James F. Lynch. Complexity classes and theories of finite models. Mathematical Systems Theory, 15:127--144, 1982.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... the class PSPACE and noninflationary fixpoint logic [Var82] cf. AV89] 1 The tight connection between descriptive and computational complexity, typically referred to as the connection between logic and complexity , was then proclaimed by Immerman [Imm87b] and studied by many researchers [Com88, Goe89, Gra84, Gra85, Gur83, Gur84, Gur88, HP84, Imm89, Lei89a, Liv82, Liv83, Lyn82, Saz80b, Saz80a, TU88]. 2 See [Imm89] for a survey. Although the relationship between descriptive and computational complexity is intimate, it is not without its problems, and the partners do have some irreconcilable differences. While computational devices work on encodings of problems, logic is applied directly to ....

J. F. Lynch. Complexity classes and theories of finite models. Mathematical System Theory, 15:127--144, 1982.

....is a function free first order formula in which at least the predicates in S; T occur. Following the traditional notion of a set of logical sentences capturing a complexity class, we can say that the set SO 9 captures the class NP (cf. Fagin1974] while the set SO 98 captures Sigma p 2 (cf. Lynch1982, Stockmeyer1977] See [Garey and Johnson1979, Johnson1990] for a definition of the classes NP and Sigma p 2 of the polynomial hierarchy) We are now ready to prove our main result, which concerns DQL boolean queries. We refer to the following useful lemmas. Let Cons( denote classical ....

J.F. Lynch. Complexity Classes and Theories of Finite Models. Mathematical Systems Theory, 15:127--144, 1982.

....of all queries definable by query expressions from L, i.e. E(L) fq : q = E) E 2 Lg. As well known, second order definability over finite structures is closely connected to computability within the polynomial hierarchy PH = Sigma P 1 [ Sigma P 2 [ Delta Delta Delta ( Sigma P 1 = NP) [Sto77, Lyn82]. For vocabulary oe denote by Sigma 1 k (oe) the second order sentences of the form (9S 1 ) 8S 2 ) Delta Delta Delta (QS k ) S 1 ; S k ) where the quantifiers alternate and is first order; we write Sigma 1 k if oe is understood. Proposition 2.1 [Fag74, Sto77, Lyn82] A ....

....P 1 = NP) Sto77, Lyn82] For vocabulary oe denote by Sigma 1 k (oe) the second order sentences of the form (9S 1 ) 8S 2 ) Delta Delta Delta (QS k ) S 1 ; S k ) where the quantifiers alternate and is first order; we write Sigma 1 k if oe is understood. Proposition 2. 1 [Fag74, Sto77, Lyn82] A database property P over R is in Sigma P k , k 1, iff there exists a Sigma 1 k (R) sentence Psi such that for each D 2 D(R) D j= Psi iff P(D) true. 3 Disjunctive datalog 3.1 Syntax A disjunctive datalog rule is a clause of the form a 1 Delta Delta Delta an t 1 ; Delta ....

J.F. Lynch. Complexity Classes and Theories of Finite Models. Mathematical Systems Theory, 15:127--144, 1982.

....this problem in unit time) in NP (cf. 23] Consequently, unless p 2 = NP (which is widely conjectured to be false) DQL is much more expressive than DATALOG : stable . Finally, the queries de nable in full second order logic are those computable within the polynomial hierarchy [34] [35]. B. Default logic Default logic has been introduced by Reiter in [11] it is one of the most extensively studied non monotonic formalisms. For a detailed treatment of this formal system, the reader is referred to [36] Interesting relations between default logic and database theory have been ....

J.F. Lynch, \Complexity Classes and Theories of Finite Models ", Mathematical Systems Theory, vol. 15, pp. 127-144, 1982.

.... Furthermore there have been results that establish an even closer connection between the arity of the quantified relations in a formula which characterizes a problem and the degree of a polynomial which bounds the running time of a nondeterministic Turing machine which solves the problem (cf. [Lyn82]) As in the case of circuit size and running time of Turing machines we are far from resolving the P vs. NP question by making use of Fagin s Theorem. On the other hand there have been a lot of results concerning the expressive power of Sigma 1 1 formulas, in which the second order ....

J. F. Lynch. Complexity classes and theories of finite models. Mathematical System Theory, 15:127--144, 1982.

....In Section 4 it is shown that unary functions are stronger than bijections. I want to thank Arnaud Durand for many very helpful discussions and Clemens Lautemann for giving the inspiration for the proof of Theorem 5. 1 for a similar connection between BinNP and quadratic time on NTMs see [Lyn82]. partial order relations arbitrary binary relations j 6= linear order relations unary functions equivalence relations graphs with bounded outdegree j 6= successor relations connected graphs with bounded degree graphs with bounded degree bijective unary ....

