T. Harju, J. Karhumki, and D. Krob. Remarks on generalized Post Correspondence Problem. Lecture Notes in Comput. Sci. 1046, pages 3948. Springer-Verlag, 1996.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... questions are left open by this research: ffl Is exponential time the best we can do when deciding marked PCP, or is there a polynomial time algorithm for the problem ffl What about decidability of strongly k marked PCP for 1 k 5 ffl What about decidability of marked generalized PCP [EKR82, HKK96] ffl The decidability status of PCP with elementary morphisms [Sal81, pp. 7277] is still open. A morphism g is elementary if it cannot be written as a composition g 2 g 1 via a smaller alphabet. Marked PCP is a subcase of elementary PCP which we have shown here to be decidable. Can our results ....

T. Harju, J. Karhum#ki, and D. Krob. Remarks on generalized Post correspondence problem. In Proceedings of 13th STACS, volume 1046 of Lecture Notes in Computer Science, pages 3948. Springer-Verlag, 1996.

....proof of Theorem 3.2 is nonconstructive. As such it is, however, if not very short, at least elementary and drastically shorter than Combinatorics of words 23 the existing decidability proofs of PCP(2) cf. EKR1] Pav] or also [HK2] in this handbook, which are about 20 pages long. As shown in [HKK] our existential proof of Theorem 3.2 can be made constructive, if an algorithm for PCP(2) or in fact for its slight generalization so called GPCP(2) for definitions cf. HK2] is known. Moreover, the arguments used in [HKK] to conclude this are short. As a conclusion from above, we know that ....

....also [HK2] in this handbook, which are about 20 pages long. As shown in [HKK] our existential proof of Theorem 3.2 can be made constructive, if an algorithm for PCP(2) or in fact for its slight generalization so called GPCP(2) for definitions cf. HK2] is known. Moreover, the arguments used in [HKK] to conclude this are short. As a conclusion from above, we know that the equality set of two binary morphisms h and g is always of one of the three different forms, namely L q , for some q 2 Q [ f1g, fff; fig or (fffl fi) for some words ff; fi; fl 2 fa; bg . Moreover, we can ....

T. Harju, J. Karhumaki and D. Krob, Remarks on generalized Post Correspondence Problem, Springer LNCS 1046, 39--48, Springer-Verlag, 1996.

.... questions are left open by this research: Is exponential time the best we can do when deciding marked PCP, or is there a polynomial time algorithm for the problem What about decidability of strongly k marked PCP for 1 k 5 What about decidability of marked generalized PCP [1, 3] The decidability status of PCP with elementary morphisms [9, pp. 72 77] is still open. A morphism g is elementary if it cannot be written as a composition g 2 g 1 via a smaller alphabet. Marked PCP is a subcase of elementary PCP which we have shown here to be decidable. Can our results ....

T. Harju, J. Karhumaki, and D. Krob. Remarks on generalized Post correspondence problem. In Proceedings of 13th STACS, volume 1046 of Lecture Notes in Computer Science, pages 39--48. Springer-Verlag, 1996.

....undecidable. The following questions remain open: ffl Is polynomial space the best we can do when deciding marked PCP or is the problem solvable even in polynomial time ffl What about decidability of strongly k marked PCP for 1 k 5 ffl What about decidability of marked generalized PCP [1,3] ffl The decidability status of PCP with elementary morphisms [9, pp. 72 77] is still open. A morphism g is elementary if it cannot be written as a composition g 2 g 1 via a smaller alphabet. Marked PCP is a subcase of elementary PCP which we have shown here to be decidable. Can our results ....

T. Harju, J. Karhumaki, and D. Krob. Remarks on generalized Post correspondence problem. In Proceedings of 13th STACS, volume 1046 of Lecture Notes in Computer Science, pages 39--48. Springer-Verlag, 1996.

....be investigated by restricting the morphisms. For example it is known that the binary PCP is decidable, i.e. the PCP is decidable for instances of size jAj 2, see [1] and [3] and it is undecidable, if jAj 7, see [7] The same bounds hold for the GPCP, see [1] and [3] for the binary case and [5] for the undecidability of the GPCP for jAj = 7. For the PCP and GPCP, the decidability status is open for the sizes between these two bounds. In this paper we shall restrict the lengths of the images. Note that it is clear that if the images are all of length one, then the PCP and the GPCP are ....

T. Harju, J. Karhumki, and D. Krob. Remarks on generalized Post Correspondence Problem. Lecture Notes in Comput. Sci. 1046, pages 3948. Springer-Verlag, 1996.

....a solution. We shall denote the instance of the GPCP by ( p 1 ; p 2 ) h; g; s 1 ; s 2 ) The pair (p 1 ; p 2 ) is called the begin words and (s 1 ; s 2 ) is called the end words. Note that also for the GPCP it is known that it is decidable, if jAj 2, see [2] and undecidable, if jAj 7, see [6]. As for the PCP, the decidability status of the GPCP is open for the alphabet size between these two bounds. The basic idea in [2] is that each instance (h; g) of the binary PCP is either (1) periodic, i.e. h(A ) u , where u 2 B , or (2) it can be reduced to an equivalent instance ....

T. Harju, J. Karhumki, D. Krob. Remarks on generalized Post Correspondence problem. Lecture Notes in Comput. Sci, vol 1046, SpringerVerlag, Berlin, 1996 pp. 39-48.

....1 = p 2 g 2 (w)s 2 : 2.1) The 6 tuple J = p 1 ; p 2 ; s 1 ; s 2 ; g 1 ; g 2 ) is called an instance of the GPCP and a word w satisfying the equation (2.1) is called a solution of J . It is known that the GPCP is decidable when j j 2, see [2] or [6] and it is undecidable when j j 7, see [8]. For the alphabets of sizes between these limits the decidability status of the GPCP is still open. For an instance I = g 1 ; g 2 ) of the PCP, let E(I) fw 2 j g 1 (w) g 2 (w)g be its equality set. Similarly, for an instance J = p 1 ; p 2 ; s 1 ; s 2 ; g 1 ; g 2 ) of the GPCP, we shall ....

T. Harju, J. Karhumaki, and D. Krob, Remarks on generalized Post Correspondence Problem, STACS'96, Lecture Notes in Comput. Sci., vol. 1046, Springer-Verlag, 1996, pp. 39-48.

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