M. Minsky. Computation: Finite and Innite Machines. Prentice Hall, 1967.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....unlabelled, asynchronous transition systems and nite elementary net systems. In order to make the proof more transparent we rst introduce an intermediate problem of domino bisimilarity and show its undecidability by a direct reduction from the undecidable halting problem for 2 counter machines [Min67]. The undecidability of the novel problem of checking domino bisimilarity seems to be interesting in its own right and does not follow from somewhat related results for domino snakes [EHM94] and domino games [Gr a90] nor from the undecidability of the classical tiling problems [BGG97] 3 2 ....

....result of the paper, i.e. the undecidability of hhp bisimilarity for nite, both labelled and unlabelled, asynchronous transition systems and elementary net systems. In Subsection 3.1 we de ne the algorithmic problem of checking domino bisimilarity. In Subsection 3. 2 we recall 2 counter machines [Min67] and in Subsection 3.3 we prove the undecidability of labelled domino bisimilarity by a reduction from the halting problem for 2 counter machines. Finally, in Subsection 3.4 we extend the undecidability result to the unlabelled case. It is worthwhile to compare our domino bisimilarity, or ....

M. Minsky. Computation: Finite and Innite Machines. Prentice Hall, 1967.

....initialised and non initialised rectangular automata. To establish theorem 5 we will reduce the reachability problem for two counter machines to the unknown rate sampling time reachability decision problem for timed automata. 4. 1 Two counter machines De nition 18 (Two counter machines [Min72] A two counter machine C is a tuple (Q; q 0 ; C 1 ; C 2 ; I) where: Q is a nite set of states, q 0 2 Q is the initial state, 11 C 1 and C 2 are two counters with values in N, I is a nite set of instructions of the form (q; q 0 2 Q, k 2 f1; 2g) 1. q : C k : C k 1; goto q ....

....problem namely the state reachability problem by: given C = Q; q 0 ; C 1 ; C 2 ; I) a two counter machine and r 2 Q, determine whether there exist c 1 ; c 2 2 N s.t. r; c 1 ; c 2 ) is reachable in SC . As two counter machines are equivalent to Turing machines we know that: Theorem 6 ( Min72] For two counter machines, the state reachability problem is undecidable. 4.2 Simulation of a two counter machine by a sampled TA We want reduce the state reachability problem for two counter machines to the unknown sampling rate reachability decision problem for TA. The reduction works as ....

Marvin Minsky. Computation: Finite and innite machines. Prentice-Hall, London, 1972.

....and others seeking to quantify the notion of complexity. The basic Chomsky classes are called, from simplest to most complex, regular, context free, context sensitive, and unrestricted; these correspond to increasingly powerful kinds of machines, at the top of which sits the Turing machine (e.g. [41]) which, according to the Church Turing thesis, is computationally universal. In fact, examples all up and down this hierarchy can be found in dynamical 2 systems theory. The languages generated by many simple hyperbolic systems are regular; this corresponds to the existence of a nite Markov ....

M. Minsky, Computation: Finite and Innite Machines. Prentice-Hall, 1967.

....work of McCulloch and Pitts [32] and Kleene [25] that nite binary state neural networks are equivalent to nite automata for processing sequentially given inputs. A somewhat interesting question here is how ecient are neural nets as representations of nite automata. The elementary constructions [33] yield a network of about 2m binary state neurons for simulating an m state automaton, and it was proved only relatively recently in [1] that at least a m log m) 1=3 ) neurons are really required in the worst case.t The upper bound was further improved to O(m 1=2 ) neurons in [22, 23] and ....

Minsky, M. L. Computation: Finite and Innite Machines. Prentice-Hall, Englewood Clis, NJ, 1967.

....free set variable X and with no set quanti ers such that for all (codes) c 2 N : c 2 WB ( 9X (c; X) Thus the desired upper bound is established. 2 Now we prove Theorem 3. The main idea lies in showing a certain (algorithmic) construction CONS which can be used for any Minsky counter machine C ([17]) with an even (for convenience) number of (nonnegative) counters c 1 ; c 2 ; c 2n ; the construction yields a pair of Petri nets N 1 , N 2 such that N 1 , N 2 are weakly bisimilar i it is true that 9x 1 8x 2 : 9x 2n 1 8x 2n : C(x 1 ; x 2 ; x 2n ) By the notation : C(x 1 ; ....

M. Minsky, Computation: Finite and Innite Machines (Prentice Hall, 1967).

.... 2; n Gamma 1) is in one of the following two forms (assuming 1 k; k 1 ; k 2 n, 1 j m) ffl c j : c j 1; goto k ffl if c j = 0 then goto k 1 else (c j : c j Gamma 1; goto k 2 ) The halting problem is undecidable even for Minsky machines with two counters initialized to zero values [27]. Any such machine M can be easily mimicked by a StExt(PA) process P (M) Delta; Q; BT) where ffl Delta contains the following rules: Delta Z j a I j :Z j , Z j a Z j Delta I j a I j :I j , I j a where j 2 f1; 2g. ffl Q = fq 1 ; q n g, where n is the number of ....

M.L. Minsky. Computation: Finite and Innite Machines. Prentice-Hall, 1967.

....the result, the technique of its proof and some consequences seem to be interesting; we discuss it in next paragraphs. The proof is based on a well known universal computing device, namely the register machines of Shepardson and Sturgis ( 26] or, equivalently, the counter machines of Minsky ([22]) The universality implies the undecidability of the relevant halting problem, which is of particular interest here. Since nonnegative counters (registers) can be regarded as places with tokens, one is naturally tempted to try to simulate a counter machine by a Petri net. But it was early ....

....is to simulate C correctly (in one of the nets) nishing in the halting state which, of course, is possible if and only if C halts for the given input values. Since the halting problem is undecidable even for a xed counter machine C with 2 counters (only the input values varying; cf. [22]) we can x the net C ; in addition, its structure only allows 2 places (corresponding to the counters) to be unbounded. In case of bisimilarity, the relevant modi cations (yielding N 1 , N 2 ) do not add unbounded places; thus we get the undecidability of the bisimilarity problem even if ....

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M. Minsky, Computation: Finite and Innite Machines (Prentice Hall, 1967).

.... 1 Introduction The relationship between discrete time recurrent neural networks (DTRNN) and nite state machines (FSM) has been explored in a number of di erent ways by many researchers in the past ten years, although this relationship has earlier roots (McCulloch Pitts 1943; Kleene 1956; Minsky 1967). All of these early papers refer to neural networks made up of threshold units (with step like activation functions) More recently, Alon et al. 1991) Indyk (1995) and Horne Hush (1996) have studied in more detail bounds on the number of threshold units necessary to implement a FSM of a ....

....will be generalized to realvalued sigmoids and to a larger class of FSM in this paper. Their proof relies on converting the deterministic nite state automaton (DFA) into a new DFA in which all states are split in as many states as there are symbols in the input alphabet of the DFA (as Minsky (1967) did) 1 An in nite value of the gain would correspond to the use of a step function; the result in this case is well known (Minsky 1967) 2 Kremer s (1995) work used Elman nets with threshold functions, not sigmoids. Kremer (1996) has recently shown that a single layer rst order ....

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Minsky, M.L. 1967. Computation: Finite and Innite Machines. Englewood Clis, NJ: Prentice-Hall, Inc. Ch: Neural Networks. Automata Made up of Parts.

.... 1 Introduction The relationship between discrete time recurrent neural networks (DTRNN) and nite state machines (FSM) has been explored in a number of di erent ways by many researchers in the past ten years, although this relationship has earlier roots (McCulloch Pitts 1943; Kleene 1956; Minsky 1967). All of these early papers refer to neural networks made up of threshold units (with step like transfer functions) More recently, Alon et al. 1991) and Horne Hush (1996) have studied in more detail bounds on the number of threshold units necessary to implement a FSM of a given number of ....

....will be generalized to realvalued sigmoids and to a larger class of FSM in this paper. Their proof relies on converting the deterministic nite state automaton (DFA) into a new DFA in which all states are split in as many states as there are symbols in the input alphabet of the DFA (as Minsky (1967) did) 1 An in nite value of the gain would correspond to the use of a step function; the result in this case is well known (Minsky 1967) 2 Kremer s (1995) work used Elman nets with threshold functions, not sigmoids. Kremer (1996) has recently shown that a single layer rst order ....

[Article contains additional citation context not shown here]

Minsky, M.L. 1967. Computation: Finite and Innite Machines. Englewood Clis, NJ: Prentice-Hall, Inc. Ch: Neural Networks. Automata Made up of Parts.

....a short term memory (STM) using context units. The context units in the input layer hold a copy of the activation of the hidden layer from the previous time step. Fully connected recurrent neural networks have been shown to be capable of implementing any arbitrary nite state automata (FSA) [49]. The network shown in Figure 3 corresponds to a single neuron. Input to the model is via nine input nodes arranged in a 3 3 grid. These nine input nodes are fully connected to an eighteen node hidden layer. Each node in the hidden layer is connected to its counterpart in an eighteen node context ....

Minsky, M. (1967). Computation: Finite and Innite Machines. Englewood Clis, NJ: Prentice{Hall.

....such as the discretization of time and signals) they de ned what we currently know as a nite state machine (FSM) Later, Kleene (1956) formalized the sets of input sequences thatled a McCulloch Pitts network to a given state and de ned what we currently know as regular sets or languages. Minsky (1967) showed that any FSM can be simulated by a discrete time recurrent neural net (DTRNN) using McCulloch Pitts units; the construction used a number of neurons proportional to the number of states in the automaton m; more recently, Alon et al. 1991) and Horne Hush (1996) have established better ....

....the corresponding FSM. Omlin Giles (1996) have proposed an algorithm for encoding deterministic nite state automata (DFA, a class of FSM) in second order recurrent neural networks which is based on a study of the xed points of the sigmoid function. Alqu ezar Sanfeliu (1995) have generalized Minsky s (1967) result to show that DFA may be encoded in Elman (1990) nets with rational (not real) sigmoid transfer functions. Kremer (1996) has recently shown that a singlelayer rst order sigmoid DTRNN can represent the state transition function of any nite state automaton. Frasconi et al. 1996) have shown ....

Minsky, M.L. 1967. Computation: Finite and Innite Machines . Englewood Clis, NJ: Prentice-Hall, Inc. Ch: Neural Networks. Automata Made up of Parts.

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M. Minsky, Computation: Finite and Innite Machines, McGraw Hill, 1967. 4

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M. Minsky, Computation: Finite and Innite Machines. Prentice-Hall, 1967.

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