D. Harel. Recurring dominoes: making the highly undecidable highly understandable. Annals of Discrete Mathematics, 24:51--72, 1985.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....g, is there a tiling of N Theta N The rules of the tiling game are 1. every point in the grid is associated with a single tile, 2. adjacent edges have the same color. Figure 2: The unbounded tiling problem. Now, the version of the UTP presented here is known to be Pi 1 complete (cf. Harel [19]) So to prove the theorem it suffices to define, for a given set of tiles T , a formula OE T in the DML language 1. its models look like grids, 2. it says that every point is covered by a tile from T , 3. and that colors match right and above neighbors, and show that OE T is satisfiable iff T ....

.... Kleene star has a Sigma 1 complete satisfiability problem; this may be proved by using a recurrent tiling problem (RTP) given a finite set of tiles T , and a tile d 1 2 T , can T tile N Theta N such that d 1 occurs infinitely often on the first row The RTP is a Sigma 1 complete problem [19]. To obtain a reduction of the RTP to satisfiability in DML plus Kleene star, we define a formula OE RT as the conjunction of the formula OE T used in the proof of Theorem 5.1 and a formula REC to be defined shortly. We use a new propositional symbol row 0 which can only be true at nodes on the ....

D. Harel. Recurring dominoes: making the highly undecidable highly understandable. In LNCS 158 (Proc. of the Conference on Foundations of Computing Theory), pages 177--194. Springer, 1983.

.... to be e#ectively axiomatizable, i.e. recursively enumerable, undecidability is lurking One bad omen is the validity of associativity, a danger sign in the arrow philosophy (cf. van Benthem 1996) But more precisely, Aiello, 2002a] shows how to e#ectively encode an undecidable tiling problem [Harel, 1983] into the complete arrow logic of the two dimensional Euclidean plane, showing that we may have gone overboard in our desire to express the truth about vectors. Thus, the Balance remains a continuing concern. 6 Concluding Remarks Our walk through space has shown modal structures wherever one ....

Harel, D. (1983). Recurring dominoes: Making the highly undecidable highly understandable. In Conference on Foundations of Computer Science, volume LNCS 158, pages 177--194. Springer.

....edges of adjacent tiles matching. The tiling problem for N N is formulated as follows: given a finite set T of tiles, does T tile N N The reduction we are looking for would be pretty simple if our frames were intransitive or the language contained the next time operators (see e.g. [10] or [19] for then we would be able to refer to the tiles on the right and above directly. As this is not the case, we will use the idea of [17] to enumerate the pairs of natural numbers and refer to the right and above pairs indirectly via special pointers. And a little trick of [25] will make ....

D. Harel. Recurring dominoes: Making the highly undecidable highly understandable. In Conference on Foundations of Computing Theory, pages 177--194. Springer, Berlin, 1983. Lec. Notes in Comp. Sci. 158.

....power to make the satis ability problem undecidable. By combining concurrent steps in a deterministic fashion, it turns out that we can encode the two dimensional grid of natural numbers N N. We can then use this encoding to reduce some undecidable tiling problems described by Wang [Wang 1961] and Harel [Harel 1985] to the problem of deterministic satis ability in our logic. This negative result was shown by Parikh [Parikh 1989] A variety of positive and negative results can be obtained in this logical framework by studying the e ect of placing suitable restrictions on dts s. For instance, we can restrict ....

Harel D 1985 Recurring dominoes: making the highly undecidable highly understandable, Ann. Disc. Math. 24:51-72.

....section we show that the membership problem for SAT n is Sigma 1 1 hard and hence not decidable. We will often omit the subscript n leaving it implicit. The Recurring Colouring Problem [Pa2] will be used to establish our negative result. Colouring problems correspond to tiling problems (see [Ha]) and in this section the colouring problem that we consider (called simply RCP) corresponds to the so called origin constrained recurring tiling problem (RTP) in [Ha] RTP was proved to be Sigma 1 1 complete by [Ha] and [Pa2] showed the recursive equivalence of RCP and RTP. In this section, ....

....Colouring Problem [Pa2] will be used to establish our negative result. Colouring problems correspond to tiling problems (see [Ha] and in this section the colouring problem that we consider (called simply RCP) corresponds to the so called origin constrained recurring tiling problem (RTP) in [Ha]. RTP was proved to be Sigma 1 1 complete by [Ha] and [Pa2] showed the recursive equivalence of RCP and RTP. In this section, we show that every instance of RCP can be reduced to the membership of a formula in SAT , thus showing that satisfiability is as hard as RCP. An instance of RCP is a ....

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Harel, D., "Recurring dominoes: making the highly undecidable highly understandable ", Ann. Disc. Math., 24 (1985) pp. 51-72.

