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First Order Logic, Fixed Point Logic and Linear Order University of Pennsylvania ScholarlyCommons IRCS Technical Reports Series Institute for Research in Cognitive Science November 1995 First Order Logic, Fixed Point Logic and Linear Order Anuj Dawar University of Pennsylvania Steven Lindell University of Pennsylvania Scott Weinstein University of Pennsylvania, [email protected] Follow this and additional works at: https://repository.upenn.edu/ircs_reports Dawar, Anuj; Lindell, Steven; and Weinstein, Scott, "First Order Logic, Fixed Point Logic and Linear Order" (1995). IRCS Technical Reports Series. 145. https://repository.upenn.edu/ircs_reports/145 University of Pennsylvania Institute for Research in Cognitive Science Technical Report No. IRCS-95-30. This paper is posted at ScholarlyCommons. https://repository.upenn.edu/ircs_reports/145 For more information, please contact [email protected]. First Order Logic, Fixed Point Logic and Linear Order Abstract The Ordered conjecture of Kolaitis and Vardi asks whether fixed-point logic differs from first-order logic on every infinite class of finite deror ed structures. In this paper, we develop the tool of bounded variable element types, and illustrate its application to this and the original conjectures of McColm, which arose from the study of inductive definability and infinitary logic on proficient classes of finite structures (those admitting an unbounded induction). In particular, for a class of finite structures, we introduce a compactness notion which yields a new proof of a ramified ersionv of McColm's second conjecture. Furthermore, we show a connection between a model-theoretic preservation property and the Ordered Conjecture, allowing us to prove it for classes of strings (colored orderings). We also elaborate on complexity-theoretic implications of this line of research. Comments University of Pennsylvania Institute for Research in Cognitive Science Technical Report No. IRCS-95-30. This technical report is available at ScholarlyCommons: https://repository.upenn.edu/ircs_reports/145 Institute for Research in Cognitive Science First Order Logic, Fixed Point Logic and Linear Order Anuj Dawar Steven Lindell Scott Weinstein University of Pennsylvania 3401 Walnut Street, Suite 400C Philadelphia, PA 19104-6228 November 1995 Site of the NSF Science and Technology Center for Research in Cognitive Science IRCS Report 95--30 First Order Logic Fixed Point Logic and Linear Order Anuj Dawar Steven Lindell Scott Weinstein Dept of Computer Science Univ of Wales Swansea Swansea SA PP UK email adawarswanseaacuk Dept of Computer Science Haverford College Haverford PA USA email slindellhaverfordedu Dept of Philosophy Univ of Pennsylvania Philadelp hi a PA USA email weinsteincisupennedu Abstract The Ordered conjecture of Kolaitis and Vardi asks whether xedp oint logic diers from rstorder logic on every innite class of nite ordered structures In this pap er wedevelop the to ol of b ounded variable elementtyp es and illustrate its application to this and the orig inal conjectures of McColm which arose from the study of inductive denability and innitary logic on procient classes of nite structures those admitting an unb ounded induction In particular for a class of nite structures weintro duce a compactness notion which yields a new pro of of a ramied version of McColms second conjecture Furthermore we show a connection b etween a mo deltheoretic preservation prop erty and the Ordered Conjecture allowing us to prove it for classes of strings colored orderings We also elab orate on complexitytheoreti c implica tions of this line of research Intro duction The extensions of rst order logic by means of xed p oint op erators in particular the least xed p oint and partial xed p oint op erators have b een much studied in recentyears in the eld of nite mo del theory This is in large measure due to their connection with complexity classes ImmermanImmandVardi Var showed that the logic LFP the extension of rst order logic with a least xed p oint op erator captures the class PTIME on ordered structures Vardi Var and Abiteb oul and VianuAV showed that the similar extension of rst or der logic with a partial xed p oint op erator PFP captures the class PSPACE on ordered structures Furthermore Abiteb oul and VianuAV showed that LFP PFP if and only if PTIME PSPACE even without the restriction to ordered structures One of the most imp ortant to ols in the analysis of the xed p oint logics is the b ounded variable innitary logic L Kolaitis and Vardi Research supp orted byEPSRCgrant GRH Partially supp orted by NSF grant CCR and the John C Whitehead faculty research fundatHaverford College Supp orted in part by NSF CCR KVb showed that on the class of