G. Rozenberg, A. Salomaa. Cornerstones of Undecidability, Prentice Hall, Englewood Cli#s, 1994.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....: i k of indices such that ff i 1 : ff i k = fi i 1 : fi i k . The list L is an instance of the PCP, and a sequence i 1 ; i k with the above property is a solution of this instance. It is well known that it is undecidable in general whether an instance of the PCP has a solution [6]. In the following encoding of the PCP, a string a 1 : am (with m 0) will be encoded as a graph consisting of m consecutive edges labelled by a 1 ; am : a 1 a 2 . am Such a graph will be depicted also as follows: a 1 : a m Let now L = h(ff 1 ; fi 1 ) ff n ....

Grzegorz Rozenberg and Arto Salomaa. Cornerstones of Undecidability. Prentice Hall, 1994.

....letter equivalent with a regular language. Corollary 2.2.7. For any context free language L, r(L) is a semilinear lan guage. Moreover, for any semilinear set S, there is a context free language L such that = r(L) For a definition of Turing machines we refer to [74] as well as to [52] 54] [68], and [70] A language L C is recursively enumerable if there exists a Turing machine :h accepting L. The language L is recursive if there exists a Turing machine :h accepting L and halting on all input words. If :h is an always halting Turing machine, then we implicitly have an algorithm to ....

....of the problem Is u primitive is the set of primitive words. Thus, instead of deciding the problem itself, we may decide the membership to the set of yes instances. Accordingly, we say that a problem is decidable if the set of its yes instances is a recursive language. We refer to [68] for a detailed discussion on decidability aspects. 21 2.2.3 Codes Let (M, be a monoid and X C M. We say that M is generated by X if M X , i.e. for any c M, there exist x, Xn X, n O, such that c x . Xn. In this case, we also say that X is a set of generators of M. We say that ....

G. Rozenberg, A. Salomaa, Cornerstones of Undecidability, Prentice Hall, 1994.

.... n 0 and (ff i ; fi i ) 2 P for i = 1; n such that ff 1 ff 2 Delta Delta Delta ff n = fi 1 fi 2 Delta Delta Delta fi n : This problem is referred to as Post s Correspondence Problem (PCP) It originates from Emil Post ( 14] an extensive recent investigation can be found in [15]. A standard method to prove undecidability of some new problem is the following. Often the equivalent formulation of PCP is used: given ff 1 ; ff 2 ; ff n ; fi 1 ; fi 2 ; fi n 2 Gamma , is there a 1 ; a 2 ; am such that ff a1 ff a2 Delta Delta Delta ff am = fi a1 ....

Rozenberg, G., and Salomaa, A. Cornerstones of Undecidability. Prentice Hall, 1994.

....it is undecidable whether F (a 1 ; an ) 0 for all a 1 ; an 2 A. Proof: For an arbitrary polynomial F over ZZ it is undecidable whether there exist a 1 ; an 2 ZZ with F (a 1 ; an ) 0. This was proved by Y. Matiyasevich in 1970 and solved Hilbert s tenth problem ([56]) Assume that a decision procedure exists deciding whether F (a 1 ; an ) 0 for all a 1 ; an 2 A for any F to be given. Since for every x 2 ZZ we have x 2 or 4 x 2 we can write x = f(a) for a 2 A and f is either the polynomial X or the polynomial 4 X . Let F be an arbitrary ....

Rozenberg, G., and Salomaa, A. Cornerstones of Undecidability. Prentice Hall, 1994.

.... problem restricted to right linear grammars is in N2EXPTIME [Bal98, BGM98] In [Bal98, BGM98] it is shown that the general satis ability problem for context free grammars logics is undecidable by reducing the empty intersection problem for context free grammars which is undecidable (see e.g. [RS94]) The proof is based on the completeness of tableaux based proof systems. Alternatively, we prove here that the general satis ability problem for linear grammar logics is undecidable and the core of our proof uses a method based on properties of formal languages only. The undecidability proof is ....

