Abstract: Symbolic substitution logic is based on optical pattern transformations This space-invariant mechanism is shown to be capable of supporting space-variant operations An optical implementation is proposed It is based on splitting an image, shifting the split images, superimposing the results, regenerating the superimposed image with an optical logic array, splitting the regenerated image, shifting the resulting images, and superimposing the shifted images Experimental results are presented Examples demonstrate how symbolic substitution logic can be used to implement Boolean logic, binary arithmetic, cellular logic, and Turing machines

Abstract: We study two classes of unbounded fan-in parallel computation, the standard one, based on unbounded fan-in ANDs and ORs, and a new class based on unbounded fan-in threshold functions. The latter is motivated by a connectionist model of the brain used in Artificial Intelligence. We are interested in the resources of time and address complexity. Intuitively, the address complexity of a parallel machine is the number of bits needed to describe an individual piece of hardware. We demonstrate that (for WRAMs and uniform unbounded fan-in circuits) parallel time and address complexity is simultaneously equivalent to alternations and time on an alternating Turing machine (the former to within a constant multiple, and the latter a polynomial). In particular, for constant parallel time, the latter equivalence holds to within a constant multiple. Thus, for example, polynomial-processor, constant-time WRAMs recognize exactly the languages in the logarithmic time hierarchy, and polynomial-word-size, constant-time WRAMs recognize exactly the languages in the polynomial time hierarchy. As a corollary, we provide improved simulations of deterministic Turing machines by constant-time shared-memory machines. Furthermore, in the threshold model, the same results hold if we replace the alternating Turing machine with the analogous threshold Turing machine, and replace the resource of alternations with the corresponding resource of thresholds. Threshold parallel computers are much more powerful than the standard models (for example, with only polynomially many processors, they can compute the parity function and sort in constant time, and multiply two integers in O(log*n) time), and appear less amenable to known lower-bound proof techniques.

Abstract: This paper contains the first concrete lower bound argument for Turing machines with one work tape and a two-way input tape (these Turing machines are often called "offline 1-tape Turing machines"). In particular we prove an optimal lower bound of Ω(n3/2/(log n)1/2) for transposing a matrix with elements of bit length ∘(logn) (where n is the length of the total input). This implies a lower bound of Ω(n3/2/(log n)1/2) for sorting on the considered type of Turing machine. We also get as corollaries the first nonlinear lower bound for the most difficult version of the two tapes — versus — one problem, and a separation of the considered type of Turing machine from that with an additional write-only output tape.

Abstract: Paul and Reischuk devised space efficient simulations of logarithmic cost random access machines and multidimensional Turing machines. We simplify their general space reduction technique and extend it to other computational models, including pointer machines, which model computations on graphs and data structures. Every pointer machine of time complexityT(n) can be simulated by a pointer machine of space complexityO(T(n)/logT(n)).

Abstract: For on-line recognition of the words in an arbitrary linear context-free language, there are known tight bounds on the time required by a deterministic multitape Turing machine. In terms of word length n , the time need never be worse than some constant times n 2 , even if only one worktape is available; and there is a linear context-free language that requires at least time proportional to n 2 /log n , no matter how many worktapes are available. Using Kolmogorov's notion of descriptional complexity as a tool, we present a simple proof of the latter result.

Abstract: Artificial Intelligence research has come under fire for failing to fulfill its promises. A growing number of AI researchers are reexamining the bases of AI research and are challenging the assumption that intelligent behavior can be fully explained as manipulation of symbols by algorithms. Three recent books -- Mind over Machine (H. Dreyfus and S. Dreyfus), Understanding Computers and Cognition (T. Winograd and F. Flores), and Brains, Behavior, and Robots (J. Albus) -- explore alternatives and open the door to new architectures that may be able to learn skills.

Abstract: Mathematical prerequisites Turing machines solvability and unsolvability formal languages recursive functions complexity theory appendix - the Turing machine simulator.

Abstract: Publisher Summary This chapter discusses the bounded arithmetic formulas and Turing machines of constant alternation. Alternating Turing machines were introduced into computational complexity theory by Chandra, Kozen, and Stockmeyer. The chapter describes the Turing machines and some of their properties. An instantaneous description (ID) of M consists of the finite strings on the tapes, the head positions (relative to the tape contents), and the current state.