....interesting linguistic constraints. However it has also crossed an important complexity boundary; as we shall now show its satis ability problem is undecidable. To prove the undecidability result it suces to give a reduction from a 0 1 hard problem to L KR2 satis ability. As is shown in [19], tiling problems provide a particularly elegant method of proving lower bounds for modal logics, so we ll use such an approach here. A tile T is a 1 1 square xed in orientation with coloured edges right(T ) left(T ) up(T ) and down(T ) taken from some denumerable set. A tiling problem takes ....

Harel, D.: 1983, Recurring dominoes: making the highly undecidable highly understandable, in Proc. of the Conference on Foundations of Computing Theory, Springer Lecture Notes in Computer Science 158, 177-194.

....models can be axiomatized in a simple (and finitary) fashion. The completeness argument is quite involved; the reason being, as we show in Section 5, that validity is not decidable. Here and in subsequent sections we make heavy use of the negative results based on domino problems due to Harel [Har85]. The results established so far are based on distributed transition systems over an infinite alphabet set. Due to the unusual mixture of modalities in our logic, there is a good deal of difference between finite and infinite alphabets. This is especially so in the presence of determinacy. This is ....

....alphabet Sigma. We begin by showing that deterministic satisfiability is undecidable, or in other words, that the set DSAT is not recursive. Various versions of the colouring problem [Parikh] will be used to establish our negative results. Colouring problems correspond to tiling problems (see [Har85]) and in this section the colouring problem that we consider (called simply CP) corresponds to the so called origin constrained tiling problem in [Har85] An instance of CP is a triple Delta = C; R; U) where C = fc 0 ; c 1 ; c k g is a finite non empty set of colours and R; U : C ( C) ....

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Harel, D. (1985), Recurring dominoes: making the highly undecidable highly understandable, Ann. Disc. Math. 24, 51-72.

....if one restricts the language to only the Priorean tense modalities and allows all models which reasonably can be called temporal (i.e. linear flows) Besides this rather technical result, we think that the main import of this paper is the method of establishing it. We use the tiling technique of [Harel, 1983]. This technique, using the undecidability result for tiling of [Robinson, 1971] has recently been used to provide very neat undecidability proofs for multi dimensional logic. See, for example, Harel, 1983] Spaan, 1993] and [Marx, 1997a] In most neat proofs, use is made of nexttime ....

....main import of this paper is the method of establishing it. We use the tiling technique of [Harel, 1983] This technique, using the undecidability result for tiling of [Robinson, 1971] has recently been used to provide very neat undecidability proofs for multi dimensional logic. See, for example, [Harel, 1983], Spaan, 1993] and [Marx, 1997a] In most neat proofs, use is made of nexttime operators which allow two dimensional structures to code tilings of the plane in a very straight forward manner. This paper shows, for the first time, that the next time temporal operators are not needed for coding ....

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D. Harel. Recurring dominoes: Making the highly undecidable highly understandable. In Conference on Foundations of Computing Theory, pages 177--194. Springer, Berlin, 1983. Lec. Notes in Comp. Sci. 158.

....to introduce the intersection operator of programs lies in the fact that it formalizes some aspects of what is known as parallelism. That is the reason why various authors have considered the following issues : decidability = complexity and axiomatization = completeness of a PDL with intersection [10, 3, 16]. From the decidability = complexity point of view, Harel [10] proves that the validity problem for a PDL with intersection, tests and deterministic atomic programs is undecidable. In the general case, non deterministic atomic programs are allowed and Danecki [3] proves that the validity problem ....

....it formalizes some aspects of what is known as parallelism. That is the reason why various authors have considered the following issues : decidability = complexity and axiomatization = completeness of a PDL with intersection [10, 3, 16] From the decidability = complexity point of view, Harel [10] proves that the validity problem for a PDL with intersection, tests and deterministic atomic programs is undecidable. In the general case, non deterministic atomic programs are allowed and Danecki [3] proves that the validity problem for a PDL with intersection and tests is decidable. From the ....

D. Harel. Recurring dominoes : making the highly undecidable highly understandable. M. Karpinski (editor), Foundations of Computation Theory. Lecture Notes in Computer Science 158, 177--194, Springer-Verlag, 1983.

....tiling is recursively inseparable from the class of dominoes that admit no tiling at all. In particular both the tiling problem and the periodic tiling problem are undecidable, in fact the tiling problem is co r.e. complete and the periodic tiling problem is r.e. complete. Theorem 3. 6 (Harel [19, 20]) The class of dominoes that admit a recurrent tiling with respect to a designated piece d 0 is Sigma 1 1 complete. Lemma 3.7 There are recursive translations from domino systems D (with distinguished piece d 0 ) to formulae D and D;d0 of L 2 such that (i) D admits a tiling if and only ....

