Delayed Theory Combination vs. Nelson-Oppen for Satisfiability Modulo Theories: A Comparative Analysis

Many approaches for Satisfiability Modulo Theory (SMT\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\mathcal({T})})$\...

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Hauptverfasser: Bruttomesso, Roberto, Cimatti, Alessandro, Franzén, Anders, Griggio, Alberto, Sebastiani, Roberto
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description Many approaches for Satisfiability Modulo Theory (SMT\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\mathcal({T})})$\end{document} rely on the integration between a SAT solver and a decision procedure for sets of literals in the background theory \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\mathcal{T}} ({\mathcal{T}}-solver$\end{document}). When \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\mathcal{T}}$\end{document} is the combination \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${{\mathcal{T}}_1\cup{\mathcal{T}}_2}$\end{document} of two simpler theories, the approach is typically handled by means of Nelson-Oppen’s (NO) theory combination schema in which two specific \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\mathcal{T}}$\end{document}-solver deduce and exchange (disjunctions of) interface equalities. In recent papers we have proposed a new approach to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$({{\mathcal{T}}_1\cup{\mathcal{T}}_2})$\end{document}, called Delayed Theory Combination (Dtc). Here part or all the (possibly very expensive) task of deducing interface equalities is played by the SAT solver itself, at the potential cost of an enlargement of the boolean search space. In principle this enlargement could be up to exponential in the number of interface equalities generated. In this paper we show that this estimate was too pessimistic. We present a comparative a
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When \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\mathcal{T}}$\end{document} is the combination \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${{\mathcal{T}}_1\cup{\mathcal{T}}_2}$\end{document} of two simpler theories, the approach is typically handled by means of Nelson-Oppen’s (NO) theory combination schema in which two specific \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\mathcal{T}}$\end{document}-solver deduce and exchange (disjunctions of) interface equalities. In recent papers we have proposed a new approach to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$({{\mathcal{T}}_1\cup{\mathcal{T}}_2})$\end{document}, called Delayed Theory Combination (Dtc). Here part or all the (possibly very expensive) task of deducing interface equalities is played by the SAT solver itself, at the potential cost of an enlargement of the boolean search space. In principle this enlargement could be up to exponential in the number of interface equalities generated. In this paper we show that this estimate was too pessimistic. We present a comparative analysis of Dtc vs. NO for SMT\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$({{\mathcal{T}}_1\cup{\mathcal{T}}_2})$\end{document}, which shows that, using state-of-the-art SAT-solving techniques, the amount of boolean branches performed by Dtc can be upper bounded by the number of deductions and boolean branches performed by NO on the same problem. We prove the result for different deduction capabilities of the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\mathcal({T}-solver)}$\end{document} and for both convex and non-convex theories.</description><identifier>ISSN: 0302-9743</identifier><identifier>ISBN: 9783540482819</identifier><identifier>ISBN: 3540482814</identifier><identifier>EISSN: 1611-3349</identifier><identifier>EISBN: 3540482822</identifier><identifier>EISBN: 9783540482826</identifier><identifier>DOI: 10.1007/11916277_36</identifier><language>eng</language><publisher>Berlin, Heidelberg: Springer Berlin Heidelberg</publisher><subject>Applied sciences ; Artificial intelligence ; Computer science; control theory; systems ; Decision Procedure ; Exact sciences and technology ; Input Formula ; Interface Equality ; Satisfiability Modulo Theory ; Truth Assignment</subject><ispartof>Lecture notes in computer science, 2006, p.527-541</ispartof><rights>Springer-Verlag Berlin Heidelberg 