Abstract: Alternating Turing machines with restrictions preventing them from returning to a previous configuration model games with rules enforcing such a restriction, for instance, the Chinese version of Go. Such restrictions do not affect the time complexity of problems for alternating Turing machines but space S on a machine with the restriction is equivalent either to time or to space exponential in S on a normal alternating machine, depending on the precise nature of the restriction.

Abstract: We use dataflow graphs to represent the computational structure, analogous to Petri nets and Turing machines, and have developed a method for analyzing the reliability of computer systems modeled as dataflow graphs. Because of the hierarchical nature of dataflow graphs, systems can be analyzed at several levels of abstraction. Reliabilities of subgraphs can be calculated using either traditional techniques or dataflow approach presented here (recursively). The reliabilities of subgraphs can then be combined leading to decomposition-aggregation approach. The time needed for an actor to complete its operation is not included in our analysis of dataflow graphs. Incorporation of the time element compounds the problem and we have not studied it yet.

Abstract: A computation of a single tape Turing machine can be simulated by a probabilistic random access machine under the logarithmic cost criterion with a better than linear speedup. This implies that there are languages of any degree of complexity on single tape Turing machines but recognised much faster by probabilistic or non-deterministic random access machines. The critical fact allowing the speed up is that the outcome of most occasions when the Turing machine head enters a block of tape can be determined, without accessing the symbols in the block, provided a small bit pattern is available which contains information on all possible such outcomes. A more complicated version of the simulation gives a similar average running time but also guarantees that the probability of exceeding this average by more than a constant factor tends to zero as the Turing machine's execution time increases.

Abstract: True self-organisation requires a new type of mathematics; the initial stages of its development are described and it is shown to be isomorphic to an earlier algebraic construction.

Abstract: Publisher Summary Recursion theory is that area of mathematical logic where one studies the qualitative aspects of computability. In complexity theory, which is part of computer science, one studies in addition quantitative aspects of computations. A number of open problems about the structure of NP where one can prove that even under the assumption PI≠NP recursion theoretic arguments will not suffice. The chapter presents polynomial time approximation schemes for some strongly NP-complete problems that arise—for example, in robotics. The chapter presents a survey for a proof of optimal lower bounds for two tapes versus one on deterministic and nondeterministic Turing machines. Results that show a substantial superiority of nondeterminism over determinism resp. co-nondeterminism over nondeterminism for one-tape Turing machines are given in the chapter. The chapter presents the proof of the desired result as the construction of a winning strategy for a two-person game.

Abstract: Much complexity-theoretic work on parallelism has focused on the class NC, which is defined in terms of logspace-uniform circuits. Yet P-uniform circuit complexity is in some ways a more natural setting for studying feasible parallelism. In this paper, P-uniform NC (PUNC) is characterized in terms of space-bounded AuxPDA's and alternating Turing Machines with bounded access to the input. We also present a general-purpose parallel computer for PUNC; this characterization leads to an easy proof that NC = PUNC iff all tally languages in P are in NC. The characterizations of PUNC lead to natural methods for modelling precomputation. We show that for many classes of interest, there is a single “universal” table which can be used in place of any table of similar size and complexity, while for certain other classes, no such “universal” table exists.

Abstract: A new and uniform technique is developed for the simulation of nondeterministic multitape Turing machines by simpler machines. For many types of restricted nondeterministic Turing machines it can now be proved that both linear time is no more powerful than real time, and multitape machines are no more powerful than machines with two tapes, one of which is a simple and normalized comparison tape. This is an improvement over all previously known simulations in terms of weaker machines. As an application we obtain that, for all such machines, the class of languages accepted in real time by multitape machines is principal and has a simple trio generator. Moreover, multitape machines with different types of tapes are hierarchically related, contrasting with the case of one-tape machines, and some important families of languages are characterized in a new way.

Abstract: A new two-person pebble game that abstracts the control structure of many parallel algorithms is defined and studied This game extends the two-person pebble game defined by Dymond and Tompa (JCSS, Vol 30, no 2, 1985, pp 149-161) in two ways: (a) the game is played on a Boolean circuit rather than on an unlabelled graph, and takes into consideration the types of the gates in the circuit, and (b) the two players' roles are completely symmetric The new game is used to study the relationship between two natural parallel complexity classes, namely LOGCFL and AC('1) LOGCFL is the class of languages log space reducible to context-free languages AC('1) is the class of languages accepted by an alternating Turing machine in space O(log n) and alternation depth O(log n) LOGCFL is a subclass of AC('1), but it is not known whether the inclusion is proper For many problems in LOGCFL the algorithms that show their membership in that class also show their membership in AC('1) However, these algorithms do not use the full power of AC('1) computations The two-person game defined here provides a model of computation in which this perceived difference can be quantified This is done by characterizing the two classes using the same measures of resources in the game model The results so obtained not only illustrate the similarity between these two classes, but also isolate a fundamental difference between them: the recognition of languages in LOGCFL does not utilize the symmetry between the two players, whereas the recognition of languages in AC('1) does A resource that captures this difference, called role switches, is identified in the game model: languages in LOGCFL use no role switches, whereas languages in AC('1) use O(log n) role switches Thus the results indicate why the two classes may not be equal As another application of this game, it is shown that the game model unifies in a single framework the proofs of the following three well known results of complexity theory: (1) Savitch's theorem that nondeterministic space S is contained in deterministic space S('2), (2) Ruzzo's NC algorithm for context-free language recognition, and (3) Borodin and Ruzzo's simulation of simultaneous space and alternation bounded alternating Turing machines by simultaneous space and time bounded alternating Turing machines Motivated by the characterizations in terms of the game, a property called semi-unboundedness is defined for the following four models: alternating Turing machines, nondeterministic auxiliary pushdown automata, bounded fan-in Boolean circuits, and unbounded fan-in Boolean circuits This is used in obtaining new characterizations of LOGCFL on these models Three of these characterizations are in terms of the same measures of resources used to characterize AC('1) on these models, and provide supporting evidence to the belief that these two classes are not equal

Abstract: Two structurally defined types of NP sets are studied. k-simple sets are defined and shown to exist in NP. Other properties of these sets are investigated. k-creative sets, as previously defined by Joseph and Young [10], are next considered. A new condition is given which implies that a set is k-creative. Several previously considered NP-complete sets are proved to be k-creative.

Abstract: We give a new characterization of the hyperarithmetic sets: a set X of integers is recursive in eα if and only if there is a Turing machine which computes X and “halts” in less than or equal to the ordinal number ωα of steps. This result represents a generalization of the well-known “limit lemma” due to J. R. Shoenfield [Sho-1] and later independently by H. Putnam [Pu] and independently by E. M. Gold [Go]. As an application of this result, we give a recursion theoretic analysis of clopen determinacy: there is a correlation given between the height α of a well-founded tree corresponding to a clopen game A ⊆ ωω and the Turing degree of a winning strategy ƒ for one of the players—roughly, ƒ can be chosen to be recursive in 0α and this is the best possible (see §4 for precise results).

Abstract: We introduce completely symmetric D2L systems and cellular automata by means of an additional restriction on the corresponding classes of symmetric devices. Then we show that completely symmetric D2L systems and cellular automata are still able to simulate Turing machine computations. As corollaries we obtain new characterizations of the recursively enumerable languages and of some space-bounded complexity classes.

Abstract: Previously, we proposed a (k, l)-neighborhood template δ-type bounded cellular acceptor (abbreviated as δ--BCA (k, l)), which is composed of a pair of converters and a configuration-reader and operates on a two-dimensional tape. Its basic properties have already been discussed. δ-BCA (k, l) is a parallel automaton, which, in a sense, is a generalization of the one-dimensional bounded cellular acceptor. This paper attempts a more detailed examination of the properties of δ-BCA (k, l), and compares its accepting power with those of other automata operating on the two-dimensional tape. The objects of comparison are the various kinds of two-dimensional finite automata and various kinds of parallel sequential array acceptors. It is known that there exists an equivalent tape-bounded Turing machine for each of these automata in the sense of the accepting power. Consequently, the comparison of the accepting power of δ-BCA (k, l) and those of two-dimensional automata amounts in a sense to the evaluation of the accepting power of δ-BCA (k, l) in terms of the tape complexity of the tape-bounded Turing machine.