D. Harel, Recurring dominoes: Making the highly undecidable highly understandable, Annals of Discrete Mathematics, 24, 1985, pp. 51--72.

....class in the pure predicate calculus. In the last thirty years they have been used to establish many undecidability results and lower complexity bounds for various systems of propositional logic, for subclasses of first order logic and for decision problems in mathematical theories (see e.g. [8, 10, 18, 23]) The original, unconstrained version of a domino problem is given by a finite set of dominoes or tiles, each of them an oriented unit square with coloured edges; the question is whether it is possible to cover the first quadrant in the Cartesian plane by copies of these tiles, without holes ....

D. Harel, Recurring dominoes: Making the highly undecidable highly understandable, Annals of Discrete Mathematics, 24 (1985), pp. 51--72.

....class in the pure predicate calculus. In the last thirty years they have been used to establish many undecidability results and lower complexity bounds for various systems of propositional logic, for subclasses of first order logic and for decision problems in mathematical theories (see e.g. [3, 10, 16, 19, 28]) The original, unconstrained version of a domino problem is given by a finite set of dominoes or tiles, each of them an oriented unit square with coloured edges. The question is whether it is possible to cover the first quadrant in the Cartesian plane by copies of these tiles, without holes ....

....domino that admit a recurrent tiling of Z Theta Z is Sigma 1 1 complete. The same applies to recurrent tilings of N Theta N . For the proof of Sigma 1 1 completeness (also of several useful variants of the recurrent domino problem) and for a number of applications in reductions see [19, 20]. Local grids. Two dimensional grids form the basis of reductions from domino problems. Let GZ and GN be the infinite two dimensional standard grids on Z Theta Z and on N Theta N, respectively, with horizontal and vertical successor relations H and V : GZ = i Z Theta Z; H;V j where H = ....

D. Harel, Recurring dominoes: Making the highly undecidable highly understandable, Annals of Discrete Mathematics, 24 (1985), pp. 51--72.

.... In this section we will prove that validity is undecidable, or to be more precise, show that the set of closed assertions A for which j= A holds is Pi 1 1 hard (see Rogers [Rog67] for a definition) The proof of undecidability is by a reduction from the Recurring Domino Problem due to Harel [Har85] Harel s problem R2) We use the problem in the following version: A recurring domino system is a tuple D = D; H;V; d 0 ; d r ) consisting of a finite set of dominoes D, rules for how dominoes can be arranged horizontally H D Theta D and vertically V D Theta D, an initial domino d 0 2 D, ....

David Harel. Recurring dominoes: Making the highly undecidable highly understandable. Ann. Disc. Math, 24:51--72, 1985.

....holds, that is, iff w 2 A. 2 Remark: For some nondeterministic Turing machines, the problem of deciding whether a word induces a repeating behaviour is not arithmetical. This fact has been used in the proof that the satisfiability problem for first order temporal logic is Sigma 1 1 complete [15]. Formula We are now going to show that the repeating behaviour of a Turing machine can be expressed in the framework of first order temporal logic over a vocabulary extended by the symbols , succ and Zero. Let us fix a Turing machine M satisfying the property of the preceding lemma. Recall ....

Harel, D. Recurring Dominoes: Making the Highly Undecidable Highly Understandable. Annals of Discrete Mathematics 24 (1985), 51--71.

....a)g. A frame for T is defined as F [ffl] B ] ffl] where is the labelled successor relation. In each of our languages it is possible to characterize this frame up to isomorphism. But, then one can encode the following recurring tiling problem, which has been shown to be undecidable in [3]. Below, we follow the definition of [12] and [13] Let Gamma be a finite set of types of tiles such that for each T 2 Gamma each side North, East, South and West is assigned a number (N(T) E(T) S(T) and W(T) resp. Let Co Gamma 4 be a set of colors such that c = T 1 ; T 2 ; T 3 ; ....

Harel, D., Recurring dominoes: making the highly undecidable highly understandable, Annals of discrete mathematics, 24 (1985) 51-72.

....from the unbounded tiling problem [5] which consists in deciding whether (a portion of) the integer grid can be tiled using a finite set of square tile types (fixed in orientation and with colored edges) in such a way that adjacent tiles have the same color on the common edge. As shown in [23], the tiling problem is well suited to show undecidability of variants of modal and dynamic logics, and the difficult part of the proof usually consists in enforcing that the tiles lie on an integer grid. To this end we exploit a query containing one inequality. Theorem 3 Let S be a schema, and ....

David Harel. Recurring dominoes: Making the highly undecidable highly understandable. Annals of Discrete Mathematics, 24:51--72, 1985.

.... expressive power to PDL, Theorem 25, first appeared explicitly in [Harel and Singerman, 1996] Theorem 26 on the undecidability of context free PDL was observed by [Ladner, 1977] Theorems 27 and 28 are from [Harel et al. 1983] An alternative proof of Theorem 28 using tiling is supplied in [Harel, 1985] ; see [Harel et al. 2000] The existence of a primitive recursive one letter extension of PDL that is undecidable was shown already in [Harel et al. 1983] but undecidability for the particular case of a , Theorem 29, is from [Harel and Paterson, 1984] Theorem 30 is from [Harel and ....

....first defined by [Moschovakis, 1974] under the name acceptable structures. In the context of logics of programs, they were reintroduced and studied in [Harel, 1979] 1 completeness of DL was first proved by Meyer, and Theorem 63 appears in [Harel et al. 1977] An alternative proof is given in [Harel, 1985] ; see [Harel et al. 2000] Theorem 65 is from [Meyer and Halpern, 1982] That the fragment of DL considered in Theorem 66 is not r.e. was proved by [Pratt, 1976] Theorem 67 follows from [Harel and Kozen, 1984] The name spectral complexity was proposed by [Tiuryn, 1986] although the ....

D. Harel. Recurring dominoes: Making the highly undecidable highly understandable. Annals of Discrete Mathematics, 24:51--72, 1985.

....simple, reductions from an instance of a domino tiling problem to instances of other problems are relatively easy to construct and comprehend. Thus, tiling problems have turned out to be quite powerful for proving undecidability and lower bounds on the complexity of various logical systems (see [6] for a survey) Domino tiling problems have also been used as alternative basic problems for reductions in the theory of NP completeness [15, 16] Less known is the family of tile connectability problems, or domino snake problems. In general, such a problem asks, given a tiling system T and two ....

....on snake problems are conceptually the same as those used for tiling problems; all are based on simulations of computations of Turing machines or register machines. Tiling problems have been very helpful in establishing lower bounds on the difficulty of other problems e.g. in logics of programs [6]. Thus, a potential direction of future work on snake problems would be to find similar applications. It should be noted that tiling problems are combinatorial in nature, while snake problems have unique geometric properties that might turn out to be useful for other applications. This direction ....

D. Harel, Recurring dominoes: making the highly undecidable highly understandable, Ann. Discrete Math. 24(5) (1985) 51-72.

....that satisfiability is Sigma 1 1 complete. The upper bound, i.e. that the problem is in Sigma 1 1 , can be established without difficulty using standard arguments, cf. HPS, Lemma 6. 3] We thus concentrate on the lower bound, carrying out a reduction from a recurring tiling problem, shown in [H2] to be Sigma 1 1 complete. We first describe the tiling problem and prove a grid like property of the set F , and then present the details of the reduction. Remark: The theorem concerns the family of all Fibonacci like sequences. However, throughout the proof we demonstrate the central issues ....

....of deciding whether a given finite set of tiles can tile G is undecidable, viz. co r.e. complete [B] Imposing recurrence conditions on the tiling (e.g. requiring that a certain tile appears infinitely often) makes these problems highly undecidable, viz. Sigma 1 1 complete, as shown in [H2]. We shall use a specialized version of the problem. For every i 0, let G i = f(j; i) j 0 j i Gamma 1g [ f(i Gamma 2; i 1)g [ f(i; j) j j i 1g: Proposition 4.2 The problem of deciding whether a given set T = fd 0 Delta Delta Delta dm g of tile types can tile G such that all the ....

[Article contains additional citation context not shown here]

D. Harel, "Recurring Dominoes: Making the Highly Undecidable Highly Understandable ", Ann. Disc. Math. 24 (1985), 51--72.

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D. Harel. Recurring dominoes: making the highly undecidable highly understandable. Annals of Discrete Mathematics, 24:51--72, 1985.

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Harel, D.: 1983, Recurring dominoes: making the highly undecidable highly understandable, in Proc. of the Conference on Foundations of Computing Theory, Springer Lecture Notes in Computer Science 158, 177-194.

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Harel, D., 1983, \Recurring dominoes: making the highly undecidable highly understandable", in Foundations of Computing Theory. Proceedings of the 1983 International FCT-Conference, Borgholm, Sweden., Marek Karpinski, ed., Springer Lecture Notes in Computer Science, 158, 177-194.

No context found.

Harel, D., 1983, "Recurring dominoes: making the highly undecidable highly understandable", in Springer Lecture Notes in Computer Science 158, 177--194.

No context found.

D. Harel. Recurring dominoes : making the highly undecidable highly understandable. M. Karpinski (editor), Foundations of Computation Theory. Lecture Notes in Computer Science 158, 177--194, SpringerVerlag, 1983.

No context found.

David Harel. Recurring dominoes: making the highly undecidable highly understandable. Ann. Disc. Math., 24:51--72, 1985.

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