{"title":"Computationally Hard Problems for Logic Programs under Answer Set Semantics","authors":"Yuping Shen, Xishun Zhao","doi":"10.1145/3676964","DOIUrl":null,"url":null,"abstract":"\n Showing that a problem is\n hard\n for a model of computation is one of the most challenging tasks in theoretical computer science, logic and mathematics. For example, it remains beyond reach to find an\n explicit\n problem that cannot be computed by polynomial size propositional formulas (PF). As a model of computation, logic programs (LP) under answer set semantics are as expressive as PF, and also\n \n \\(\\mathtt{NP}\\)\n \n -complete for satisfiability checking. In this paper, we show that the PAR problem is hard for LP, i.e., deciding whether a binary string contains an odd number of\n \n \\(1\\)\n \n ’s requires\n exponential\n size logic programs. The proof idea is first to transform logic programs into equivalent boolean circuits, and then apply a probabilistic method known as\n random restriction\n to obtain an exponential lower bound. Based on the main result, we generalize a sufficient condition for identifying hard problems for LP, and give a separation map for a logic program family from a computational point of view, whose members are all equally expressive and share the same reasoning complexity.\n","PeriodicalId":0,"journal":{"name":"","volume":"36 21","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"94","ListUrlMain":"https://doi.org/10.1145/3676964","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
Showing that a problem is
hard
for a model of computation is one of the most challenging tasks in theoretical computer science, logic and mathematics. For example, it remains beyond reach to find an
explicit
problem that cannot be computed by polynomial size propositional formulas (PF). As a model of computation, logic programs (LP) under answer set semantics are as expressive as PF, and also
\(\mathtt{NP}\)
-complete for satisfiability checking. In this paper, we show that the PAR problem is hard for LP, i.e., deciding whether a binary string contains an odd number of
\(1\)
’s requires
exponential
size logic programs. The proof idea is first to transform logic programs into equivalent boolean circuits, and then apply a probabilistic method known as
random restriction
to obtain an exponential lower bound. Based on the main result, we generalize a sufficient condition for identifying hard problems for LP, and give a separation map for a logic program family from a computational point of view, whose members are all equally expressive and share the same reasoning complexity.