On Turing Machines Deciding According to the Shortest Computations

In this paper we propose and analyse from the computational complexity point of view several new variants of nondeterministic Turing machines. In the first such variant, a machine accepts a given input word if and only if one of its shortest possible computations on that word is accepting; on the other hand, the machine rejects the input word when all the shortest computations performed by the machine on that word are rejecting. We are able to show that the class of languages decided in polynomial time by such machines is PNP[log]. When we consider machines that decide a word according to the decision taken by the lexicographically first shortest computation, we obtain a new characterization of PNP. A series of other ways of deciding a language with respect to the shortest computations of a Turing machine are also discussed.

A Formal Analysis of the Concept of Behavioral Individuation of Mental States in the Functionalist Framework

The functionalist theory of mind proposes to analyze mental states in terms of internal states of Turing machine, and states of the machine’s tape and head. In the paper, I perform a formal analysis of this approach. I define the concepts of behavioral equivalence of Turing machines, and of behavioral individuation of internal states. I prove a theorem saying that for every Turing machine T there exists a Turing machine T’ which is behaviorally equivalent to T, and all of whose internal states of T’ can be behaviorally individuated. Finally, I discuss some applications of this theorem to computational theories of mind.

Quantum information distance

In this paper, we give a definition for quantum information distance. In the classical setting, information distance between two classical strings is developed based on classical Kolmogorov complexity. It is defined as the length of a shortest transition program between these two strings in a universal Turing machine. We define the quantum information distance based on Berthiaume et al.’s quantum Kolmogorov complexity. The quantum information distance between qubit strings is defined as the length of the shortest quantum transition program between these two qubit strings in a universal quantum Turing machine. We show that our definition of quantum information distance is invariant under the choice of the underlying quantum Turing machine.

Interactive Computation and Artificial Epistemologies

What is algorithmic thought? It is not possible to address this question without first reflecting on how the Universal Turing Machine transformed symbolic logic and brought to a halt the universality of mathematical formalism and the biocentric speciation of thought. The article draws on Sylvia Wynter’s discussion of the sociogenic principle to argue that both neurocognitive and formal models of automated cognition constitute the epistemological explanations of the origin of the human and of human sapience. Wynter’s argument will be related to Gilbert Simondon’s reflections on ‘technical mentality’ to consider how socio-techno-genic assemblages can challenge the biocentricism and the formalism of modern epistemology. This article turns to ludic logic as one possible example of techno-semiotic languages as a speculative overturning of sociogenic programming. Algorithmic rules become technique-signs coinciding not with classic formalism but with interactive localities without re-originating the universality of colonial and patriarchal cosmogony.

Verification of composed web service using Synthesized Nondeterministic Turing Model (SNTMM) with Multiple Tapes and Stacks

To verify the composed Web services, a general view of what traits of a service need to be identified is still lacking. The existing verification model did not address any mechanism for getting alternative services if we failed to reach the desired service and partially concentrated on the reachability problem for a deterministic and non-deterministic system in sequential. This paper proposes a Synthesised Non-deterministic Turing Machine Model (SNTMM) by combining the Multistacked Non-deterministic Turing Machine (MSNTM) model and Multitaped Non-deterministic Turing Machine (MTNTM) model to verify the composed Web services for both deterministic and non-deterministic systems in parallel. The deceased transition and departed service marking algorithm have been proposed to address each participated service’s reachability in composing service for all possible input in parallel. This article shows an example to demonstrate the meticulousness of the model. The experimental results show that the performance of the proposed model is measured efficiently

Space-bounded OTMs and REG ∞

An important theorem in classical complexity theory is that REG = LOGLOGSPACE, i.e., that languages decidable with double-logarithmic space bound are regular. We consider a transfinite analogue of this theorem. To this end, we introduce deterministic ordinal automata (DOAs) and show that they satisfy many of the basic statements of the theory of deterministic finite automata and regular languages. We then consider languages decidable by an ordinal Turing machine (OTM), introduced by P. Koepke in 2005 and show that if the working space of an OTM is of strictly smaller cardinality than the input length for all sufficiently long inputs, the language so decided is also decidable by a DOA, which is a transfinite analogue of LOGLOGSPACE ⊆ REG; the other direction, however, is easily seen to fail.

Undecidability in quantum thermalization

The investigation of thermalization in isolated quantum many-body systems has a long history, dating back to the time of developing statistical mechanics. Most quantum many-body systems in nature are considered to thermalize, while some never achieve thermal equilibrium. The central problem is to clarify whether a given system thermalizes, which has been addressed previously, but not resolved. Here, we show that this problem is undecidable. The resulting undecidability even applies when the system is restricted to one-dimensional shift-invariant systems with nearest-neighbour interaction, and the initial state is a fixed product state. We construct a family of Hamiltonians encoding dynamics of a reversible universal Turing machine, where the fate of a relaxation process changes considerably depending on whether the Turing machine halts. Our result indicates that there is no general theorem, algorithm, or systematic procedure determining the presence or absence of thermalization in any given Hamiltonian.