B.Jack Copeland (University of Canterbury, New Zealand)
This paper sketches the history of the idea of a hypermachine - a machine able to generate functions that the universal Turing machine cannot. The paper also examines the origins of the myth that Turing, or Church, or both, somehow showed such machines to be an impossibility.
Carol Cleland (University of Colorado at Boulder)
Intuitively, an effective procedure is a way of getting something done in a finite number of distinct steps. Put another way, what makes a procedure effective is the consequences of following it. As I have argued, the consequences of following a procedure are not determined by its instruction set. This is particularly significant in the case of mundane procedures (recipes, methods, etc.) and computer programs, where the consequences of following a procedure crucially depend upon causal processes. Yet the role of causal processes is usually ignored in theoretical discussions of the computational capacities of concrete computers. I identify the Turing machine model of effective procedures as the source of the problem, contending that it collapses the process which makes a procedure effective into the following (implementation) of the procedure. I show that there are fundamental differences between causal processes and implementations of procedures, and argue that failure to respect these differences has led to a number of puzzles and mistaken views about the capacities of concrete computers.
Mark Hogarth (Cambridge University)
The concept of computability takes on a new appearance in the light of some recent investigations in modern physics. The picture that emerges encompasses that advanced by Turing and his followers, but is richer and more revealing. It is also more satisfying in that the pure and physical aspects of computability are clearly disentangled, and the long-recognised difficulties with the status of the Church-Turing thesis vanish since, without any theoretical detriment, the thesis itself is seen to play no role.
Hava Siegelmann (Technion, Haifa)
The term "Neural Computation" is frequently used, but the associated research field has many faces. While neuro-scientists mainly look for models to explain experimental data, computer scientists consider variants of the classical Turing model in which the additions are reminiscent of particular properties of the brain. The gap between the models considered by these communities is immense: The theory of computational complexity requires the assumption of discrete computation and does not allow for other types of computational paradigms.
In this work, we consider a basic neural network model: finite number of neurons, recurrent interconnection, continuous activation function, and real numbers as weights. This model is considered "analog" for both the use of real numbers as weights which makes the phase space continuous, and the continuity of its activation function. In computational terms, this type of continuous flow (due to its activation function) is definitely a restriction on our model. There are no discrete flow commands such as "If $z$ is greater than $0$, compute one thing, otherwise continue in another computation path".
Given the above, how does the analog network compare with the digital computer in terms of computational capabilities? We show that although the network is continuous, cannot branch on values, does not have a memory unit in the classical sense, and utilizes only a fixed number of neurons, it still can compute anything that a digital machine does. Furthermore, due to its real-valued weights, it is inherently richer than the digital computational model and can compute functions which no digital computer can. The network, although transcending the Turing model, is still sensitive to resource constraints and still computes only a small subclass of all possible functions. It thus constitutes a well-defined super-Turing computational model.
An analogy to the Church-Turing thesis of computability is proposed, but to the realm of analog computation: "No possible abstract analog device can have more computational capabilities (up to polynomial time) than Analog Recurrent Neural Networks". This can be interpreted as suggesting that the neural network be considered a standard model in the realm of analog computation, functioning in a role similar to that of the Turing machine in the realm of digital computation.
An alternative interpretation is possible, in which the extra power is not due to the use of real weights, but rather due to the ability of the machine to tune its internal parameters. This is in contradiction to the Turing model, which is inherently static.
(We note that our results consider the theoretical issue of computational capabilities only: Practical issues such as convergence of the learning algorithm and availability of training data may prevent a network from learning functions that it can in principle represent).
Mike Stannett (Controller, Noisefactory (http://noisefactory.co.uk/))
Because computation concerns the measurement of physical values and the workings of physical devices, the question "what can be computed?" depends ultimately on the underlying model of physics one assumes to be true. In a Newtonian world where time moves forward one click after another, we naturally expect to find computation taking place in well defined steps, and resulting models include the ever-familiar model of Turing computation. Allowing quantum indeterminacy into the model, but keeping the clockwork, we encounter quantum computation. Similarly, in a Newtonian world where time is thought to pass continuously rather than in jumps, we have the analog computation central to Pour-ElŐs researches. Rather than ask what computation would be like in this physics or that, we ask whether we can introduce universal physical structure as a parameter; our interest is a general unified model of computation which will work in any version of physics - you tell the model what your physics looks like and the model tells you whatŐs computable. Since the power of computational systems seems to be inherently tied to the assumed nature of time, we have generated a model of computation over arbitrary temporal structures, and are looking to extend it to include other physical ÔgivensŐ like space and curvature.
