I. Stewart. Complete problems for symmetric logspace involving free groups. Information Processing Letters, 40:263-267, 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....exist rules (a i c; da i 1 ; 2 R. Thus (a i ; a i 1 ) 2 T . ut The preceding lemma is the key for proving that the UWP for the class of 2homogeneous STSs is in SL. In general it is quite dicult to prove that a problem is contained in SL. A useful strategy developed in [12] and applied in [25, 26] is based on a logical characterization of SL. In the following we consider nite structures of the form A = f0; n 1g; 0; max; s; R) Here max = n 1, and R and s are binary relations on f0; n 1g, where s(x; y) holds if and only if y = x 1. The logic FO posSTS is the set of all ....

I. Stewart. Complete problems for symmetric logspace involving free groups. Information Processing Letters, 40:263-267, 1991.

....group generated by S. Let S = fs; s Gamma1 js 2 Sg, where s Gamma1 denotes the inverse of s. Let S denote the set of all finite words over S. Let U = fu 1 ; um g S , where m 2 N and let x 2 S . Problem: Is x 2 hUi That is, is x in the subgroup of F generated by U Reference: [AM84b, Ste89] Hint: Stewart reported an error in the result contained in [AM84b] The reduction in [AM84b] is a generic one from a normal form Turing machine. However, it reduces a Turing machine computation to a version of GWP where S is a countably infinite set [Ste89] Stewart shows using the Nielsen ....

....of F generated by U Reference: AM84b, Ste89] Hint: Stewart reported an error in the result contained in [AM84b] The reduction in [AM84b] is a generic one from a normal form Turing machine. However, it reduces a Turing machine computation to a version of GWP where S is a countably infinite set [Ste89]. Stewart shows using the Nielsen reduction algorithm that this problem is still in P . Stewart calls this P complete problem the generalized word problem for countably generated free groups (GWPC) He shows that GWPC is log space reducible to GWP thus proving GWP is P complete as well [Ste89] ....

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I. A. Stewart. Complete problems for symmetric logspace involving free groups. Technical Report 300, University of Newcastle upon Tyne, 1989. To appear, Information Processing Letters.

....group generated by S. Let S = fs; s 01 js 2 Sg, where s 01 denotes the inverse of s. Let S 3 denote the set of all finite words over S. Let U = fu 1 ; um g S 3 , where m 2 N and let x 2 S 3 . Problem: Is x 2 hUi That is, is x in the subgroup of F generated by U Reference: [AM84b, Ste89] Hint: Stewart reported an error in the result contained in [AM84b] The reduction in [AM84b] is a generic one from a normal form Turing machine. However, it reduces a Turing machine computation to a version of GWP where S is a countably infinite set [Ste89] Stewart shows using the Nielsen ....

....of F generated by U Reference: AM84b, Ste89] Hint: Stewart reported an error in the result contained in [AM84b] The reduction in [AM84b] is a generic one from a normal form Turing machine. However, it reduces a Turing machine computation to a version of GWP where S is a countably infinite set [Ste89]. Stewart shows using the Nielsen reduction algorithm that this problem is still in P . Stewart calls this P complete problem the generalized word problem for countably generated free groups (GWPC) He shows that GWPC is log space reducible to GWP thus proving GWP is P complete as well [Ste89] ....

[Article contains additional citation context not shown here]

I. A. Stewart. Complete problems for symmetric logspace involving free groups. Technical Report 300, University of Newcastle upon Tyne, 1989. To appear, Information Processing Letters.

....s 2 Sigmag. A word over Sigma is a string of symbols from Sigma [ Sigma Gamma1 ; the length of the word is the length of the string. A word w over Sigma is reduced if there are no consecutive occurrences in w of the symbols s and s Gamma1 , or s Gamma1 and s, for any s 2 Sigma. See [26] for further details. Problem: Is w 2 hW i For a finite set of reduced words W = fw 1 ; w t g over Sigma, hW i denotes the group generated by w 1 ; w t . Reference: Stewart [26] Hint: The logical characterization of SL by Immerman as (FO posSTC) 11] is used to show ....

....occurrences in w of the symbols s and s Gamma1 , or s Gamma1 and s, for any s 2 Sigma. See [26] for further details. Problem: Is w 2 hW i For a finite set of reduced words W = fw 1 ; w t g over Sigma, hW i denotes the group generated by w 1 ; w t . Reference: Stewart [26]. Hint: The logical characterization of SL by Immerman as (FO posSTC) 11] is used to show GWPC(2) is in SL. For the completeness proof see [26, Corollary on page 267] Remarks: GWPC(2) is called the Generalized Word Problem for finitelygenerated subgroups of Countably generated free groups ....

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Iain A. Stewart. Complete problems for symmetric logspace involving free groups. Information Processing Letters, 40(5):263--267, December 1991.

.... translations [58] the method of quantifier elimination has been sketched above) Other problems have been shown to be complete for various complexity classes (including NSYMLOG and the classes Sigma p i and Pi p i of the Polynomial Hierarchy) via quantifier free projections (see, e.g. [36, 54, 55, 57, 68]) Complete problems for complexity classes consisting of monotone problems via monotone quantifier free projections, have been given in [40, 62, 67] The apparent anomaly that HP [FO s ] NP and 3COL [FO s ] L NP , yet both logics are formed by augmenting FO s with operators ....

I.A. Stewart, Complete problems for symmetric logspace involving free groups, Inform. Process. Lett. 40 (1991) 263--267.

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Iain A. Stewart, Complete problems for symmetric logspace involving free groups. Information Processing Letters 40(5) (1991), 263--267.

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