2006</rights><rights>2007 INIST-CNRS</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><relation>Lecture Notes in Computer Science</relation></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/11916277_36$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/11916277_36$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>309,310,779,780,784,789,790,793,4050,4051,27925,38255,41442,42511</link.rule.ids><backlink>$$Uhttp://pascal-francis.inist.fr/vibad/index.php?action=getRecordDetail&amp;idt=19131403$$DView record in Pascal Francis$$Hfree_for_read</backlink></links><search><contributor>Voronkov, Andrei</contributor><contributor>Hermann, Miki</contributor><creatorcontrib>Bruttomesso, Roberto</creatorcontrib><creatorcontrib>Cimatti, Alessandro</creatorcontrib><creatorcontrib>Franzén, Anders</creatorcontrib><creatorcontrib>Griggio, Alberto</creatorcontrib><creatorcontrib>Sebastiani, Roberto</creatorcontrib><title>Delayed Theory Combination vs. Nelson-Oppen for Satisfiability Modulo Theories: A Comparative Analysis</title><title>Lecture notes in computer science</title><description>Many approaches for Satisfiability Modulo Theory (SMT\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\mathcal({T})})$\end{document} rely on the integration between a SAT solver and a decision procedure for sets of literals in the background theory \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\mathcal{T}} ({\mathcal{T}}-solver$\end{document}). When \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\mathcal{T}}$\end{document} is the combination \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${{\mathcal{T}}_1\cup{\mathcal{T}}_2}$\end{document} of two simpler theories, the approach is typically handled by means of Nelson-Oppen’s (NO) theory combination schema in which two specific \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\mathcal{T}}$\end{document}-solver deduce and exchange (disjunctions of) interface equalities. In recent papers we have proposed a new approach to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$({{\mathcal{T}}_1\cup{\mathcal{T}}_2})$\end{document}, called Delayed Theory Combination (Dtc). Here part or all the (possibly very expensive) task of deducing interface equalities is played by the SAT solver itself, at the potential cost of an enlargement of the boolean search space. In principle this enlargement could be up to exponential in the number of interface equalities generated. In this paper we show that this estimate was too pessimistic. 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We prove the result for different deduction capabilities of the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\mathcal({T}-solver)}$\end{document} and for both convex and non-convex theories.</description><subject>Applied sciences</subject><subject>Artificial intelligence</subject><subject>Computer science; control theory; systems</subject><subject>Decision Procedure</subject><subject>Exact sciences and technology</subject><subject>Input Formula</subject><subject>Interface Equality</subject><subject>Satisfiability Modulo Theory</subject><subject>Truth Assignment</subject><issn>0302-9743</issn><issn>1611-3349</issn><isbn>9783540482819</isbn><isbn>3540482814</isbn><isbn>3540482822</isbn><isbn>9783540482826</isbn><fulltext>true</fulltext><rsrctype>book_chapter</rsrctype><creationdate>2006</creationdate><recordtype>book_chapter</recordtype><recordid>eNpNkDtPwzAUhc1LopRO_AEvDAwpvrZjO2xVeUqFDpQ5smMHDGkc2aVS_j2pChJ3ucP5zhk-hC6ATIEQeQ1QgKBSlkwcoDOWc8IVVZQeohEIgIwxXhyhSSHVXwbFMRoRRmhWSM5O0SSlTzIcAyWKfITqW9fo3lm8-nAh9nge1sa3euNDi7dpil9ck0KbLbvOtbgOEb8OWaq9Nr7xmx4_B_vdhH3bu3SDZ7uJTscB2zo8a3XTJ5_O0Umtm-Qmv3-M3u7vVvPHbLF8eJrPFlkHEkSWSyWsLRTkoqLOqooLUzkQOa2Udg5YDcIwwipqRU6MBBDcMsONA-VsrdgYXe53O50q3dRRt5VPZRf9Wse-HPQx4IQN3NWeS0PUvrtYmhC-Ugmk3Iku_4lmP5cnavU</recordid><startdate>2006</startdate><enddate>2006</enddate><creator>Bruttomesso, Roberto</creator><creator>Cimatti, Alessandro</creator><creator>Franzén, Anders</creator><creator>Griggio, Alberto</creator><creator>Sebastiani, Roberto</creator><general>Springer Berlin Heidelberg</general><general>Springer</general><scope>IQODW</scope></search><sort><creationdate>2006</creationdate><title>Delayed Theory Combination vs. Nelson-Oppen for Satisfiability Modulo Theories: A Comparative Analysis</title><author>Bruttomesso, Roberto ; Cimatti, Alessandro ; Franzén, Anders ; Griggio, Alberto ; Sebastiani, Roberto</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-p1716-5786dd98156c2ed8c46bce1652c8aee13f16b303c2d650b71164d3b4be18edf83</frbrgroupid><rsrctype>book_chapters</rsrctype><prefilter>book_chapters</prefilter><language>eng</language><creationdate>2006</creationdate><topic>Applied sciences</topic><topic>Artificial intelligence</topic><topic>Computer science; control theory; systems</topic><topic>Decision Procedure</topic><topic>Exact sciences and technology</topic><topic>Input Formula</topic><topic>Interface Equality</topic><topic>Satisfiability Modulo Theory</topic><topic>Truth Assignment</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Bruttomesso, Roberto</creatorcontrib><creatorcontrib>Cimatti, Alessandro</creatorcontrib><creatorcontrib>Franzén, Anders</creatorcontrib><creatorcontrib>Griggio, Alberto</creatorcontrib><creatorcontrib>Sebastiani, Roberto</creatorcontrib><collection>Pascal-Francis</collection></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Bruttomesso, Roberto</au><au>Cimatti, Alessandro</au><au>Franzén, Anders</au><au>Griggio, Alberto</au><au>Sebastiani, Roberto</au><au>Voronkov, Andrei</au><au>Hermann, Miki</au><format>book</format><genre>bookitem</genre><ristype>CHAP</ristype><atitle>Delayed Theory Combination vs. Nelson-Oppen for Satisfiability Modulo Theories: A Comparative Analysis</atitle><btitle>Lecture notes in computer science</btitle><seriestitle>Lecture Notes in Computer Science</seriestitle><date>2006</date><risdate>2006</risdate><spage>527</spage><epage>541</epage><pages>527-541</pages><issn>0302-9743</issn><eissn>1611-3349</eissn><isbn>9783540482819</isbn><isbn>3540482814</isbn><eisbn>3540482822</eisbn><eisbn>9783540482826</eisbn><abstract>Many approaches for Satisfiability Modulo Theory (SMT\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\mathcal({T})})$\end{document} rely on the integration between a SAT solver and a decision procedure for sets of literals in the background theory \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\mathcal{T}} ({\mathcal{T}}-solver$\end{document}). When \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\mathcal{T}}$\end{document} is the combination \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${{\mathcal{T}}_1\cup{\mathcal{T}}_2}$\end{document} of two simpler theories, the approach is typically handled by means of Nelson-Oppen’s (NO) theory combination schema in which two specific \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\mathcal{T}}$\end{document}-solver deduce and exchange (disjunctions of) interface equalities. In recent papers we have proposed a new approach to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$({{\mathcal{T}}_1\cup{\mathcal{T}}_2})$\end{document}, called Delayed Theory Combination (Dtc). Here part or all the (possibly very expensive) task of deducing interface equalities is played by the SAT solver itself, at the potential cost of an enlargement of the boolean search space. In principle this enlargement could be up to exponential in the number of interface equalities generated. In this paper we show that this estimate was too pessimistic. We present a comparative analysis of Dtc vs. NO for SMT\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$({{\mathcal{T}}_1\cup{\mathcal{T}}_2})$\end{document}, which shows that, using state-of-the-art SAT-solving techniques, the amount of boolean branches performed by Dtc can be upper bounded by the number of deductions and boolean branches performed by NO on the same problem. We prove the result for different deduction capabilities of the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\mathcal({T}-solver)}$\end{document} and for both convex and non-convex theories.</abstract><cop>Berlin, Heidelberg</cop><pub>Springer Berlin Heidelberg</pub><doi>10.1007/11916277_36</doi><tpages>15</tpages><oa>free_for_read</oa></addata></record>
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subjects Applied sciences
Artificial intelligence
Computer science
control theory
systems
Decision Procedure
Exact sciences and technology
Input Formula
Interface Equality
Satisfiability Modulo Theory
Truth Assignment
title Delayed Theory Combination vs. Nelson-Oppen for Satisfiability Modulo Theories: A Comparative Analysis
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