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n \n\n
\n Automotive Safety Integrity Level (ASIL)
<\/strong>of the system element s<\/em>j<\/sub><\/td>\nWeight\u00a0<\/strong>w<\/em>j<\/sub>
<\/strong>of the system element s<\/em>j<\/sub><\/td>\n<\/tr>\n\n None \/ QM<\/td>\n 1<\/td>\n<\/tr>\n \n ASIL A<\/td>\n 2<\/td>\n<\/tr>\n \n ASIL B<\/td>\n 4<\/td>\n<\/tr>\n \n ASIL C<\/td>\n 6<\/td>\n<\/tr>\n \n ASIL D<\/td>\n 8<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n 2.3 Structural Complexity Reduction<\/strong><\/h3>\n
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3. Summary<\/h2>\n
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References<\/h3>\n