G. Rozenberg and A. Salomaa. Cornerstones of Undecidability. Prentice Hall, 1994.

....1.2 On the edge of undecidability The PCP is undecidable in the general case and it is one of the most important undecidable problems, since it is very useful in proving other undecidability results. For a more extensive and general treatment of undecidability, we refer to Rozenberg and Salomaa [8]. By restricting the PCP we can investigate the border between decidability and undecidability. In the restricted cases some further assumptions are posed to the morphisms. For example, if we assume that the size of an instance is two, i.e. the domain alphabet A is binary, then the PCP is ....

G. Rozenberg and A. Salomaa, Cornerstones of Undecidability, Prentice Hall, 1994.

....nition 4.4. A sequence x 2 N is incompressible or chaotic when there is a constant c such that C(x(n) n c for all n, where x(n) x 1 x 2 : xn . The great technical advantage of working with pre x algorithms comes from a result in Information Theory called Kraft s inequality (1949) [37]. It says that every pre x free language L over f0; 1g satis es X w2L 2 jwj 1: The following generalization of it is crucial [7, 9] Theorem 4 (Kraft Chaitin Theorem) Let W f0; 1g N be a recursively enumerable set, that is the range of a partial recursive function : N f0; 1g ....

....inputs we have 1. Therefore, 0 1 and it is also called the halting probability with respect to U , that is, is the probability that the machine U stops when fed with an input p chosen at random (i.e. by tossing an honest coin) It can be shown that have the follwing curious properties [37]. First of all, it is an incompressible number, hence (Martin L of) random. 27 In spite of being noncomputable, can be estimated and it is known that 0:00106502 0:217643. Also, if the pre x n) of size n is known then we can decide all halting problems codi able in less than n bits. It ....

G. Rozenberg and A. Salomaa, Cornerstones of Undecidability, Prentice Hall (1994).

....and (ff i ; fi i ) 2 P for i = 1; n such that ff 1 ff 2 Delta Delta Delta ff n = fi 1 fi 2 Delta Delta Delta fi n : This problem is referred to as Post s Correspondence Problem (PCP) y . It originates from Emil Post (Post (1946) an extensive recent investigation can be found in Rozenberg and Salomaa (1994). A standard method to prove undecidability of some new problem is the following. Start with an arbitrary instance P of PCP. Using this instance P , construct an instance of the new problem such that PCP has a solution for P if and only if the constructed instance of the new problem has a ....

G. Rozenberg and A. Salomaa. Cornerstones of Undecidability. Prentice Hall, 1994.

....argument given as an input. We shall give here a short account on the well known undecidability result that states that it is undecidable for a given Turing machine whether its computation halts on a given input word. For a more complete treatment the reader is referred to Rozenberg and Salomaa [21]. Our denition of a Turing machine is not the usual one. However, it is rather immediate that this denition of Turing machine has equivalent accepting power than the one given in the literature, see e.g. 21] A Turing machine, TM for short, is a 8 tuple M = Q; Sigma; Theta; R; q 0 ; #; X; q h ....

....input word. For a more complete treatment the reader is referred to Rozenberg and Salomaa [21] Our denition of a Turing machine is not the usual one. However, it is rather immediate that this denition of Turing machine has equivalent accepting power than the one given in the literature, see e.g. [21]. A Turing machine, TM for short, is a 8 tuple M = Q; Sigma; Theta; R; q 0 ; #; X; q h ) where Q is the nite set of states, Sigma and Theta are the nite input and tape alphabets, respectively, with Sigma Theta, R is a set of transition rules, q 0 2 Q is the initial state, # 2 Theta ....

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G. Rozenberg and A. Salomaa, Cornerstones of Undecidability, Prentice Hall, 1994.

....section we shall give a short account on semantics of FOPL. We shall discuss about model theory, satisfiability (truth in some L model) and validity (truth in all L models) The formal definition of L model is presented after these notions. 2 For details on formal languages and grammars see e.g. [4]. 4 A formula is said to be SATISFIABLE, if it can be true in some situation, i.e. in some L model. For instance, the formula 9x 0 #P 0 #x 0 ## is satisfiable. If the predicate P 0 is interpreted to be an even natural number , the formula is satisfied when x is 2,4,6. In this case, the set ....

