[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-665453-105":59,"doc-detail-665453-en":130},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":123,"head_meta":125,"extra_data":127,"updated_unix":129},105,"en","verification-of-arithmetic-functions-with-binary-moment-diagrams","Verification of Arithmetic Functions with Binary Moment Diagrams","","Binary Moment Diagrams (BMDs) provide a canonical symbolic representation for linear functions, extending Binary Decision Diagram ideas beyond Boolean ranges. They enable embedding arbitrary functions from Boolean variables into real, rational, or integer values, supporting word-level data-path modeling. The approach verifies arithmetic circuits hierarchically: first build word-level specifications for key blocks, then prove overall word-level functionality. Multipliers up to 62-bit word sizes are verified, and BMDs can express Boolean functions with comparable complexity to BDDs.",{"@graph":69,"@context":122},[70,84,105],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":40,"@type":76,"position":81},"https://docshare.wps.com/document/research-report/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/verification-of-arithmetic-functions-with-binary-moment-diagrams/665453/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/verification-of-arithmetic-functions-with-binary-moment-diagrams/665453.png","ImageObject",300,407,{"name":92,"@type":93},"Ophelia","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-30","2026-10-29",true,{"@type":102,"interactionType":103,"userInteractionCount":19},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What are Binary Moment Diagrams (BMDs) used for in this paper?","Question",{"text":112,"@type":113},"BMDs represent linear functions symbolically, enabling functions over non-Boolean numeric ranges to be modeled and manipulated at the word level.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"How does the proposed hierarchical verification approach work?",{"text":117,"@type":113},"It verifies basic circuit blocks by showing their bit-level implementations satisfy word-level specifications, then verifies that composing block functions matches the system’s word-level specification.",{"name":119,"@type":110,"acceptedAnswer":120},"What types and sizes of multipliers were verified?",{"text":121,"@type":113},"The technique verified multipliers with word sizes up to 62 bits.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},665453,1793304262,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":19,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":139,"language":140,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":67,"update_tm":129,"read_time":144},7971461741311,"https://ap-avatar.wpscdn.com/avatar/74000253aff267980c6?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779345379180704826","Veri􀀌cation of Arithmetic Functions with Binary Moment Diagrams􀀃  \nRandal E. Bryant  \nYirng-An Chen  \nCarnegie Mellon University  \nPittsburgh, PA 15213  \nMay 31, 1994  \nAbstract  \nBinary Moment Diagrams (BMDs) provide a canonical representations for linear functions similar to the way Binary Decision Diagrams (BDDs) represent Boolean functions. Within the class of linear functions, we can embed arbitary functions from Boolean variables to real, rational, or integer values. BMDs can thus model the functionality of data path circuits operating over word level data. Many important functions, including integer multiplication, that cannot be represented e􀀎ciently at the  \nbit level with BDDs have simple representations at the word level with BMDs. Furthermore, BMDs can represent Boolean functions with around the same complexity as BDDs.  \nWe propose a hierarchical approach to verifying arithmetic circuits, here basic building blocks are 􀀌rst shown to implement a word-level speci􀀌cation. The overall circuit functionality is then veri􀀌ed at the word level. Multipliers with word sizes of up to 62 bits have been veri􀀌ed by this technique.  \nKeywords: Formal veri􀀌cation, binary decision diagrams, arithmetic circuits, multipliers  \n1. Introduction  \nBinary Decision Diagrams (BDDs) have proved successful for representing and manipulating Boolean functions symbolically [4] in a variety of application domains. Building on this success, there have been several e􀀋orts to extend the BDD concept to represent functions over Boolean variables, but having non-Boolean ranges, such as integers or real numbers  \n􀀃 This research is sponsored by the Wright Laboratory, Aeronautical Systems Center, Air Force Materiel Command, USAF, and the Advanced Research Projects Agency (ARPA) under grant number F33615-93-1- 1330.  \n1  \n\n|  | Boolean | Numeric |  |\n| --- | --- | --- | --- |\n|  |  | Terminal | Edge-Weighted |\n| Pointwise | BDD | MTBDD, ADD | EVBDD |\n| Moment | FDD | BMD | *BMD |\n\nTable 1: Categorization of Graphical Function Representations.  \n[1, 7, 8 , 15, 17 ] . This class of functions is sometimes termed \\pseudo-Boolean\" [12] . Many tasks can be expressed in terms of operations on such functions, including integer linear programming, matrix manipulation, spectral transforms, and word-level digital system analysis. To date, the proposed representations for these functions have proved too fragile for routine application|too often the data structures grow exponentially in the number of variables.  \nIn this paper we propose a new representation called Multiplicative Binary Moment Diagrams (*BMDs) that improve on previous methods. *BMDs incorporate two novel features: they are based on a decomposition of a linear function in terms of its \\moments,\" and they have weights associated with their edges which are combined multiplicatively. These features have as heritage ideas found in previous function representations, namely the Reed-Muller decomposition used by Functional Decision Diagrams (FDDs) [9, 14 ], and the additive edge weights found in Edge-Valued Binary Decision Diagrams (EVBDDs) [15] . The relations between the various representations are described more fully below.  \n*BMDs are particularly e􀀋ective for representing digital systems at the word level, where sets of binary signals are interpreted as encoding integer (􀀌xed point) or rational (􀀍oating point) values. Common integer and 􀀍oating point encodings have e􀀎cient representations as *BMDs, as do operations such as addition and multiplication. *BMDs can also represent Boolean functions as a special case, with size comparable to BDDs.  \n*BMDs can serve as the basis for a hierarchical methodology for verifying circuits such as multipliers. At the low level, we have a set of building blocks such as add steppers, Booth steppers, and carry save adders described at both the bit level (as combinational circuits) and at the word level (as algebraic expressions) . Using a methodology proposed by Lai an","cbCaihm53HGEjUyc","https://ap.wps.com/l/cbCaihm53HGEjUyc","pdf",319688,37,"English","# Introduction\n## Background: BDDs and extending to non-Boolean ranges\n## Proposed representation: Multiplicative BMDs\n## Hierarchical verification methodology\n# Graphical Function Representations\n## Categorization by range and decomposition\n## Example decompositions (MTBDD vs BMD)","[{\"question\":\"What are Binary Moment Diagrams (BMDs) used for in this paper?\",\"answer\":\"BMDs represent linear functions symbolically, enabling functions over non-Boolean numeric ranges to be modeled and manipulated at the word level.\"},{\"question\":\"How does the proposed hierarchical verification approach work?\",\"answer\":\"It verifies basic circuit blocks by showing their bit-level implementations satisfy word-level specifications, then verifies that composing block functions matches the system’s word-level specification.\"},{\"question\":\"What types and sizes of multipliers were verified?\",\"answer\":\"The technique verified multipliers with word sizes up to 62 bits.\"}]","Verification of Arithmetic Functions with Binary Moment Diagrams | PDF",93]