Post, E., 1946. A variant of a recursively unsolvable problem. Bull. AMS 52, 264--268.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....# i m The list P is called an instance of PCP, the string # i 1 # i m a solution for P . We write (#, #) P if (#, #) # i , # i ) for some 1 # i # n. Without loss of generality we require P to be non empty. PCP is known to be undecidable even in the case of a two letter alphabet (Post [15]) Matiyasevich and Senizergues [12] showed that PCP is undecidable even when restricted to instances consisting of seven pairs. An obvious breadth first search procedure yields the semi decidability of PCP and hence the complement of PCP is not semi decidable. For preliminaries on rewriting the ....

E. Post. A variant of a recursively unsolvable problem. Bulletin of the American Mathematical Society, 52:264--268, 1946.

....for i = 1, k. A solution to the correspondence is any non empty word w = w 1 wn over the alphabet . k such that uw = vw , where uw = uw1 . uwn . This correspondence problem is known to be undecidable: there is no algorithm that decides if a given instance has a solution [Pos46] It is easy to see that the problem remains undecidable when the alphabet # contains only two letters. The problem is also known to be undecidable for k = 7 pairs [MS96] but is decidable for k = 2 pairs; the decidability of the cases 2 k 7 is yet unresolved. We are now ready to state our ....

E. L. Post. A variant of a recursively unsolvable problem. Bulletin of the American Mathematical Society, 52:264268, 1946.

....the first order theory of nonstructural subtyping is undecidable for any type language with a binary type constructor and the bottom element (or dually, the top element #) The formula we exhibit is in ###### fragment. The proof is via a reduction from the Post s Correspondence Problem (PCP) [32] to a first order formula of nonstructural subtyping. Since PCP is undecidable [32] the first order theory of non structural subtyping is undecidable as well. The proof follows the framework of Treinen [40] and is inspired by the proof of undecidability of the first order theory of ordering ....

....with a binary type constructor and the bottom element (or dually, the top element #) The formula we exhibit is in ###### fragment. The proof is via a reduction from the Post s Correspondence Problem (PCP) 32] to a first order formula of nonstructural subtyping. Since PCP is undecidable [32], the first order theory of non structural subtyping is undecidable as well. The proof follows the framework of Treinen [40] and is inspired by the proof of undecidability of the first order theory of ordering constraints over feature trees [25] Recall that an instance of PCP is a finite set of ....

E.L. Post. A Variant of a Recursively Unsolvable Problem. Bull. of the Am. Math. Soc., 52, 1946.

....3.1 Undecidability of with General Key Boxes We prove that satisfiability of w.r.t. key boxes is undecidable for a large class of concrete domains if we allow complex to occur in key definitions. The proof is by a reduction of the well known undecidable Post Correspondence Problem [43, 27]. Definition 6 (PCP) An instance P of the Post Correspondence Problem is given by a finite, non empty list (# 1 , r 1 ) # k , r k ) of pairs of words over some alphabet #. A sequence of integers i 1 , i m , with m 1, is called a solution for P i# # i 1 # i m = r i 1 r ....

E. M. Post. A variant of a recursively unsolvable problem. Bulletin of the American Mathematical Society, 52:264--268, 1946.

....a solution of the system S iff l i 1 Delta Delta Delta l i m = r i 1 Delta Delta Delta r i m . It is well known that the Post Correspondence Problem, i.e. the question whether there exists a solution for a given system, is in general undecidable if the alphabet contains at least two symbols [Pos46]. Elements of the alphabet Sigma will be represented as unary function symbols and a word w = a 1 Delta Delta Delta a n over Sigma thus becomes a term a 1 (a 2 ( a n (ffl) where ffl is a constant corresponding to the empty word. So, composition 4 of words is associative since ....

E. M. Post. A variant of a recursively unsolvable problem. Bull. Am. Math. Soc., 52:264--268, 1946.

....= 1; k. A solution to the correspondence is any non empty word w = w 1 w n over the alphabet f1; kg such that uw = v w , where uw = uw 1 : uwn . This correspondence problem is known to be undecidable: there is no algorithm that decides if a given instance has a solution [Pos46] It is easy to see that the problem remains undecidable when the alphabet contains only two letters. The problem is also known to be undecidable for k = 7 pairs [MS96] but is decidable for k = 2 pairs; the decidability of the cases 2 k 7 is yet unresolved. We are now ready to state our ....

E. L. Post. A variant of a recursively unsolvable problem. Bulletin of the American Mathematical Society, 52:264-268, 1946.

....unary key boxes. Finally, we identify a concrete domain such that ALCO(D) concept satisfiability (without key boxes) is already NEXPTIME hard. Undecidability of ALCK(D) concept satisfiability w.r.t. general key boxes is proved by reduction of the undecidable Post Correspondence Problem (PCP) [Post, 1946] . Definition 5. An instance P of the PCP is given by a finite, non empty list (# 1 , r 1 ) # k , r k ) of pairs of words over some alphabet #. A sequence of integers i 1 , i m , with m 1, is called a solution for P iff # i 1 # i m = r i 1 r i m . The problem is to ....