G. Rozenberg & A. Salomaa. Cornerstones of Undecidability. Prentice-Hall, 1994.

....in [104] however, they are currently superseded by pumping lemmas based on other tools (see for example [149] 34 CHAPTER 1. COMPLEXITY 1.4.5 Trade O#s In this section we give some examples of trade o#s between program size and computational complexities. We start with an example, discussed in [125], of a language having a very low program size complexity, but a fairly high computational complexity. Let # i ,i=0,1, be an ordering of all regular expressions over the alphabet A. A positive integer i is saturated i# the regular language denoted by # i equals A # . A real number r =0.a 0 a ....

G. Rozenberg, A. Salomaa, Cornerstones of Undecidability, Prentice--Hall, 1994.

.... the standard framework for mathematics we will never know which of the two algorithms actually is the one that computes F . The deepest result marking the limits of mathematics was discovered by Godel [18] we will present it in the stronger, information theoretic variant due to Chaitin [9, 7, 37]: An n bit formal axiomatic system cannot enable one to exhibit any specific object with program size complexity greater than n c. 34 . There exists an exponential diophantine equation 35 P (n, x, y 1 , y 2 , y m ) 0, such that every n bit formal axiomatic system cannot enable ....

G. Rozenberg, A. Salomaa. Cornerstones of Undecidability, Prentice-Hall, 1994.

....n 0 and (ff i ; fi i ) 2 P for i = 1; n such that ff 1 ff 2 Delta Delta Delta ff n = fi 1 fi 2 Delta Delta Delta fi n : This problem is referred to as Post s Correspondence Problem (PCP) 2 . It originates from Emil Post ( 14] an extensive recent investigation can be found in [15]. A standard method to prove undecidability of some new problem is the following. 2 Often the equivalent formulation of PCP is used: given ff 1 ; ff 2 ; ff n ; fi 1 ; fi 2 ; fi n 2 Gamma , is there a 1 ; a 2 ; am such that ff a1 ff a2 Delta Delta Delta ff am = fi ....

Rozenberg, G., and Salomaa, A. Cornerstones of Undecidability. Prentice Hall, 1994.

....to x within a fixed recursive time bound. It is straightforward to show that K is useful, while no recursive or random sequence is useful. It is well known that the halting probability of a universal self delimiting Turing machine (Chaitin s Omega number, see Chaitin [9] Rozenberg and Salomaa [20], Calude [4] is random, but K is not; Omega and K contain the same quantity of information but codified in vastly different ways. As we noted before, K is useful but Omega is not useful in the sense of Juedes, Lathrop, and Lutz [13] However, when one is interested in approximating sequences ....

G. Rozenberg, A. Salomaa. Cornerstones of Undecidability, Prentice-Hall, Englewood Cliffs, 1994.

....of but not known through human reason . Indeed, if we know some reasonably long prefix of #, say the first 10000 bits, then we are able to decide of formal systems F and well formed formulas #, both of a reasonable size, whether # is provable, refutable or independent in F , Rozenberg and Salomaa [38]. A brief outline of the contents of the paper follows. Very little previous knowledge is required on part of the reader; Salomaa [39] may be consulted if need arises. A proof for the undecidability of the halting problem, based on algorithmic information theory, is given in Section 2. We believe ....

....information is contained in the halting problem The easiest way to measure it is to switch from the deterministic point of view to the probabilistic one. Put all programs in a bag and think of P i as a binary string. For technical reasons (see, for instance, Calude [11] Rozenberg and Salomaa [38]) we may assume that the resulting set of strings is prefix free, i.e. no string in the set is a proper prefix of another one. In other words, every program is self delimiting: Its total length (say, in bits) is given by the program itself. Real programming languages are self delimiting as they ....

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G. Rozenberg, A. Salomaa. Cornerstones of Undecidability, Prentice Hall. [in press]

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G. Rozenberg, A. Salomaa. Cornerstones of Undecidability, Prentice Hall, Englewood Cli#s, 1994.

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Rozenberg, G.; Salomaa, A. "Cornerstones of Undecidability" Prentice Hall 1994.

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