Sera no Amoroso and Yale N. Patt. Decision procedures for surjectivity and injectivity of parallel maps for tessellation structures. Journal of Computer and System Sciences, 6:448-464, 1972. 13  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Automata and Block Permutations are synchronous and massively parallel mappings. A Cellular Automaton A is reversible if and only if GA is bijective and there is another ca B such that A = GB . Such an automaton B is called the inverse of A. Reversible ca are denoted r ca. Amoroso and Patt [1] give an algorithm which decides whether a 1 dimensional cellular automata is reversible or not. Kari [5] proves that the reversibility of ca is undecidable in greater dimensions. By construction, block permutations are reversible. One simply uses the inverse permutation on the same partition to ....

S. Amoroso and Y. Patt. Decision procedure for surjectivity and injectivity of parallel maps for tessellation structure. Journal of Computer and System Sciences, 6:448464, 1972.

.... ca (r ca) have been investigated from the 60s: the equivalence between bijectivity and injectivity by Moore and Myhill [11,14] in the 70s: the equivalence of reversibility and bijectivity by Hedlund [5] and Richardson [15] and the decidability of reversibility in dimension 1 of Amoroso and Patt [1]; to its undecidability in higher dimension by Kari [6,7] in 1990. The computing power of r ca as well as their simulation powers was particularly investigated in [18] Bennett [2] proved that reversible Turing machines could simulate any Turing Machine. In 1977, Tooeoli [16] proved that r ca of ....

....if and only if its global function G is bijective and G is the global function of some ca (pca,bca) We denote r ca (r bca, r pca) the class of reversible ca (pca,bca) The main decidability result is: Theorem 1. The reversibility of ca is decidable in dimension 1 (Amoroso and Patt 1972 [1]) but it is undecidable for higher dimension (Kari 1990 [6,7] Whereas for pca and bca, the following lemmas hold in any dimension and states that as far as reversibility is concerned, bca and pca fundamentally dioeer from ca. Recall that bijectivity for ca is equivalent to reversibility [5,15] ....

S. Amoroso and Y. Patt. Decision procedure for surjectivity and injectivity of parallel maps for tessellation structure. Journal of Computer and System Sciences, 6:448464, 1972.

....will in general have exponential space complexity. Note, though, that B(ae) can be used to determine surjectivity (as well as openness and injectivity) in quadratic time. The fast algorithm uses products of semiautomata rather than the power automaton construction needed for minimization, see [1] and [22] On the other hand, there is no hope of obtaining a strict classification of cellular automata along the lines of Wolfram s classes by means of easily computable parameters: questions relating to the Wolfram hierarchy are in general highly undecidable, even if we restrict our attention ....

....the 1 cycles are disjoint it suffices to establish our claim for just one such cycle, say, D. Pick a state q 2 D of rank less than s. We will use q as the base point and D as the base cycle. The deletion operator is representable by our hypothesis. The shift operator here is simply represented by [1], since D does not contain the anchor point q 0 . Claim 3: Assume that Q s is controllable where s 1. Then Qs is also controllable. All the states in Q oe either lie on one of the 1 cycles dealt with in claim 2, or they lie on a 0 cycle that intersects one of these 1 cycles. Again, by ....

S. Amoroso and Y. N. Patt. Decision procedures for surjectivity and injectivity of parallel maps for tesselation structures. Journal of Computer and Systems Sciences, 6:448--464, 1972.

....S is O(j j 2r ) O(n 2 ) 2 9 4 Conclusion It would be interesting to generalize results concerning reversibility of a linear classical CA for the well formedness of an LQCA. For example a necessary condition for reversibility is the notion of balancedness of the local transition function [1], which means that every state has the same number of preimages. How does balancedness generalizes to the quantum model It remains open, as stated also by Watrous, whether a QTM can simulate an LQCA with reasonable slowdown. Partitioned linear quantum cellular automata This appendix treats a ....

S. Amoroso and Y. Patt, Decision Procedures for Surjectivity and Injectivity of Parallel Maps for Tessellation Structures, Journal of Computer and System Sciences 6, 448-464, 1972.