nite structures LFP and PFP can b e seen as fragments of L Moreover L has an elegantcharacterization in terms of p ebble games which has proved an extremely useful to ol in the analysis of the expressivepower of the xed p oint logics The logics LFP and PFP are b oth extensions of rst order logic and indeed they are prop er extensions on the class of all nite structures and on the class of ordered nite structures It also follows from the result of Abiteb oul and Vianu that if we can separate these two logics on any class of nite structures C then wewould separate PTIME from PSPACE On the other hand one can construct innite classes of structures on which the logics are equivalent and b oth of them indeed even L collapse to rst order logic Kolaitis and Vardi KVa initiated an investigation of which classes of structures C have the prop erty that LFP and L collapse to rst order logic on C They proved a conjecture of McColm McC showing that L collapses to FO if and only if every p ositive rstorder induction is b ounded Gurevich Immerman and Shelah GIS refuted another conjecture due to McColm by constructing a class of structures on which LFP collapses to FO but L do es conjectured the following weaker version of not Kolaitis and Vardi KVb McColms conjecture which remains op en Conjecture KolaitisVardi On every innite class of ordered structures thereisapolynomial time computable query that is not rst order denable In this pap er we discuss McColms conjectures relating them to nite vari able elementtyp es as intro duced in DLW a notion of compactness for classes of nite structures and a preservation prop erty In particular we relate this preservation prop erty to Conjecture allowing us to prove it for classes of strings linear orders with unary relations We also comment on the complexitytheo retic implications of Conjecture Parts of the material in this pap er app eared in preliminary form in Daw Section covers the background material on xed p oint logics innitary logics and elementtyp es Section relates inductive denitions and McColms conjectures to b ounded variable elementtyp es compactness and preservation prop erties Section discusses the relation b etween the preservation prop erties and Conjecture while Section relates this conjecture to questions in com plexity theory Background We assume the standard denitions of a rst order language or signature and a structure interpreting it Unless otherwise mentioned all structures we will b e dealing with are assumed to have nite universe and all signatures are assumed to b e nite and relational that is to consist of nitely many relation symb ols We write F to denote the class of all nite structures of signature and O to denote the class of ordered nite structures ie O is the collection of structures in F whichinterpret the binary relation symbol as a linear fg order An nary query over a class of structures C is a map Q sending each structure A C to an nary relation over A which satises the following condition for all A B C if f is an isomorphism from A onto B then QBf QA We will write FO LFP etc b oth to denote logics ie sets of formulas and the classes of queries that are expressible in the resp ective logics Wesay a logic L col lapses to another logic L over a class of structures C if and only if the collection of restrictions of queries in L to C is included in the collection of restrictions of queries in L to C Inductive and Innitary Logics Let R x x b e a rstorder formula On a structure A denes the k A A op erator R fha a ijhAR ij a a gIf is an Rp ositive k k A formula is monotone Wemay view as determining an induction on A the A m m The closure stages of which are dened as follows A A A A m m ordinal of on A denoted jjjj is the least m such that The A A A th y can b e uniformly dened over all m stage of the induction determined b m structures by a rstorder formula whichwe denote by The set inductively is the least xed p oint of the op erator that dened by on A denoted A A m is where m jjjj If s is a k tuple of elements of A and s A A A A m we use jsj to denote the least m suchthats The stage comparison query A for denoted is the query which assigns to each structure A the k ary relation dened as follows jsj js j s s s s A A where s and s are k tuples of elements of A We write LFP for the extension of rstorder logic with the lfp op eration which uniformly determines the least xed p ointofanRp ositiveformula That is for any Rp ositiveformula lfpR x x is a formula of LFP and k A j lfpR x x s if and only if s We will need the following k A basic result ab out inductive denabilitywhich is a sp ecial case of Moschovakiss Stage Comparison Theorem see Mos Theorem Mos Let beanR positive rstorder formula The stage comparison
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