[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-267925-105":59,"doc-detail-267925-en":131},{"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":124,"head_meta":126,"extra_data":128,"updated_unix":130},105,"en","sound-probabilistic-sat-with-projection-improved-method","Sound Probabilistic #SAT with Projection - Improved Method","","A sound probabilistic estimation technique is developed for counting the number of models of a Boolean formula under projection. The method supports quantitative program analyses by encoding reachability and deriving cardinalities of sets of reachable program states. The APPROXMC-P tool improves the base confidence behavior of APPROXMC, reduces repetitions for higher confidence, and decreases SAT solver queries while combining probabilistic model counting with general projection scopes.",{"@graph":69,"@context":123},[70,84,106],{"@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/sound-probabilistic-sat-with-projection-improved-method/267925/",{"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/sound-probabilistic-sat-with-projection-improved-method/267925.png","ImageObject",300,407,{"name":92,"@type":93},"\tWilliam","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-30","2026-09-14",true,{"@type":102,"interactionType":103,"userInteractionCount":105},"InteractionCounter",{"@type":104},"ViewAction",12,{"@type":107,"mainEntity":108},"FAQPage",[109,115,119],{"name":110,"@type":111,"acceptedAnswer":112},"What does “sound probabilistic #SAT with projection” estimate?","Question",{"text":113,"@type":114},"It estimates the number of models of a Boolean formula after projecting it onto a given variable scope, with sampling error bounds provided a priori.","Answer",{"name":116,"@type":111,"acceptedAnswer":117},"Why is projection needed for quantitative program analysis?",{"text":118,"@type":114},"#SAT alone cannot directly reason about program reachability; projection corresponds to computing image or preimage information for relations encoded by the formula.",{"name":120,"@type":111,"acceptedAnswer":121},"What are the main improvements introduced by APPROXMC-P?",{"text":122,"@type":114},"APPROXMC-P improves APPROXMC performance by raising base confidence and reducing the number of repetitions for higher confidence, while also lowering the number of SAT solver queries per repetition.","https://schema.org",{"og:url":83,"og:type":125,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":127,"canonical":83},"index,follow",{"doc_id":129,"site_id":62},267925,1789426560,{"code":4,"msg":5,"data":132},{"doc_id":129,"user_id":133,"nickname":92,"user_avatar":134,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":135,"file_id":136,"file_url":137,"file_type":138,"file_size":139,"view_count":105,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":140,"language":141,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":142,"faqs":143,"seo_title":144,"seo_description":67,"update_tm":130,"read_time":145,"preview_page_count":4},5909887254083,"https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c","Sound Probabilistic \\#SAT with Projection  \nVladimir Klebanov  \n[klebanov@kit.edu](klebanov@kit.edu)  \nAlexander Weigl [weigl@kit.edu](weigl@kit.edu)  \nJörg Weisbarth [joerg.bretten@web.de](joerg.bretten@web.de)  \nInstitute for Theoretical Informatics  \nKarlsruhe Institute of Technology  \nGermany  \nWe present an improved method for a sound probabilistic estimation of the model count of a boolean formula under projection. The problem solved can be used to encode a variety of quantitative program analyses, such as concerning security of resource consumption. We implement the technique and discuss its application to quantifying information flow in programs.  \n1 Introduction  \nThe \\#SAT problem is concerned with counting the number of models of a boolean formula. Since \\#SATis a computationally difficult problem, not only exact but also approximative solutions are of interest. A powerful approximation method is probabilistic approximation, making use of random sampling. We call a probabilistic approximation method sound when the probability and magnitude of the samplingrelated error can be bounded a priory. A prominent recent development in sound probabilistic \\#SAT is APPROXMC [1] .  \nIn this paper we present APPROXMC-P, an improved method and a tool for sound probabilistic \\#SAT with projection. Just as APPROXMC, which it enhances, APPROXMC-P belongs to the category of (e ; d ) counters. In this context, the parameter e represents the tolerance and 1 􀀀 d the confidence of the result. For example, choosing e = 0:1 and d = 0:14 implies that the computed result provably lies with a probability of 86% between the 0:9-fold and the 1:1-fold of the correct result. Both parameters can be configured by the user.  \nAn enticing application of \\#SAT solvers is quantitative program analysis, which often requires establishing cardinality of sets defined in terms of the set of reachable program states. Reducing an analysis to \\#SAT has the advantage that one can use a variety of established reasoning techniques. At the sametime, it is reasonably easy to represent behavior of intricate low-level programs in boolean logic. Yet,\\#SAT alone is typically not sufficient for this purpose—one needs a way to reason about program reachability. In logic, this reasoning corresponds to projection. If a boolean formula encodes a relation (e.g., a transition relation on states), then computing the image or the preimage of the relation is a projection operation.  \nAPPROXMC-P takes as input a boolean formula in conjunctive normal form together with a projection scope (a set of variables) and the parameters e and d and estimates the number of models of the formula projected on the given scope. Of course, by setting the scope to encompass all variables in the logical signature, one can use APPROXMC-P as a non-projecting \\#SAT solver.  \nThe specific contributions of this paper are the following:  \nFirst, we materially improve the performance of APPROXMC, in particular its base confidence. Probabilistic counters meet confidence demands above the base confidence by repeating the estimation. We reduce the number of repetitions for confidence values above 0.6 by about an order of magnitude on  \nM. Tribastone and H. Wiklicky (Eds.): International Workshop on Quantitative Aspects of Programming Languages and Systems (QAPL'16) EPTCS 227, 2016, pp. 15–29, doi:10.4204/EPTCS.227.2  \n􀀍c V. Klebanov, A. Weigl & J. Weisbarth  \n16 Sound Probabilistic \\#SAT with Projection  \naverage. Furthermore, we reduce, for one repetition, the number of SAT solver queries by at least 15% . A detailed comparison is presented in Section 3.1.  \nSecond, we combine probabilistic model counting with projection, even though the ideas behind this combination are not completely new. A particular special case has previously appeared in [2] in the context of uniform model sampling. There, the formula j is treated by considering only its projection on the independent support. An independent support is a subset of","cbCaiuurrLRtNNT4","https://ap.wps.com/l/cbCaiuurrLRtNNT4","pdf",287593,15,"English","# Introduction\n## #SAT and sound probabilistic estimation\n## Projection in quantitative program analysis\n## APPROXMC-P inputs and problem setting\n## Contributions of the paper\n## Logical Foundations","[{\"question\":\"What does “sound probabilistic #SAT with projection” estimate?\",\"answer\":\"It estimates the number of models of a Boolean formula after projecting it onto a given variable scope, with sampling error bounds provided a priori.\"},{\"question\":\"Why is projection needed for quantitative program analysis?\",\"answer\":\"#SAT alone cannot directly reason about program reachability; projection corresponds to computing image or preimage information for relations encoded by the formula.\"},{\"question\":\"What are the main improvements introduced by APPROXMC-P?\",\"answer\":\"APPROXMC-P improves APPROXMC performance by raising base confidence and reducing the number of repetitions for higher confidence, while also lowering the number of SAT solver queries per repetition.\"}]","Sound Probabilistic #SAT with Projection - Improved Method | PDF",38]