E. M. Post. A variant of a recursively unsolvable problem. Bull. of the Amer. Math. Soc., 52:264--268, 1946.

....in [23] It is established that this problem is equivalent to the halting problem of 2 counter machines (which is undecidable) The 3 and 4 subformula problems remain open. 23 7.4. The Proof via Post This proof [28] is not based on the Conway functions but on the better known Post problem [42]. The Post Problem. Let us consider a finite alphabet Sigma. A Post correspondence system over Sigma is a non empty finite set S = f(l i ; r i ) j i 2 [1; Delta Delta Delta ; m]g where the l i , r i are words over Sigma. A non empty sequence of indices 1 i 1 ; Delta Delta Delta ; i n ....

Post E.M. "A Variant of a Recursively Unsolvable Problem" Bulletin of American Mathematics Society. n

....for the languages in the the first class, even if we fix an underling constraint system with a finite domain, but decidable for the languages in the second class for arbitrary constraint systems. The undecidability result is obtained by a reduction from the Post s correspondence problem [13]. The decidability result is obtained by a reduction to Buchi automata [2] following similar results in [17] and [10] establishing the finite state representability of rep processes. The expressiveness gaps illustrated above may look surprising to those acquainted with the # calculus, because the ....

....FD[1] i.e. D = 1. In sections 6 we shall give an input output preserving encoding from rec p into the parameterless recursion language rec d . Therefore, rec d is undecidable as well. Our proof of undecidability will proceed by a reduction from the Post s correspondence problem (PCP) [13]. Let us recall the following definition. Definition 6. Post s Correspondence Problem) A PCP instance is a tuple (W,V ) where W = vn are two set of words over the alphabet 1 . A solution to this instance is a sequence of indexes i 0 , i m in I = n s.t. w ....

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E. L. Post. A variant of a recursively unsolvable problem. Bulletin of the American Mathematical Society, 52:264--268, 1946.

.... , equality and integer constants. It then applies the solution of Hilbert s 10th problem [9] The main idea is to use polynomials within predicates. We think that this part of the proof maybe of little relevance for practical stylesheets. The other proofs reduce Post s correspondence problem [16]. The first of these uses the child axis, equality, disjunction, conjunction, and the nconc function that concatenates all the string values of a set of nodes. The second uses the child and next sibling axis, equality, disjunction, conjunction and the conc function that concatenates two strings. ....

E. Post. A variant of a recursively unsolvable problem.

.... it is undecidable whether there exists some natural number 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 ....

Post, E. A variant of a recursively unsolvable problem. Bulletin of the American Mathematical Society 52 (1946). 17

....following way: for all of the implications X ) Y and all instances of PCP we construct a TRS that always satisfies Y , and either satisfies X if and only if the PCP instance admits a solution, or satisfies X if and only if the PCP instance admits no solution. Since PCP is known to be undecidable ([18]) this proves relative undecidability of the implication X ) Y . The main part of the paper consists of constructions of such TRSs parametrized by PCP instances and corresponding proofs of the above mentioned properties. In the next section this is done for the confluence hierarchy and in Sect. ....

E. Post, A Variant of a Recursively Unsolvable Problem, Bulletin of the American Mathematical Society 52 (1946) 264--268.

....Automata Theory and Combinatorics on Words 1 Introduction Let A and B be two nite alphabets and h; g be two morphisms h; g : A B . The Post Correspondence Problem, PCP for short, is to determine if there exists a nonempty word w 2 A such that h(w) g(w) It was proved by Post [8] that this problem is undecidable in general. Such a word w that h(w) g(w) is called a solution of the instance (h; g) of the PCP. The size of the instance (h; g) is the size of the domain alphabet A. Another important problem is the generalized PCP, GPCP for short. It consists of two morphisms ....

E. Post. A variant of a recursively unsolvable problem. Bull. of Amer. Math. Soc., 52:264268, 1946.

....The situation for RSRL is quite di#erent. Theorem 3.1 Given a sentence # and a finite RSRL interpretation I, it is in general not decidable, if I models #. We prove this theorem by coding Post correspondence problems in finite RSRL structures. Let # be an alphabeth. A Post correspondence system [10] is a finite set P of ordered pairs of nonempty strings; that is P is a finite subset of # # . Here is a simple example: Let # = a, b and P = a, ab) ba, a) A match of P is any string w # # # such that, for some n 0 and some (not necessarily distinct) pairs (u 1 , v 1 ) ....

Post, E., A variant of a recursively unsolvable problem, Bulletin of the AMS 52 (1946), pp. 264--268.

....: am g , is there a sequence ffi 1 ; ffi k 2 f1; ng such that p ffi 1 1 1 1 p ffi k = q ffi 1 1 1 1 q ffi k We identify the problem and the set P ; and say P is solvable if there exists such a sequence. It is well known that solvability of P is undecidable even when m = 2[17]. Thus, if we find for each P over a fixed alphabet a CTRS R in the class C so that R has the property OE if and only if P has a solution (or so that R has the property OE if and only if P has no solutions) then it follows that whether OE holds for a CTRS in C is undecidable. In the succeeding ....