Hebrew University of Jerusalem
The Hypercomputation Thesis (HT) is the claim that some machines (perhaps only in the abstract) compute functions that cannot be computed by a universal Turing-machine. If HT is true, then three important questions immediately present themselves:
- Are there any constraints on computable functions? Are there functions that are not computable?
- Is Church-Turing Thesis false?
- Can physical systems carry out hypercomputations?
In this paper we primarily address questions (2) and (3). We first argue that the truth of HT does not falsify the Church-Turing Thesis. We then examine the relations between the Church-Turing Thesis and some physical forms of it (e.g., Gandy (1980), Wolfram (1985), Sieg and Byrnes (1999)). Finally, we explain why these physical forms of the Church-Turing Thesis might be false.
Gert-Jan Lokhorst (Erasmus University, Rotterdam)
Hava Siegelmann has proven that analog recurrent neural nets are, in general, more powerful than the universal Turing machine. This result makes one wonder whether human beings may perhaps be regarded as super-Turing machines. However, there are three reasons to reject the latter hypothesis:
- Physics suggests that there can be no super-Turing machines in the real world (Bekenstein bound; speculations about the ultimate discreteness of space-time);
- The most sophisticated models of the brain which are currently available (with spiking neurons and a generous amount of noise) imply that brains are less powerful than finite automata (results by Maass and others); and
- It is doubtful whether the claim that system X is a super-Turing machine is empirically meaningful.
I will discuss whether these reasons are really convincing, and argue that they are not.
I am nevertheless reluctant to give up the venerable old hypothesis (which has been defended by Kolmogorov, Burks, Minsky, Nelson, and others) that man is, at best, a finite automaton. I do not think that I am a super Turing machine, if only out of modesty. But I am only speaking for myself here; I do not know about you.
Gualtiero Piccinini (University of Pittsburgh)
The focus of this paper is on TuringŐs ideas about machines, mathematical methods of proof, and intelligence. By the late 1930s, Kurt Godel and other logicians, including Turing himself, had shown that no finite set of rules could be used to generate all true mathematical statements. Yet according to Turing, there was no upper bound to the number of mathematical truths provable by intelligent human beings, for they could invent new rules and methods of proof. So, the output of a human mathematician, for Turing, was not a computable sequence (i.e., one that could be generated by a Turing machine). Since computers only contained a finite number of instructions (or programs), one might argue, they could not reproduce human intelligence.
Turing called this the mathematical objection to his view that machines can think. Logico-mathematical reasons, stemming from his own work, helped to convince Turing that it should be possible to reproduce human intelligence, and eventually compete with it, by developing the appropriate kind of digital computers. He felt it should be possible to program a computer so that it could learn or discover new rules, overcoming the limitations imposed by the incompleteness and undecidability results in the same way in which human mathematicians presumably do.
Benjamin Wells (University of San Francisco)
As the last dissertation Alfred Tarski signed, my thesis involved showing that equational theories may be decidable if the number of variables is restricted, yet undecidable if arbitrary equations are considered (there are now several nice papers along these lines by six or seven authors). Tarski always encouraged me to write this up (and I still am!) by saying that it was a good lesson for mathematicians as well as logicians in making sure they got the quantifiers in the right order and paid some attention to requirements of formality in math. But toward the end of the process, late one early morning, he asked me if I knew why he liked this problem. I replied that "it was a good lesson ... ," but he said, "Sure. But thatŐs not why I like it. It is because your equational theory is decidable even if it is not recursive." I protested, "But you need an oracle ... ." He would have none of it, saying that he invented the notion of decidable theory, and this was one.
In this talk I shall present the context for his bold statement, his informal argument in support, routes to formalization at lower complexity, and speculation on what he may have meant.
If you want to prepare, try B. Wells, "Pseudorecursive Varieties of Semigroups - I", International Journal of Algebra and Computation, vol.6 (1996), pp.457-510.
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