....to the corollary above, in general, one can make public the RCA A used in encryption without sacrificing the secrecy of the inverse RCA A Gamma1 used in decryption. Note the requirement that the RCA used is at least two dimensional, since all one dimensional RCA can effectively be inverted [1]. To build a practical public key cryptosystem from the idea above one has to give a procedure for constructing RCA that are difficult to invert. We demostrate, using an example, that if several RCA that are easy to invert are composed, the resulting RCA can be hard to invert. Example. Let us ....

S. Amoroso and Y. Patt, Decision Procedures for Surjectivity and Injectivity of Parallel Maps for Tessellation Structures, J. Comput. System Sci. 6 (1972), 448-- 464.

....Moore s and Myhill s results originated a lengthy Garden of Eden debate (see references in [61] which brought to light a number of subtle issues somehow related to invertibility. But invertibility was explicitly addressed only in 1972, in seminal papers by Richardson[60] and Amoroso and Patt[2]. #4 After that, theoretical work on invertibility in cellular automata proliferated[3,61,54,46 48,90,35] In spite of that work, however, for many #4 Unbeknownst to those authors, systems that are in essence one dimensional cellular automata had already been studied in an abstract ....

....school. T. Toffoli, N. H. Margolus = Invertible Cellular Automata 4 years the most interesting ica actually exhibited remained an extremely simple minded one (the longest orbit is of period two ) discovered by Patt through brute force enumeration[56] Ica continued to appear to be quite rare [2]. Not only rare, but also simple minded On the basis of various kinds of circumstantial evidence (cf. e.g. 63] Burks conjectured that an ica cannot be computation universal[11] note that that was at a time when computation universality was being turned up under almost every stone) and ....

[Article contains additional citation context not shown here]

Amoroso, Serafino, and Y. N. Patt, "Decision Procedures for Surjectivity and Injectivity of Parallel Maps for Tessellation Structures," J. Comp. Syst. Sci. 6 (1972), 448--464. #25 More formally, the uniformity group of the structural description is a proper subgroup of that of the functional behavior.

....[9, 8] Richardson proved in 1972 [10] that if a CA realizes a bijective function, then there exists another CA called its inverse that realizes the inverse function. The same year, Amoroso and Patt proved that the reversibility (or the surjectivity) of one dimensional CA is decidable [1]. One dimensional CA work on a bi infinite line of cells, two dimensional CA work on a plane tiled with square cells, etc: This work was partially supported by the Esprit Basic Research Action Algebraic and Semantical Methods In Computer Science and by the PRC Math ematique et ....

S. Amoroso and Y.N. Patt. Decision procedures for surjectivity and injectivity of parallel maps for tesselation structures. J. Comp. Syst. Sci., 6:448--464, 1972.

.... erasable configurations for the existence of nonconstructible configurations is a necessary and sufficient condition [11] In the seventies the constructibility nonconstructibility problem was investigated in a series of papers in connection with the surjectivity injectivity of CA mappings [12 15] and invertibility of CA [16, 17] To date, two important results concerning the nonconstructibility problem have been obtained: the undecidability of the surjectivity (absence of nonconstructible configurations) problem for multi dimensional CA proved in [18] and the classification of 3 site ....

Amoroso S. and Patt Y. Decision procedure for surjectivity and injectivity of parallel maps for tessellation structures. J.Comput. Syst. Sci. 6(1972), 448464.

....operator OE : hq; pi 7 hq; Gammapi. 3 This interest is not abating; see, for instance, 55, 7, 8, 10] to light a number of subtle issues somehow related to invertibility. But invertibility was explicitly addressed only in 1972, in seminal papers by Richardson[60] and Amoroso and Patt[2]. 4 After that, theoretical work on invertibility in cellular automata proliferated[3, 61, 54, 46, 47, 48, 90, 35] In spite of that work, however, for many years the most interesting ica actually exhibited remained an extremely simpleminded one (the longest orbit is of period two ) discovered ....

.... 61, 54, 46, 47, 48, 90, 35] In spite of that work, however, for many years the most interesting ica actually exhibited remained an extremely simpleminded one (the longest orbit is of period two ) discovered by Patt through brute force enumeration[56] Ica continued to appear to be quite rare [2]. Not only rare, but also simple minded On the basis of various kinds of circumstantial evidence (cf. e.g. 63] Burks conjectured that an ica cannot be computationuniversal [11] note that that was at a time when computation universality was being turned up under almost every stone) and soon ....