E. Post. A variant of recursively unsolvable problem. Bulletin of the American Mathematical Society, 52:264--268, 1946.

....1 and indices i 1 , i N # 1, n for which u i 1 . u i N = v i 1 . v i N Post s correspondence problem is trivially decidable for one letter alphabets but is undecidable when the alphabet contains more than one letter; for a proof of this result see the original paper [Post46] or [HU69] The decidability of PCP(n) does not depend on the cardinality of A when #A # 2 but depends on the number of pairs of words. The number of pairs of words for which the problem is undecidable has been successively improved. It is now known that the case of n = 7 pairs is undecidable, ....

E. L. Post. A variant of a recursively unsolvable problem. Bull. Amer. Math. Soc., 52:264--268, 1946.

....observed, QCMs can effectively be simulated by TCMs; so the results hold for these machines as well. In contrast, emptiness is undecidable for PCMs and TCMs with multiple pushdown stacks (finite crossing worktapes) In fact, it follows from the undecidability of the Post Correspondence Problem [27] that emptiness is undecidable for machines with only two pushdown stacks, even if the stacks are restricted to making only one alternation from non popping to non pushing. Example 2: Consider again the producer consumer system given in Figure 1 and the QCM M that is constructed from it by ....

E. Post, "A variant of a recursively unsolvable problem," Bull. Am. Math. Soc., 52 (1946) 264--268.

....the rst order theory of nonstructural subtyping is undecidable for any type language with a binary type constructor and the bottom element (or dually, the top element ) The formula we exhibit is in the 989898 fragment. The proof is via a reduction from the Post s Correspondence Problem (PCP) [33] to a rst order formula of nonstructural subtyping. Since PCP is undecidable [33] the rst order theory of non structural subtyping is undecidable as well. The proof follows the framework of Treinen [41] and is inspired by the proof of undecidability of the rst order theory of ordering ....

....with a binary type constructor and the bottom element (or dually, the top element ) The formula we exhibit is in the 989898 fragment. The proof is via a reduction from the Post s Correspondence Problem (PCP) 33] to a rst order formula of nonstructural subtyping. Since PCP is undecidable [33], the rst order theory of non structural subtyping is undecidable as well. The proof follows the framework of Treinen [41] and is inspired by the proof of undecidability of the rst order theory of ordering constraints over feature trees [26] Recall that an instance of PCP is a nite set of ....

E.L. Post. A Variant of a Recursively Unsolvable Problem. Bull. of the Am. Math. Soc., 52, 1946.

.... of application domains [5, 13] inverse roles are present in most expressive Description Logics [5, 10] and the role forming constructor is a natural counterpart to the concept forming concrete domain constructor [8] By introducing a NExpTime complete variant of the Post Correspondence Problem [17, 9], we identify a large class of concrete domains D such that reasoning with each of the above three extensions of ALC(D) separately) is NExpTime hard. This dramatic increase in complexity is rather surprising since, from a computational point of view, all of the proposed extensions look harmless. ....

....concepts to be restricted without further notice. Note that the set of restricted ALCRPI(D) concepts is closed under negation, and, hence, subsumption can be reduced to satis ability. 3 A NExpTime complete Variant of the PCP The Post Correspondence Problem (PCP) as introduced 1946 by Emil Post [17], is an undecidable problem frequently employed in undecidability proofs. In this section, we de ne a NExpTime complete variant of the PCP together with a concrete domain P that is suitable for reducing PCPs to the satis ability problem of Description Logics with concrete domains. De nition 8 ....

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E. M. Post. A variant of a recursively unsolvable problem. Bull. Am. Math. Soc., 52:264-268, 1946.

.... h(w) g(w) The pair (h; g) is called an instance of the PCP and a word w 2 A a solution of the instance (h; g) if h(w) g(w) The set of all solutions, E(h; g) fw 2 A j h(w) g(w)g; is called the equality set of the instance (h; g) The PCP is undecidable in this general form, see [6]. Also the restrictions of the PCP have received much attention. For example, if jAj 2 then the problem is decidable, see [1] or [3] for a somewhat shorter proof. On the other hand, if jAj 7, then the PCP is undecidable, see [5] Here we consider in nite solutions of an instance (h; g) of the ....

E. Post. A variant of a recursively unsolvable problem. Bull. of Amer. Math. Soc., 52:264-268, 1946.

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Post, E., 1946. A variant of a recursively unsolvable problem. Bull. AMS 52, 264--268.

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E. Post. A variant of a recursively unsolvable problem. Bull. AMS, 52:264--268, 1946.

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Emil Post. A variant of a recursively unsolvable problem. Bulletin American Mathematical Society, 52:264--268, 1946.

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E. L. Post. A variant of a recursively unsolvable problem. Bulletin of the American Mathematical Society, 52:264--268, 1946.

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E. Post. A variant of a recursively unsolvable problem. Bulletin of the AMS, 52:264--268, 1946.

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