New bounds for the distributed firing synchronization problem
Complex phenomena in a wide variety of areas such as the simulation of physical systems, biologic... more Complex phenomena in a wide variety of areas such as the simulation of physical systems, biological modeling, image processing, and language recognition can be modeled using networks of very simple machines called finite automata. The interest in studying cellular automata is not in the description of the local state transitions of each machine, but rather in studying the global behavior across the entire system. The topic of this thesis is the firing synchronization problem. The goal is a state synchronization, that is, to define the automata in the network so that all machines enter a special fire state at the same time. The problem requires that all automata in the network be identical, and that any solution to the problem must cause all automata to fire simultaneously and for the first time at the end of the computation. The work presented in this thesis falls into two categories. The first set of results improve previously published solutions to variants of the firing synchronization problem. The improvements are not in the amount of time needed for synchronization, but instead in the number of states needed to represent machines. Since the number of states represents the complexity of the solution, modifying solutions to use fewer states improves them by making them simpler. The second set of results concerns lower bounds for the firing synchronization problem. As with the previous results, the concern is not in bounding the amount of time needed to synchronize, but instead in finding bounds on the number of states used by the automaton. This work not only produces new bounds on two variants of the problem, but also provides evidence that the ultimate resolution of the state lower bound may need a different set of techniques than have been previously used.
Proceedings of the twentieth annual ACM symposium on Theory of computing - STOC '88, 1988
Let A be a (possibly infinite) graph, and locate a processor p at every vertex of A. Each process... more Let A be a (possibly infinite) graph, and locate a processor p at every vertex of A. Each processor p hzs a finite set Q of states. If d is Ihe clegrcc th;tc pfoccssor p lies in A , then its neighbors arc v,(p),. * ,vd(p). The state of each processor is a function of the states of its neighbors. More precisely, the state of p at time I, St (p .t) i!; updated as St(p,t+l) = Tranr(.9(V,(P),f), '. . Sf(v,(p:,t)) where Tram : Qd +Q is the transition function. Such objects are called celhhr spaces, tessalations, iterative arrays of finite automata, systolic arrays, or networks of computers. They have been extensively studied [vN] [HI IK] ['E!) as they are a natural model of parallel universal computers, and of many physical and biologicxl phenomena. We will examine the cluestion of how to make such networks resistant to errors.-' Supported in part by National Science Foundation ~;rants DCR 8407256. CCR 87-10078. CCR 8746518, and by AFOSR c'ontract 87-04co Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for d rect commercial advantage, the ACM copyright notice and the title of the publication and its date appear, a.nd notice is given that copying is by permission of the Association for Computing Machinery. To copy otherwise, or 1o republish, rcquircs a fee and/or spxfic permission.
Both the bottleneck counting argument [7, 8] and Razborov's approximation method [1, 4, 12] ... more Both the bottleneck counting argument [7, 8] and Razborov's approximation method [1, 4, 12] have been used to prove exponential lower bounds for monotone circuits. We show that under the monotone circuit model for every proof by the approximation method, there is a bottleneck counting proof and vice versa. We also illustrate the elegance of the bottleneck counting technique with a simple self-explained example: the proof of a (previously known) lower bound for the 3-CLIQUEn problem by the bottleneck counting argument. Keywords: Computational complexity; Circuit complexity; Monotone circuit complexity 1 Introduction Razborov's proof of an exponential lower bound on the size of monotone Boolean circuits to detect cliques in a graph [12] [1], represented a breakthrough in the theory of monotone circuit complexity. The proof introduced the method of approximation. The method is roughly as follows. Consider two sets of test inputs, a positive (the output is 1) and a negative on...