[Article contains additional citation context not shown here]

Amoroso, Serafino, and Y. N. Patt, "Decision procedures for surjectivity and injectivity of parallel maps for tessellation structures," J. Comp. Syst. Sci. 6 (1972), 448--464.

....4. Clearly, GM j GN on S Z iff M j N on (S Theta S) The equivalence problem of singlevalued transducer is shown to be decidable in [8] M and N are (length preserving) transducers. 2 The injectivity and surjectivity are well known decidable problems for 1 dimensional CA, see [1, 4]; we will extend these results to GCA. Note that because of the shiftinvariance, the injectivity of GCA M on S Z clearly implies the injectivity of GM on S . The converse is less obvious but holds, too. The case k 1; c 6= oe k (c) GM (c) GM (oe k (c) d seems to violate only the ....

S. Amoroso and Y. N. Patt, Decision procedures for surjectivity and injectivity of parallel maps for tessellation structures, J. of Comp. and System Sciences 6, 448-464 (1972).

....S is O(j Sigmaj 2r ) O(n 2 ) 4. CONCLUSION It would be interesting to generalize results concerning reversibility of a linear classical CA for the well formedness of an LQCA. For example a necessary condition for reversibility is the notion of balancedness of the local transition function [1], which means that every state has the same number of preimages. How does balancedness generalizes to the quantum model It remains open, as stated also by Watrous, whether a QTM can simulate an LQCA with reasonable slowdown. ACKNOWLEDGMENTS We are thankful to St ephane Boucheron, Bruno Durand, ....

S. Amoroso and Y. Patt, Decision Procedures for Surjectivity and Injectivity of Parallel Maps for Tessellation Structures, Journal of Computer and System Sciences 6, 448--464, 1972.

....can also be used to study the reversibility. A more detailed elaboration will replace this informal discussion) 4 Future Research . Study the injectivity and surjectivity of GCAs. It is expected that the methods developed for CAs can also be used to give answers for one dimensional GCAs ([1], 6] Use regular expressions to describe configurations. Apply the methods, found in Wolfram [7] to the class of GCAs. What is the connection between the time and the space evolution in GCAs . Study the connection between CAs and GCAs. Is is possible, given an arbitrary CA, to find ....

S. Amoroso and Y. N. Patt. Decision procedures for surjectivity and injectivity of parallel maps for tesselation structures. Journal of Computer and Systems Sciences, 6:448--464, 1972.

....are invertible models [5, 8] Cellular automata represent one of the best models of parallel computation; the study of invertibility in cellular automata is of great interest in modelling physics. Several theoretical results concerning invertibility in cellular automata have been presented ([2, 9, 10, 12, 13, 15, 18]) some leading to open questions. ffl In [18] the existence of a peculiar class (residual class) of cellular automata had been predicted but, until now, no such cellular automata had been exhibited. Here we explicitly construct a cellular automaton in this class, i.e. a cellular automaton that ....

....1 ; Morita and Harao [12] proved the existence of computation universal cellular automata in the one dimensional case. For many years a major challenge has been deciding whether or not a given cellular automaton is invertible. For the one dimensional case Amoroso and Patt proved that Theorem 2. 2 [2] There is an effective procedure for deciding whether or not an arbitrary onedimensional cellular automaton, given in terms of a local map, is invertible. In other words, the class of invertible one dimensional cellular automata is recursive. Concerning multidimensional cellular automata, the ....

Amoroso, S., Patt, Y. N., (1972) "Decision procedures for surjectivity and injectivity of parallel maps for tesselation structures", J. Comp. Syst. Sci 6, 448--464.

....bijection. Hedlund [6] and Richardson [17] proved that any function over S Z d which commutes with any shift and which is continuous for the product topology is the global function of some ca. As a consequence, for any ca it is enough to be one to one to be reversible. In 1972, Amoroso and Patt [1] proved that ca reversibility is decidable in dimension 1. In 1990, Kari [9] proved that it is not decidable any more in dimension 2 and above. In 1977, Tooeoli [18] proved that any ca can be simulated by a reversible ca (r ca) one dimension higher and that there are r ca of dimension 2 and above ....

S. Amoroso and Y. Patt. Decision procedure for surjectivity and injectivity of parallel maps for tessellation structure. Journal of Computer and System Sciences, 6:448464, 1972.