J. F. Lynch. Complexity classes and theories of finite models. Mathematical System Theory, 15:127--144, 1982.

.... There is a distance respecting isomorphism ff from G 1 Gamma H 1 to G 2 Gamma H 2 (i.e. d(x; H 1 ) d(ff(x) H 2 ) for every x 2 G 1 Gamma H 1 ) 5 For the corresponding modified set game, Fagin defines an equivalent logic in [Fag96] 3 The Power of Partial Orders It was shown by Lynch [Lyn82, Lyn92] that, for every k 1, k ary NP captures at least all sets of strings that are accepted by a nondeterministic Turing machine in time O(n k ) In particular, BinNP captures nondeterministic quadratic time on strings. It is an interesting open problem to find a (natural) graph problem that is ....

J. F. Lynch. Complexity classes and theories of finite models. Mathematical System Theory, 15:127--144, 1982.

....the spoiler to change the topology of the given structure strongly. As we want to close the gap between logics that capture complexity classes and logics that allow non expressibility results, we go one step further and consider linear time on Turing machines. It was already shown by Lynch in [Lyn82a, Lyn92] that NTIME(n k ) is captured by ESO logic, in which quantification is restricted to k ary relations (k ary ESO, for short) and that NTIME(n) is captured by monadic ESO logic with addition. That a class is captured by a logic shall mean that every set in the class is characterized by a formula ....

J. F. Lynch. Complexity classes and theories of finite models. Mathematical System Theory, 15:127--144, 1982.

....) Theorem 4.2, combined with Theorem 4.1 (b) and (c) shows that SO is much more expressive than mSO, and SO9( is similarly more expressive than mSO9( A seemingly smaller change to mSO9 also results in a leap of expressiveness from the regular languages to the level of NP. The paper [ Lynch, 1982 ] showed that if we consider mSO9( instead of mSO9( for strings) then the resulting class contains NTIME[n] and hence contains many NP complete languages, such as Graph Three Colorability. 5 Interactive Models and Complexity Classes 5.1 Interactive Proofs In Section 2.2 of Chapter 27, we ....

J. Lynch. Complexity classes and theories of finite models. Math. Sys. Thy., 15:127-- 144, 1982.

....formula. In order to make the above statements more precise we need some standard definitions from descriptive computational complexity that we introduce in a simplified manner which mixes together syntactical and semantical notions. For a rigorous treatment see [Fag74] Gra84] Imm89] Lyn82] Definition 2.1 A finite type is a finite sequence of nonnegative integers. Given a finite type T = n 1 ; n 2 ; n k ) a finite T structure is a (k 1) tuple F = X; f 1 ; f 2 ; f k ) where X is a non empty finite set called the domain of the structure F and, for all i, f i ....

J. Lynch. Complexity classes and theories of finite models. Mathematical Systems Theory, 15:127--144, 1982.

....an isomorphic copy of G and some isolated vertices. A set A of graphs is called weakly expressible by a formula in the presence of padding, if is able to distinguish between (sufficiently) padded versions of graphs from A and padded versions of graphs that are not in A. From results of Lynch [Lyn82, Lyn92] it can be easily concluded that (essentially) every NP set of graphs is weakly expressible by an existential monadic second order (Mon Sigma 1 1 ) formula with polynomial padding and built in addition. In particular, NP 6= coNP if and only if there is a coNP set of graphs that is not weakly ....

....of G and at least f(jGj) Gamma jGj isolated vertices. We say that a formula weakly expresses a set A of graphs with f(n) padding, if holds for all f(n) padded versions of graphs from A and does not hold for any f(n) padded version of a graph that is not in A. From the results of Lynch in [Lyn82, Lyn92] (see also [GO96] it can easily be concluded that every (well behaved) NP set A of graphs is weakly expressible by a Mon Sigma 1 1 formula with polynomial padding, in the presence of built in addition. Intuitively, this holds because Mon Sigma 1 1 is able, in the presence of addition, to use ....

[Article contains additional citation context not shown here]

J. F. Lynch. Complexity classes and theories of finite models. Mathematical System Theory, 15:127--144, 1982.

....formulas over . Clearly, FOL( SOL( For OE 2 SOL( we denote by Mod(OE) the class of finite structures A such that (A) j= OE. Let F = F ( SOL( A class of finite (ordered) structures K is F definable if it is of the form Mod(OE) for some OE 2 F . It is well known, [Lyn82], that K is SOL definable iff K is in the polynomial hierarchy PH. We investigate the expressive power of second order logic over finite structures, when two limitations are imposed: On the number of alternations of quantifiers and on the arity of the second order variables. A preliminary ....

....We look only at formulas in doubly prenex normal form where the first order part is of the form 8 x9 y and then count only second order quantification and restrict the arities of the second order variables. We leave it to the reader to restate our results From this point of view. Fact 8 (Lynch [Lyn82]) If K is in NP and recognizable in NT IME(n d ) then K is definable in SAA(d; 1) existential) and even AA(d; c) for some c 2 N. Remark. The converse of fact 8 is not true. The reason is, that even in AA(d; c) we count only blocks of existential (universal) quantifiers. If a converse were ....

J.F. Lynch. Complexity classes and theories of finite models. Mathematical Systems Theory, 15:127--144, 1982.

....is in accordance with known results about decision problems. However, whereas in the presence of a linear order on inputs, the proofs there are quite involved ( dR87, AF90] here the extension of the proof to ordered structures is rather straightforward. In the light of Lynch s results in [Ly82], it would be interesting to see if it can even be extended to addition structures. 11 The chromatic index of a graph is the chromatic number of the corresponding edge graph. Acknowledgements I would like to thank Elias Dahlhaus for his insightful comments and suggestions. ....

James F. Lynch, Complexity Classes and Theories of Finite Models. Math. Syst. Theory 15, pp. 127--144.

.... class PSPACE and noninflationary fixpoint logic [Var82] see also [Imm82] 1 The tight connection between descriptive and computational complexity, typically referred to as the connection between logic and complexity , was then proclaimed by Immerman [Imm87b] and studied by many researchers [Com88, Goe89, Gra84, Gra85, Gur83, Gur84, Gur88, HP84, Imm89, Lei89a, Liv82, Liv83, Lyn82, Saz80b, Saz80a, TU88]. 2 See [Imm89] for a survey. Although the relationship between descriptive and computational complexity is intimate, it is not without its problems, and the partners do have some irreconcilable differences. While computational devices work on encodings of problems, logic is applied directly ....

J. F. Lynch. Complexity classes and theories of finite models. Mathematical System Theory, 15:127--144, 1982.

....formula. In order to make the above statements more precise we need some standard definitions from descriptive computational complexity that we introduce in a simplified manner which mixes together syntactical and semantical notions. For a rigorous treatment see [Fag74] Gra84] Imm89] Lyn82] Definition 2.1 A finite type is a finite sequence of non negative integers. Given a finite type T = n 1 ; n 2 ; n k ) a finite T structure is a (k 1) tuple F = X; f 1 ; f 2 ; f k ) where X is a non empty finite set called the domain of the structure F and for all i, f i ....

J. Lynch. Complexity classes and theories of finite models. Mathematical Systems Theory, 15:127--144, 1982.

.... the class PSPACE and noninflationary fixpoint logic [Var82] cf. AV89] 1 The tight connection between descriptive and computational complexity, typically referred to as the connection between logic and complexity , was then proclaimed by Immerman [Imm87b] and studied by many researchers [Com88, Goe89, Gra84, Gra85, Gur83, Gur84, Gur88, HP84, Imm89, Lei89a, Liv82, Liv83, Lyn82, Saz80b, Saz80a, TU88]. 2 See [Imm89] for a survey. Although the relationship between descriptive and computational complexity is intimate, it is not without its problems, and the partners do have some irreconcilable differences. While computational devices work on encodings of problems, logic is applied directly to ....

J. F. Lynch. Complexity classes and theories of finite models. Mathematical System Theory, 15:127--144, 1982.

....of natural structures. Then K is in NP iff K is definable by an existential formula of Second Order Logic. In our terminology the set of existential formulas of Second Order Logic captures NP. This is a special case of a more general theorem due to J. Lynch, generalizing a result of Stockmeyer [Lyn82] noting that NP is just one level in the polynomial hierarchy PH. Recall, cf. GJ79] that PH is the union of the complexity classes Sigma n P and Pi n P , where Sigma 0 P = Pi 0 P = P, Sigma 1 P is NP and Pi 1 P is Co Gamma NP. In general, Sigma k 1 P is defined as the class ....

....accurate question is how to give LC a natural syntax. For the case where C(T) equals the recursive sets of natural structures such a characterization is easy and was given in section 2.1. For second order logic and some of its sub logics complexity classes have been identified in [Fag75] and [Lyn82] as we have seen in section 2.1. In [Imm88] the logic FOL[DTC] was introduced which is regular in our sense. FOL[DTC] is an extension of first order logic, where operators are added expressing the deterministic transitive closure for binary relations over k tuples for every k 2 IN. This ....

J.F. Lynch. Complexity classes and theories of finite models. Mathematical Systems Theory, 15:127--144, 1982.

No context found.

James F. Lynch. Complexity classes and theories of finite models. Mathematical Systems Theory, 15:127--144, 1982.

No context found.

James Lynch, "Complexity Classes and Theories of Finite Models," Math. Sys. Theory 15, 1982, (127-144).

No context found.

J. Lynch, Complexity classes and theories of finite models, Math. Systems Theory 15, 1982, pp. 127--144.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC