Neil Immerman. Languages that capture complexity classes. SIAM Journal of Computing, 16(4):760--778, August 1987.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....the base type b in addition to the equality test. Assume that the test for connectivity of graphs of type fb Theta bg is not definable in RL g) Then TC is different from NLOGSPACE. 2 By changing a query from connectivity to a DLOGSPACE complete one (e.g. deterministic transitive closure [24]) we can obtain a similar result showing that nondefinability of deterministic transitive closure in RL g) implies the separation between TC and DLOGSPACE. Note that there are other known cases when expressivity bounds for query languages cannot be proved without separating complexity ....

N. Immerman. Languages that capture complexity classes. SIAM Journal on Computing 16 (1987), 760-- 778.

....and descriptive complexity theory. For example, extending rst order logic FO with a vectorized sequence of Lindstr om quanti ers corresponding to the problem TC is an elementary mechanism by which one can augment FO with a (limited) means of recursion. In what is now a seminal result, Immerman [8, 9] showed that the resulting logic, transitive closure logic ( TC) FO] restricted to the class of ordered structures captures exactly the complexity class NL (with the corollary that NL is closed under complementation) Also, Fagin [5] used the problem STRCONN, i.e. the class of nite ....

....B j= So for every A 2 A j= 9X 1 9X 2 : 9X q where the X i are chosen to be the free second order variables of . Similarly, for every B 62 B j= 9X 1 9X 2 : 9X q : Hence, 9X 1 9X 2 : 9X q is a mon ( formula which de nes 4 Playing our game In [8], Immerman exhibited a particularly strong normal form result for formulae of transitive closure logic on the class of ordered structures. This normal form result was established by induction on the symbolic complexity of a formula. In fact, his case bycase analysis goes through even on unordered ....

N. Immerman, Languages that capture complexity classes, SIAM Journal of Computing 16 (1987) 760-778.

....whether existential second order formulae are less expressive than full second order formulae. At this point it is tempting to give a logical description of other complexity classes. The four results below give the flavor of this theory. The reader is referred to the original article of Immerman [31] for more information on this topic. Let L (resp. NL, PSPACE) be the class of sets accepted by a deterministic Turing machine in logarithmic space (resp. by a non deterministic Turing machine in logarithmic space, by a deterministic Turing machine in polynomial space) The following inclusions are ....

....by F LFP the set of formulae expressible using F plus the operator LFP . 16 is definable in F 1 (S) LFP if and only if it belongs to the class P . These results are stated when formulae are interpreted on finite words only. For an extension of these results to finite structures see Immerman [31] and the beautiful survey of Fagin [25] 3.3. Monadic second order Considering only monadic second order formulae on words is a much more drastic restriction. Theorem 3.8. Buchi [12] Elgot [22] Let L be a subset of A ) The following conditions are equivalent: 1) L is definable in ....

N. Immerman, 1987, Languages that capture complexity classes, SIAM J. Comput. 16, 760--778.

....Proposition 9 Every query on nite constraint databases that is computable in NL is expressible in L PATH . Proof : Directed reachability is the following query: given a directed graph G, compute the set of all pairs (v; w) of vertices of G such that there is a path from v to w. It was proved in [19] that directed reachability is complete for NL under rst order reductions. Since it can easily be seen that the class of nite database queries de nable in L PATH is closed under rst order reductions, it suces to prove that directed reachability is expressible in L PATH . Finite directed ....

N. Immerman. Languages that capture complexity classes. SIAM J. Comput. 16 (1987), 760-778.

.... an ordering) the PSPACE computable queries are precisely the queries expressible in While (renamed PFP in [AbV91] ffl In [V82] and [I83] it was proven that the PTIME computable queries are precisely the queries expressible in (positive elementary) fixed point logic (given an ordering) ffl In [I87], it was proven that the NLOGSPACE computable queries are precisely the queries expressible in transitive closure logic (given an ordering) These and similar results appearing in the 1980s formed the beginning of what has become Descriptive Complexity Theory, which is described in [I99] ....

....following recursion. ffl All First Order formulas are in FO pos TC. ffl If is an FO pos TC formula with an even number of arguments, then TC[ is in FO pos TC. ffl If and are in FO pos TC, then so are , 9x , and 8y . This is an important logic: Theorem 1. 2 (Immerman [I87]) In the presence of an ordering, the logic FO pos TC expressible queries are precisely the NLOGSPACE computable queries. It is quirkier than FO LFP: Remark 1.2 In general, the class of FO pos TC expressible queries is not closed under negation ( GrM96] however, in the presence of an ....

N. Immerman, Languages that capture complexity classes, SIAM J. Computing 16 (1987), 760--778.

.... restricted Henkin quantifiers capture NL [9] over ordered structures, while standard Henkin quantifiers capture Theta over ordered structures [22] transitive closure logic captures NL over ordered structures, while deterministic transitive closure logic captures L over ordered structures [30, 25]; and, various extensions of first order logic by quantifiers such as the Hamiltonian Path Quantifier capture NP or Theta 2 (i.e. logspace with an NP oracle) over ordered structures, 53, 54, 22] The relationship of oracles, generalized quantifiers and subprograms was also investigated in ....

....Struct(oe) be problems. Pi is L reducible to Sigma if there exist an interpretation I of into oe, s.t. for all A 2 Struct( A 2 Pi iff I(A) 2 Sigma where k is the arity of I. By restricting the logic L for the interpretations, we obtain low level reductions: A quantifier free reduction [30] is a reduction whose defining formulas are quantifier free. Logic Programs. A logic program is a finite collection of clauses (also called rules) of the form A B 1 ; Bm (1) where A is an atom and the B i s are literals in a first order language. We use : for the negation (as failure) ....

N. Immerman. Languages that Capture Complexity Classes. SIAM Journal of Computing, 16:760--778, 1987.

....From Proposition 2.2 and the hypothesis, it follows that ESO(Q) captures REG; it is wellknown that REG is closed under complementation. Corollary 12.6 ESO(9 8) is the unique maximal regular ESO prefix class which does not capture REG. Note that Theorem 12.3, Corollary 12.6 and the results in [25] imply that model checking for regular ESO prefix classes which do not capture REG is in low levels of AC . From the dichotomy theorem for model checking (Theorem 10.5) we thus obtain the following dichotomy theorem for closure under complementation. Theorem 12.7 (Dichotomy Theorem for ....

N. Immerman. Languages that Capture Complexity Classes. SIAM Journal of Computing, 16:760--778, 1987.

....[1] In this case, for each D in D , P(D) can be partitioned into just two strata: D and P. 3 Semantics and complexity of DATALOG TM queries We assume that the reader is familiar with the basic notions of complexity classes [18,27] and of query language complexity evaluation (see for example [2, 3, 6, 8, 17, 19 21, 36, 41]) Definition 1. A (bound) DATALOG TM query Q is a pair (v G, P) where G is a ground literal, called query goal, P is a DATALOG program and v is a symbol denoting the type of semantics adopted for the query: for nondeterministic semantics, 3 for possible semantics, V for certain semantics ....

Immerman, N.: Languages that Capture Complexity Classes. SIAM J Comput 16(4), 76(b778, 1987.

No context found.

Neil Immerman. Languages that capture complexity classes. SIAM Journal of Computing, 16(4):760--778, August 1987.

No context found.

N. Immerman. Languages that capture complexity classes. SIAM Journal of Computing, 16(4):760--778, Aug. 1987.

No context found.

N. Immerman. Languages that capture complexity classes. SIAM Journal of Computing, 16(4):760--778, Aug. 1987.

No context found.

N. Immerman. Languages that capture complexity classes. SIAM Journal of Computing, 16:760--778, 1987.

No context found.

N. Immerman. Languages that capture complexity classes. SIAM Journal of Computing, 16:760--778, 1987.

No context found.

N. Immerman, Languages that capture complexity classes, SIAM Journal of Computing 16 (1987), 760--778.

No context found.

N. Immerman, Languages that capture complexity classes, SIAM Journal of Computing 16 (1987), 760--778.

No context found.

N. Immerman. Languages that capture complexity classes. SIAM Journal of Computing, 16:760--778, 1987.

No context found.

N. Immerman, Languages that Capture Complexity Classes, SIAM J. Comput. 16 (1987), 760--778.

No context found.

N. Immerman. Languages that capture complexity classes. SIAM Journal of Computing, 16:760-- 778, 1987.

No context found.

N. Immerman, "Languages That Capture Complexity Classes", SIAM J. Computing 16 (1987) 760--778.

No context found.

N. Immerman, "Languages that Capture Complexity Classes,' SIAM J. Comput. 16:4 (1987), 760 -- 778; MR 88j:68051.

No context found.

N. Immerman. Languages that capture complexity classes. SIAM J. of Computing, 16:4:760--778, 1987.

No context found.

N. Immerman, Languages that capture complexity classes, SIAM Journal on Computing, 16 (1987), pp. 760--778.

No context found.

N. Immerman, "Languages that capture complexity classes," SIAM J. Comput. 16 (1987) 760--778.

No context found.

N. Immerman. Languages that capture complexity classes. SIAM Journal on Computing, 16:760--778, 1987.

No context found.

N. Immerman. Languages that capture complexity classes. SIAM Journal of Computing, 16:760--778, 1987.

First 50 documents  Next 50

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