Advanced Collective Communication in WDM Optical Rings
PDF and gzipped PostScript formats via anonymous FTP from the area ftp.cs.unibo.it:/pub/TR/UBLCS ... more PDF and gzipped PostScript formats via anonymous FTP from the area ftp.cs.unibo.it:/pub/TR/UBLCS or via WWW at
Vehicular Ad Hoc Networks (VANETs) are attracting the attention of researchers, industry, and gov... more Vehicular Ad Hoc Networks (VANETs) are attracting the attention of researchers, industry, and governments for their potential of significantly increasing the safety level on the road. In order to understand whether VANETs can actu-ally realize this goal, in this paper we analyze the dynamics of multi-hop emergency message dissemination in VANETs. Under a probabilistic wireless channel model that accounts for interference, we derive lower bounds on the probability that a car at distance d from the source of the emergency message correctly receives the message within time t. Be-sides d and t, this probability depends also on 1-hop channel reliability, which we model as a probability value p, and on the message dissemination strategy. Our bounds are derived for an idealized dissemination strategy which ignores inter-ference, and for two provably near-optimal dissemination strategies under protocol interference. The bounds derived in the first part of the paper are used to carefully ana...
We present two solutions to the firing synchronization problem on the ring, an 8-state minimal-ti... more We present two solutions to the firing synchronization problem on the ring, an 8-state minimal-time solution and a 6-state non-minimal-time solution. Both solutions use fewer states than the previous best-known minimal-time automaton, a 16-state solution due to Culik. We also give the first lower bounds on the number of states needed for solutions to the ring firing synchronization problem. We show that there is no 3-state solution and no 4-state, symmetric, minimal-time solution for the ring. 1
In this pa per we prese nt improve d bounds on the comple xity of solutions to the ¿ ring synchro... more In this pa per we prese nt improve d bounds on the comple xity of solutions to the ¿ ring synchroniza tion problem. In the ¿ ring synchronization proble m we cons ider a one-dime ns ional a rray of n identica l ¿ nite a utoma ta . Initia lly all a utoma ta a re in the same s ta te e xcept for one automaton which is des igna te d a s the initia tor for the synchroniza tion. Our re sults hold for the origina l problem, whe re the initia tor may be located a t eithe r e ndpoint, and for the va ria nt whe re any one of the automata ma y be the initiator, ca lle d the ge nera lize d problem. In both ca ses , the goa l is to de ¿ ne the set of s ta te s a nd trans ition rule s for the a utoma ta so tha t a ll machine s e nte r a specia l ¿ re s tate s imultane ous ly and for the ¿ rs t time during the ¿ na l round of the computation. In our work we improve the cons truction for the bes t known minima l-time solution to the ge nera lize d problem by re ducing the number of s ta te s nee de...
Non-Minimal Time Solutions for the Firing Sychronization Problem
The firing synchronization problem concerns a one-dimensional array of $n$ finite automata. All a... more The firing synchronization problem concerns a one-dimensional array of $n$ finite automata. All automata are identical, except the ones on either end of the array, one of which is the initiator for the synchronization. The goal is to define the set of states and transition rules for the automata so that all machines enter a special fire state for the first time and simultaneously at some time $t(n)$. The generalized firing synchronization problem differs from the original problem in that the initiator of the synchronization can be located anywhere in the array. In this paper we present two non-minimal time solutions to the firing synchronization problem. The first is a 6-state solution to the original problem which allows the initiator to be located at either the right or left endpoint. This automaton has two fewer states than R.~Balzer''s 8-state minimal-time automaton \cite{b}, the minimal-time automaton with the smallest number of states that allows the initiator to be lo...
On some central problems in computational complexity
Page 1. f > " u u t u l w K ø { ÿ { Q " # I Q ( ' ' µ ... more Page 1. f > " u u t u l w K ø { ÿ { Q " # I Q ( ' ' µ µ ] y w u Ï w s Ì Ç y u l l u Ç u ö # í ì ø ì ( f " f y l w Ï w u y ø µ Å Ì t u Ð µ o ...
Abstract. It is likely that reliable and fast space-bounded probabilistic acceptors are less powe... more Abstract. It is likely that reliable and fast space-bounded probabilistic acceptors are less powerful than nondeterministic ones. We consider a restricted model of space-bounded probabilistic computation, the random analog of a model studied in [CR]. We show that maze traversal ...
New bounds for the distributed firing synchronization problem
Complex phenomena in a wide variety of areas such as the simulation of physical systems, biologic... more Complex phenomena in a wide variety of areas such as the simulation of physical systems, biological modeling, image processing, and language recognition can be modeled using networks of very simple machines called finite automata. The interest in studying cellular automata is not in the description of the local state transitions of each machine, but rather in studying the global behavior across the entire system. The topic of this thesis is the firing synchronization problem. The goal is a state synchronization, that is, to define the automata in the network so that all machines enter a special fire state at the same time. The problem requires that all automata in the network be identical, and that any solution to the problem must cause all automata to fire simultaneously and for the first time at the end of the computation. The work presented in this thesis falls into two categories. The first set of results improve previously published solutions to variants of the firing synchronization problem. The improvements are not in the amount of time needed for synchronization, but instead in the number of states needed to represent machines. Since the number of states represents the complexity of the solution, modifying solutions to use fewer states improves them by making them simpler. The second set of results concerns lower bounds for the firing synchronization problem. As with the previous results, the concern is not in bounding the amount of time needed to synchronize, but instead in finding bounds on the number of states used by the automaton. This work not only produces new bounds on two variants of the problem, but also provides evidence that the ultimate resolution of the state lower bound may need a different set of techniques than have been previously used.
Proceedings of the twentieth annual ACM symposium on Theory of computing - STOC '88, 1988
Let A be a (possibly infinite) graph, and locate a processor p at every vertex of A. Each process... more Let A be a (possibly infinite) graph, and locate a processor p at every vertex of A. Each processor p hzs a finite set Q of states. If d is Ihe clegrcc th;tc pfoccssor p lies in A , then its neighbors arc v,(p),. * ,vd(p). The state of each processor is a function of the states of its neighbors. More precisely, the state of p at time I, St (p .t) i!; updated as St(p,t+l) = Tranr(.9(V,(P),f), '. . Sf(v,(p:,t)) where Tram : Qd +Q is the transition function. Such objects are called celhhr spaces, tessalations, iterative arrays of finite automata, systolic arrays, or networks of computers. They have been extensively studied [vN] [HI IK] ['E!) as they are a natural model of parallel universal computers, and of many physical and biologicxl phenomena. We will examine the cluestion of how to make such networks resistant to errors.-' Supported in part by National Science Foundation ~;rants DCR 8407256. CCR 87-10078. CCR 8746518, and by AFOSR c'ontract 87-04co Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for d rect commercial advantage, the ACM copyright notice and the title of the publication and its date appear, a.nd notice is given that copying is by permission of the Association for Computing Machinery. To copy otherwise, or 1o republish, rcquircs a fee and/or spxfic permission.
Both the bottleneck counting argument [7, 8] and Razborov's approximation method [1, 4, 12] ... more Both the bottleneck counting argument [7, 8] and Razborov's approximation method [1, 4, 12] have been used to prove exponential lower bounds for monotone circuits. We show that under the monotone circuit model for every proof by the approximation method, there is a bottleneck counting proof and vice versa. We also illustrate the elegance of the bottleneck counting technique with a simple self-explained example: the proof of a (previously known) lower bound for the 3-CLIQUEn problem by the bottleneck counting argument. Keywords: Computational complexity; Circuit complexity; Monotone circuit complexity 1 Introduction Razborov's proof of an exponential lower bound on the size of monotone Boolean circuits to detect cliques in a graph [12] [1], represented a breakthrough in the theory of monotone circuit complexity. The proof introduced the method of approximation. The method is roughly as follows. Consider two sets of test inputs, a positive (the output is 1) and a negative on...
Advanced Collective Communication in WDM Optical Rings
PDF and gzipped PostScript formats via anonymous FTP from the area ftp.cs.unibo.it:/pub/TR/UBLCS ... more PDF and gzipped PostScript formats via anonymous FTP from the area ftp.cs.unibo.it:/pub/TR/UBLCS or via WWW at
Vehicular Ad Hoc Networks (VANETs) are attracting the attention of researchers, industry, and gov... more Vehicular Ad Hoc Networks (VANETs) are attracting the attention of researchers, industry, and governments for their potential of significantly increasing the safety level on the road. In order to understand whether VANETs can actu-ally realize this goal, in this paper we analyze the dynamics of multi-hop emergency message dissemination in VANETs. Under a probabilistic wireless channel model that accounts for interference, we derive lower bounds on the probability that a car at distance d from the source of the emergency message correctly receives the message within time t. Be-sides d and t, this probability depends also on 1-hop channel reliability, which we model as a probability value p, and on the message dissemination strategy. Our bounds are derived for an idealized dissemination strategy which ignores inter-ference, and for two provably near-optimal dissemination strategies under protocol interference. The bounds derived in the first part of the paper are used to carefully ana...
We present two solutions to the firing synchronization problem on the ring, an 8-state minimal-ti... more We present two solutions to the firing synchronization problem on the ring, an 8-state minimal-time solution and a 6-state non-minimal-time solution. Both solutions use fewer states than the previous best-known minimal-time automaton, a 16-state solution due to Culik. We also give the first lower bounds on the number of states needed for solutions to the ring firing synchronization problem. We show that there is no 3-state solution and no 4-state, symmetric, minimal-time solution for the ring. 1
In this pa per we prese nt improve d bounds on the comple xity of solutions to the ¿ ring synchro... more In this pa per we prese nt improve d bounds on the comple xity of solutions to the ¿ ring synchroniza tion problem. In the ¿ ring synchronization proble m we cons ider a one-dime ns ional a rray of n identica l ¿ nite a utoma ta . Initia lly all a utoma ta a re in the same s ta te e xcept for one automaton which is des igna te d a s the initia tor for the synchroniza tion. Our re sults hold for the origina l problem, whe re the initia tor may be located a t eithe r e ndpoint, and for the va ria nt whe re any one of the automata ma y be the initiator, ca lle d the ge nera lize d problem. In both ca ses , the goa l is to de ¿ ne the set of s ta te s a nd trans ition rule s for the a utoma ta so tha t a ll machine s e nte r a specia l ¿ re s tate s imultane ous ly and for the ¿ rs t time during the ¿ na l round of the computation. In our work we improve the cons truction for the bes t known minima l-time solution to the ge nera lize d problem by re ducing the number of s ta te s nee de...
Non-Minimal Time Solutions for the Firing Sychronization Problem
The firing synchronization problem concerns a one-dimensional array of $n$ finite automata. All a... more The firing synchronization problem concerns a one-dimensional array of $n$ finite automata. All automata are identical, except the ones on either end of the array, one of which is the initiator for the synchronization. The goal is to define the set of states and transition rules for the automata so that all machines enter a special fire state for the first time and simultaneously at some time $t(n)$. The generalized firing synchronization problem differs from the original problem in that the initiator of the synchronization can be located anywhere in the array. In this paper we present two non-minimal time solutions to the firing synchronization problem. The first is a 6-state solution to the original problem which allows the initiator to be located at either the right or left endpoint. This automaton has two fewer states than R.~Balzer''s 8-state minimal-time automaton \cite{b}, the minimal-time automaton with the smallest number of states that allows the initiator to be lo...
On some central problems in computational complexity
Page 1. f > " u u t u l w K ø { ÿ { Q " # I Q ( ' ' µ ... more Page 1. f > " u u t u l w K ø { ÿ { Q " # I Q ( ' ' µ µ ] y w u Ï w s Ì Ç y u l l u Ç u ö # í ì ø ì ( f " f y l w Ï w u y ø µ Å Ì t u Ð µ o ...
Abstract. It is likely that reliable and fast space-bounded probabilistic acceptors are less powe... more Abstract. It is likely that reliable and fast space-bounded probabilistic acceptors are less powerful than nondeterministic ones. We consider a restricted model of space-bounded probabilistic computation, the random analog of a model studied in [CR]. We show that maze traversal ...
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