....say A is a reversible (or injective) CA (RCA) iff F is one to one. 2 It is in general hard to design an RCA using the framework of a usual CA. Because, it has been known that the problem whether a given 2 D CA is reversible is undecidable [23] For the case of 1 D CA it is known to be decidable [3]. But designing a 1 D RCA is still not easy. Gamma Gamma Gamma Gamma Gamma Gamma Gamma Gamma Gamma Gamma Gamma Gamma Gamma Gamma Gamma Gamma Gamma Gamma Gamma Gamma Gamma Gamma Gamma Gamma Gamma Gamma Gamma Gamma ....

Amoroso, S. and Patt, Y.N., Decision procedures for surjectivity and injectivity of parallel maps for tessellation structures, J. Comput. Syst. Sci., 6, 448--464 (1972).

....; x 21 ; xm 0 1 ) f(x 12 ; x 21 ; x 22 ; xm 0 1 ; xm 0 2 ) 3) In general we can chunk by a factor n to obtain a cellular automaton with state set S 0 = S n and local map f 0 of arity m 1 n . If we have an invertible cellular automaton of radius 1, it is shown in [1] that the radius of the inverse is not necessarily 1. Using a chunking process, however, we can reduce the inverse cellular automaton also to one with radius 1. Thus when considering reversible cellular automata of radius 1, we can assume without loss of generality that the inverse is also of ....

S. Amaroso and Y.N. Patt. Decision procedures for surjectivity and injectivity of parallel maps for tessellation structures. J. Comp. Syst. Sci., 6:448--464, 1972.

.... ca (r ca) have been investigated from the 60s: the equivalence between bijectivity and injectivity by Moore and Myhill [11,14] in the 70s: the equivalence of reversibility and bijectivity by Hedlund [5] and Richardson [15] and the decidability of reversibility in dimension 1 of Amoroso and Patt [1]; to its undecidability in higher dimension by Kari [6,7] in 1990. The computing power of r ca as well as their simulation powers was particularly investigated in [18] Bennett [2] proved that reversible Turing machines could simulate any Turing Machine. In 1977, Tooeoli [16] proved that r ca of ....

....and only if its global function G is bijective and G Gamma1 is the global function of some ca (pca,bca) We denote r ca (r bca, r pca) the class of reversible ca (pca,bca) The main decidability result is: Theorem 1. The reversibility of ca is decidable in dimension 1 (Amoroso and Patt 1972 [1]) but it is undecidable for higher dimension (Kari 1990 [6,7] Whereas for pca and bca, the following lemmas hold in any dimension and states that as far as reversibility is concerned, bca and pca fundamentally dioeer from ca. Recall that bijectivity for ca is equivalent to reversibility [5,15] ....

S. Amoroso and Y. Patt. Decision procedure for surjectivity and injectivity of parallel maps for tessellation structure. Journal of Computer and System Sciences, 6:448464, 1972.

....and Block Permutations are synchronous and massively parallel mappings. A Cellular Automaton A is reversible if and only if GA is bijective and there is another ca B such that G Gamma1 A = GB . Such an automaton B is called the inverse of A. Reversible ca are denoted r ca. Amoroso and Patt [1] give an algorithm which decides whether a 1 dimensional cellular automata is reversible or not. Kari [5] proves that the reversibility of ca is undecidable in greater dimensions. By construction, block permutations are reversible. One simply uses the inverse permutation on the same partition to ....

S. Amoroso and Y. Patt. Decision procedure for surjectivity and injectivity of parallel maps for tessellation structure. Journal of Computer and System Sciences, 6:448464, 1972.

No context found.

Sera no Amoroso and Yale N. Patt. Decision procedures for surjectivity and injectivity of parallel maps for tessellation structures. Journal of Computer and System Sciences, 6:448-464, 1972. 13

No context found.

S. Amoroso and Y. N. Patt. Decision procedures for surjectivity and injectivity of parallel maps for tesselation structures. Journal of Computer and Systems Sciences, 6:448-464, 1972.

No context found.

Amoroso, S. and Patt, Y.N., Decision procedures for surjectivity and injectivity of parallel maps for tessellation structures, J. Comput. Syst. Sci., 6, 448--464 (1972).

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC