[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-266561-105":59,"doc-detail-266561-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","hard-and-easy-distributions-of-sat-problems","Hard and Easy Distributions of SAT Problems","","Large-scale experiments in satisfiability testing compare how instance generation affects the difficulty faced by SAT procedures. Prior work noted that satisfiability of random formulas can seem surprisingly easy, but an appropriate choice of distribution and parameter settings enables the construction of random formulas that are genuinely hard. The study establishes predictable mixtures of easy and hard cases and provides a benchmark to evaluate and stress-test satisfiability-testing methods in practical research.",{"@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/hard-and-easy-distributions-of-sat-problems/266561/",{"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/hard-and-easy-distributions-of-sat-problems/266561.png","ImageObject",300,407,{"name":92,"@type":93},"wps_ap_test_251126_0180","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-09-19","2026-09-14",true,{"@type":102,"interactionType":103,"userInteractionCount":14},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"How do the experiments relate instance distribution to SAT difficulty?","Question",{"text":112,"@type":113},"The experiments show that by selecting a right distribution of SAT instances and suitable parameter values, random formulas can be generated to produce either easy or hard cases for satisfiability testing.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"Why is average-case evaluation of SAT difficult?",{"text":117,"@type":113},"Average-case results depend on the assumed distribution of instances, and earlier findings can be invalidated when the distribution used contains mostly easy instances.",{"name":119,"@type":110,"acceptedAnswer":120},"Which SAT procedure is used for the tests and why?",{"text":121,"@type":113},"The tests use the Davis-Putnam procedure because it is closely related to resolution, a central reasoning method in AI, and because many empirical SAT studies use refinements of this approach, supporting comparisons.","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},266561,1789408933,{"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":14,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":34,"language":139,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":140,"faqs":141,"seo_title":142,"seo_description":67,"update_tm":129,"read_time":143},8796095027276,"https://avatar.qwps.com/avatar/d3BzX2FwX3Rlc3RfMjUxMTI2XzAxODA=","# Hard and Easy Distributions of SAT Problems\n\nDept.of Computing ScienceAT&T Bell LaboratoriesDept.of Computer ScienceSimon Fraser UniversityMurray Hill,NJ07974University of TorontoBurnaby,Canada V5A 1S6selman@research.att.comToronto,Canada M5S 1A4mitchell@cs.sfu.cahector@ai.toronto.edu  \n## Abstract\n\nWe report results from large-scale experiments insatisfiability testing.As has been observed byothers,testing the satisfiability of random formu-las often appears surprisingly easy.Here we showthat by using the right distribution of instances,and appropriate parameter values,it is possibleto generate random formulas that are hard,thatis,for which satisfiability testing is quite diflicult.Our results provide a benchmark for the evalua-tion of satisfiability-testing procedures.  \n## Introduction\n\nMany computational tasks of interest to AI,to the ex-tent that they can be precisely characterized at all,can be shown to be NP-hard in their most generalform.However,there is fundamental disagreement,atleast within the AI community,about the implicationsof this.It is claimed on the one hand that since theperformance of algorithms designed to solve NP-hardtasks degrades rapidly with small increases in inputsize,something will need to be given up to obtain ac-ceptable behavior.On the other hand,it is arguedthat this analysis is irrelevant to AI since it based onworst-case scenarios,and that what is really needed isa better understanding of how these procedures per-form“on average”.  \nThe first computational task shown to be NP-hard,by Cook(1971),was propositional satisfiability orSAT:given a formula of the propositional calculus,de-cide if there is an assignment to its variables that makesthe formula true according to the usual rules of inter-pretation.Subsequent tasks have been shown to beNP-hard by proving they are at least as hard as SAT.Roughly,a task is NP-hard if a good algorithm for itwould entail a good algorithm for SAT.Unlike manyother NP-hard tasks(see Garey and Johnson(1979)fora catalogue),SAT is of special concern to Ai becauseof its direct relationship to deductive reasoning(i.e.,  \nDavid MitchellBart SelmanHector Levesque*  \ngiven a collection of base facts ∑,a sentence a may bededuced iff ∑U{-a}is not satisfiable).Many otherforms of reasoning,including default reasoning,diag-nosis,planning and image interpretation,also makedirect appeal to satisfiability.The fact that these usu-ally require much more than the propositional calculussimply highlights the fact that SAT is a fundamentaltask,and that developing SAT procedures that workwell in AI applications is essential.  \nWe might ask when it is reasonable to use a soundand complete procedure for SAT,and when we shouldsettle for something less.Do hard cases come up often,or are they always a result of strange encodings tailoredfor some specific purpose?One difficulty in answeringsuch questions is that there appear to be few applica-ble analytical results on the expected difficulty of SAT(although see below).It seems that,at least for thetime being,we must rely largely on empirical results.  \nA number of papers(some discussed below)haveclaimed that the difficulty of SAT on randomly gen-erated problems is not so daunting.For example,anoften-quoted result(Goldberg,1979;Goldberg et al.1982)suggests that SAT can be readily solved “on av-erage\"in O(n²)time.This does not settle the questionof how well the methods will work in practice,but atfirst blush it does appear to be more relevant to AIthan contrived worst cases.  \nThe big problem is that to examine how well a pro-cedure does on average one must assume a distributionof instances.Indeed,as we will discuss below,Francoand Paull(1983)refuted the Goldberg result by show-ing that it was a direct consequence of their choice ofdistribution.It's not that Goldberg had a clever al-gorithm,or that the problem is easy,but that theyhad used a distribution with a preponderance of easyinstances.That is,from the space of all problem in-stances,","cbCaibmF0T4wvXIL","https://ap.wps.com/l/cbCaibmF0T4wvXIL","pdf",1068893,"English","# Abstract\n# Introduction\n## Background on NP-hardness and SAT\n## Average-case versus worst-case analysis\n## Instance distributions and sampling bias\n## Motivation for predictable easy/hard benchmarks\n## Davis-Putnam SAT procedure\n## Paper organization","[{\"question\":\"How do the experiments relate instance distribution to SAT difficulty?\",\"answer\":\"The experiments show that by selecting a right distribution of SAT instances and suitable parameter values, random formulas can be generated to produce either easy or hard cases for satisfiability testing.\"},{\"question\":\"Why is average-case evaluation of SAT difficult?\",\"answer\":\"Average-case results depend on the assumed distribution of instances, and earlier findings can be invalidated when the distribution used contains mostly easy instances.\"},{\"question\":\"Which SAT procedure is used for the tests and why?\",\"answer\":\"The tests use the Davis-Putnam procedure because it is closely related to resolution, a central reasoning method in AI, and because many empirical SAT studies use refinements of this approach, supporting comparisons.\"}]","Hard and Easy Distributions of SAT Problems | PDF",18]