[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-1-en-105":3,"doc-seo-252338-105":53,"doc-detail-252338-en":127},{"code":4,"msg":5,"data":6},0,"success",[7,14,19,24,29,34,39,44,49],{"id":8,"doc_module":9,"doc_module_name":10,"category_name":11,"show_sort_weight":12,"slug":13},11,1,"Template","Presentations",90,"presentations",{"id":15,"doc_module":9,"doc_module_name":10,"category_name":16,"show_sort_weight":17,"slug":18},12,"Resumes",80,"resumes",{"id":20,"doc_module":9,"doc_module_name":10,"category_name":21,"show_sort_weight":22,"slug":23},14,"Invoices",70,"invoices",{"id":25,"doc_module":9,"doc_module_name":10,"category_name":26,"show_sort_weight":27,"slug":28},15,"Posters",60,"posters",{"id":30,"doc_module":9,"doc_module_name":10,"category_name":31,"show_sort_weight":32,"slug":33},16,"Social Media",50,"social-media",{"id":35,"doc_module":9,"doc_module_name":10,"category_name":36,"show_sort_weight":37,"slug":38},17,"Forms",40,"forms",{"id":40,"doc_module":9,"doc_module_name":10,"category_name":41,"show_sort_weight":42,"slug":43},18,"Letters",30,"letters",{"id":45,"doc_module":9,"doc_module_name":10,"category_name":46,"show_sort_weight":47,"slug":48},21,"Paper Templates",5,"papers-templates",{"id":50,"doc_module":9,"doc_module_name":10,"category_name":51,"show_sort_weight":4,"slug":52},158,"General","general-158",{"code":4,"msg":54,"data":55},"ok",{"site_id":56,"language":57,"slug":58,"title":59,"keywords":60,"description":61,"schema_data":62,"social_meta":120,"head_meta":122,"extra_data":124,"updated_unix":126},105,"en","efficient-minimization-of-fully-testable-2-spp-networks","Efficient Minimization of Fully Testable 2-SPP Networks","","The paper presents a heuristic algorithm for minimizing 2-SPP networks, defined as three-level EXOR-AND-OR forms with EXOR gates restricted to fan-in 2. It builds on exact covering-based methods for unrestricted and 2-SPP cases, which have worst-case exponential complexity. By extending the ESPRESSO expandirredundant-reduce paradigm, the approach iterates local minimization and solution reshaping. An EXOR-irredundant notion is introduced to establish full testability and correctness, supported by extensive experiments with affordable runtimes.",{"@graph":63,"@context":119},[64,80,102],{"@type":65,"itemListElement":66},"BreadcrumbList",[67,71,74,77],{"item":68,"name":69,"@type":70,"position":9},"https://docshare.wps.com","Home","ListItem",{"item":72,"name":10,"@type":70,"position":73},"https://docshare.wps.com/template/",2,{"item":75,"name":11,"@type":70,"position":76},"https://docshare.wps.com/template/presentations/",3,{"item":78,"name":59,"@type":70,"position":79},"https://docshare.wps.com/template/efficient-minimization-of-fully-testable-2-spp-networks/252338/",4,{"url":78,"name":59,"@type":81,"image":82,"author":87,"headline":59,"publisher":90,"fileFormat":93,"inLanguage":57,"description":61,"dateModified":94,"datePublished":95,"encodingFormat":93,"isAccessibleForFree":96,"interactionStatistic":97},"DigitalDocument",{"url":83,"@type":84,"width":85,"height":86},"https://docshare.wps.com/thumbnails/efficient-minimization-of-fully-testable-2-spp-networks/252338.png","ImageObject",442,249,{"name":88,"@type":89},"Adam","Person",{"url":68,"name":91,"@type":92},"DocShare","Organization","application/pdf","2026-10-30","2026-09-13",true,{"@type":98,"interactionType":99,"userInteractionCount":101},"InteractionCounter",{"@type":100},"ViewAction",7,{"@type":103,"mainEntity":104},"FAQPage",[105,111,115],{"name":106,"@type":107,"acceptedAnswer":108},"What problem does the paper address?","Question",{"text":109,"@type":110},"The paper addresses heuristic minimization of 2-SPP networks, i.e., three-level EXOR-AND-OR forms with EXOR gates limited to fan-in 2.","Answer",{"name":112,"@type":107,"acceptedAnswer":113},"Why aren’t exact minimization methods used?",{"text":114,"@type":110},"Exact procedures reduce minimization to covering problems and can have worst-case exponential complexity, making them infeasible for many industrial benchmarks.",{"name":116,"@type":107,"acceptedAnswer":117},"How does the proposed heuristic improve solutions?",{"text":118,"@type":110},"It extends the ESPRESSO expand/irredundant/reduce cycle with operations specialized for 2-SPP forms, iterating local minimization and reshaping until no further improvement is found.","https://schema.org",{"og:url":78,"og:type":121,"og:title":59,"og:site_name":91,"og:description":61},"article",{"robots":123,"canonical":78},"index,follow",{"doc_id":125,"site_id":56},252338,1790450006,{"code":4,"msg":5,"data":128},{"doc_id":125,"user_id":129,"nickname":88,"user_avatar":130,"doc_module":9,"category_id":8,"category_name":11,"doc_title":59,"doc_description":61,"doc_content":131,"file_id":132,"file_url":133,"file_type":134,"file_size":135,"view_count":101,"is_deleted":4,"is_public":9,"is_downloadable":9,"audit_status":9,"page_count":136,"language":137,"language_code":57,"site_id":56,"html_lang":57,"table_of_contents":138,"faqs":139,"seo_title":140,"seo_description":61,"update_tm":141,"read_time":73},1374404737137,"https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d","Efficient Minimization of Fully Testable 2-SPP Networks  \nAnna Bernasconi Department of Computer Science University of Pisa, Italy [annab@di.unipi.it](annab@di.unipi.it)  \nRolf Drechsler Institute of Computer Science University of Bremen, Germany [drechsle@informatik.uni-bremen.de](drechsle@informatik.uni-bremen.de)  \nValentina Ciriani  \nDepartment of Information Technologies University of Milano, Italy [ciriani@dti.unimi.it](ciriani@dti.unimi.it)  \nTiziano Villa DIEGM University of Udine, Italy [villa@uniud.it](villa@uniud.it)  \nAbstract  \nThe paper presents a heuristic algorithm for the minimization of 2-SPP networks, i.e., three-level EXOR-ANDOR forms with EXOR gates restricted to fan-in 2. Previous works had presented exact algorithms for the minimization of unrestricted SPP networks and of 2-SPP networks. The exact minimization procedures were formulated as covering problems as in the minimization of SOP forms and had worst-case exponential complexity. Extending the expandirredundant-reduce paradigm of the ESPRESSO heuristic, we propose a minimization algorithm for 2-SPP networks that iterates local minimization and reshape of a solution until further improvement. We introduce also the notion of EXOR-irredundant to prove that OR-AND-EXOR irredundant networks are fully testable and guarantee that our algorithm yields OR-AND-EXOR irredundant solutions. Wereport a large set of experiments showing impressive highquality results with affordable run times, handling also examples whose exact solutions could not be computed.  \n1. Introduction  \nMainstream logic synthesis concentrates on two extremes: two-level logic and unrestricted multi-level logic. The former has been studied in great depth both from the theoretical and practical viewpoint, resulting in exact and heuristic automatic minimizers of industrial quality, such as ESPRESSO [2] . For the latter we do not have yet a complete exact characterization, but a robust theory of algebraic and Boolean division triggered the development of efficient heuristic tools, such as SIS [13] .  \nIn-between there are interesting restricted forms of multi-level logic, of which three-level logic attracted the attention of many researchers, as surveyed in [4] . Here we  \ncontinue the investigation of three-level EXOR-AND-OR forms, introduced in [9, 4] . They are a direct generalization of AND-OR forms, obtained generalizing cubes to pseudocubes where literals in cubes may be replaced by EXOR factors in pseudocubes. Pseudocubes have been shown in [4] to correspond to affine spaces over the Boolean vector space  ,    . The repeated union of pseudocubes yields prime pseudocubes, an extension of primes for SOP; once prime pseudocubes are computed, exact minimization of EXOR-AND-OR forms is reduced to the solution of a covering table, as in case of SOP forms. To be technologically feasible, EXOR-AND-OR forms are restricted to EXOR factors with at most  literals. In this paper we will discuss only forms with , called 2-SPP forms [5] .  \nAlthough exact methods for SPP minimization perform well on many examples [4, 5], they are not affordable for all industrial benchmarks, therefore we must give up exact minimization for heuristic one, mirroring what has been done for SOP minimization [11, 2] . In SOP heuristic minimization the solution of the covering table is replaced by the iteration of a sequence of EXPAND, IRREDUNDANT COVER, REDUCE operations: EXPAND replaces each implicant in the cover with a largest prime containing it and eventually other cubes of the cover (or parts of them); IRREDUNDANT COVER removes a maximal set of redundant implicants; REDUCE replaces each prime implicant by a smallest implicant that covers all the relatively essential vertices of the prime implicant (the vertices not contained in any other cube of the given representation) . These operations are performed heuristically, i.e., the order of expanding and reducing implicants matters with respect to the final quality. The REDU","cbCaidjfqCd47muF","https://ap.wps.com/l/cbCaidjfqCd47muF","pdf",217693,6,"English","# Introduction\n## Restricted three-level EXOR-AND-OR forms\n## ESPRESSO-style heuristic minimization\n## Operations for 2-SPP forms and iterative loop\n## Testability under the stuck-at fault model (SAFM)","[{\"question\":\"What problem does the paper address?\",\"answer\":\"The paper addresses heuristic minimization of 2-SPP networks, i.e., three-level EXOR-AND-OR forms with EXOR gates limited to fan-in 2.\"},{\"question\":\"Why aren’t exact minimization methods used?\",\"answer\":\"Exact procedures reduce minimization to covering problems and can have worst-case exponential complexity, making them infeasible for many industrial benchmarks.\"},{\"question\":\"How does the proposed heuristic improve solutions?\",\"answer\":\"It extends the ESPRESSO expand/irredundant/reduce cycle with operations specialized for 2-SPP forms, iterating local minimization and reshaping until no further improvement is found.\"}]","Efficient Minimization of Fully Testable 2-SPP Networks | PDF",1789261134]