International Journal of Foundations of Computer Science, 2015
In the late nineteen sixties it was observed that the r.e. languages form an infinite proper hier... more In the late nineteen sixties it was observed that the r.e. languages form an infinite proper hierarchy [Formula: see text] based on the size of the Turing machines that accept them. We examine the fundamental position of the finite languages and their complements in the hierarchy. We show that for every finite language L one has that L, [Formula: see text] for some [Formula: see text] where m is the length of the longest word in L, c is the cardinality of L, and [Formula: see text]. If [Formula: see text], then [Formula: see text] for some [Formula: see text]. We also prove that for every n, there is a finite language Ln with [Formula: see text] such that [Formula: see text] but Ln, [Formula: see text] for some [Formula: see text]. Several further results are shown that how the hierarchy can be separated by increasing chains of finite languages. The proofs make use of several auxiliary results for Turing machines with advice.
Representation and Reality in Humans, Other Living Organisms and Intelligent Machines, 2017
Cognitive processes are often modelled in computational terms. Can this still be done if only min... more Cognitive processes are often modelled in computational terms. Can this still be done if only minimal assumptions are made about any sort of representation of reality? Is there a purely knowledge-based theory of computation that explains the key phenomena which are deemed to be computational in both living and artificial systems as understood today? We argue that this can be done by means of techniques inspired by the modelling of dynamical systems. In this setting, computations are defined as curves in suitable metaspaces and knowledge is generated by virtue of the operation of the underlying mechanism, whatever it is. Desirable properties such as compositionality will be shown to fit naturally. The framework also enables one to formally characterize the computational behaviour of both knowledge generation and knowledge recognition. The approach may be used in identifying when processes or systems can be viewed as being computational in general. Several further questions pertaining to the philosophy of computing are considered.
ABSTRACT We study a versatile model of evolving interactive computing: lineages of automata. A li... more ABSTRACT We study a versatile model of evolving interactive computing: lineages of automata. A lineage consists of a sequence of interactive finite automata, with a mechanism of passing information from each automaton to its immediate successor. Lineages enable a definition of a suitable complexity measure for evolving systems. We show several complexity results, including a hierarchy result.
Recent research in theoretical physics on 'Malament-Hogarth space-times' indicates that so-called... more Recent research in theoretical physics on 'Malament-Hogarth space-times' indicates that so-called relativistic computers can be conceived that can carry out certain classically undecidable queries in finite time. We observe that the relativistic Turing machines which model these computations recognize precisely the Δ 2 −sets of the Arithmetical Hierarchy. In a complexity-theoretic analysis, we show that the (infinite) computations of S(n)-space bounded relativistic Turing machines are equivalent to (finite) computations of Turing machines that use a S(n)-bounded advice f , where f itself is computable by a S(n)-space bounded relativistic Turing machine. This bounds the power of polynomial-space bounded relativistic Turing machines by TM/poly. We also show that S(n)-space bounded relativistic Turing machines can be limited to one or two relativistic phases of computing.
... The previous results can be seen as applications of computability theory to artificial life s... more ... The previous results can be seen as applications of computability theory to artificial life systems. The main result explaining the emergence of the super-Turing computing potential within the respective systems certainly justifies the approach and points to the increasing role that ...
In MANETs and similar link-free networks of communicating objects there is no central authority f... more In MANETs and similar link-free networks of communicating objects there is no central authority for naming and connection management. Autoconfiguration of network nodes is therefore desirable and, building on approaches in IPv6, a number of 'zero configuration' networking protocols have been proposed for this case. Typically these protocols do not easily scale and have difficulty with network partitioning and merging. In this paper we propose a number of novel, decentralized techniques for name resolution in zero configuration protocols that are more flexible and yet lead to name extensions of smallest possible length, assuming that objects sufficiently mix within their ranges. Finally, the techniques are converted into a fully decentralized, scalable autoconfiguration protocol for use in ad hoc networks with directional antennas. The protocol is 'lazy' in the sense that name resolution is postponed until the moment that conflicts prevent the correct functioning of the communication structure.
The ÿrst attempts concerning formalization of the notion of fuzzy algorithms in terms of Turing m... more The ÿrst attempts concerning formalization of the notion of fuzzy algorithms in terms of Turing machines are dated in late 1960s when this notion was introduced by Zadeh. Recently, it has been observed that corresponding so-called classical fuzzy Turing machines can solve undecidable problems. In this paper we will give exact recursion-theoretical characterization of the computational power of this kind of fuzzy Turing machines. Namely, we will show that fuzzy languages accepted by these machines with a computable t-norm correspond exactly to the union 0 1 ∪ 0 1 of recursively enumerable languages and their complements. Moreover, we will show that the class of polynomially time-bounded computations of such machines coincides with the union NP ∪ co-NP of complexity classes from the ÿrst level of the polynomial hierarchy.
ABSTRACT We develop a model of computation as an unbounded process, measuring complexity by the n... more ABSTRACT We develop a model of computation as an unbounded process, measuring complexity by the number of observed behavioural changes during the computation. In a natural way, the model brings effective unbounded computation up to the second level of the Arithmetical Hierarchy, unifying several earlier concepts like trial-and-error predicates and relativistic computing. The roots of the model can be traced back to the circular a-machines already distinguished by Turing in 1936. The model allows one to introduce nondeterministic unbounded computations and to formulate an analogue of the PP-versus-NPNP question. We show that under reasonable assumptions, the resource-bounded versions of deterministic and nondeterministic unbounded computation have equal computational power but that in general, the corresponding complexity classes are different (Pmind⊊NPmind)(Pmind⊊NPmind).
We present a new model of parallel computation called the "array processing machine" or APM (for ... more We present a new model of parallel computation called the "array processing machine" or APM (for short). The APM was designed to closely model the architecture of existing vector-and array processors, and to provide a suitable unifying framework for the complexity theory of parallel combinatorial and numerical algorithms. It is shown that every problem that is solvable in polynomial space on an ordinary, sequential random access machine can be solved in parallel polynomial time on an APM (and vice versa). The relationship to other models of parallel computation is discussed.
The computational potential of artificial living systems can be studied without knowing the algor... more The computational potential of artificial living systems can be studied without knowing the algorithms that govern their behavior. Modeling single organisms by means of socalled cognitive transducers, we will estimate the computational power of AL systems by viewing them as conglomerates of such organisms. We describe a scenario in which an artificial living (AL) system is involved in a potentially infinite, unpredictable interaction with an active or passive environment, to which it can react by learning and adjusting its behaviour. By making use of sequences of cognitive transducers one can also model the evolution of AL systems caused by 'architectural' changes. Among the examples are 'communities of agents', i.e. by communities of mobile, interactive cognitive transducers. Most AL systems show the emergence of a computational power that is not present at the level of the individual organisms. Indeed, in all but trivial cases the resulting systems possess a super-Turing computing power. This means that the systems cannot be simulated by traditional computational models like Turing machines and may in principle solve noncomputable tasks. The results are derived using non-uniform complexity theory.
... Page 8. VIII Organization s s Program Committee Jiˇrı Wiedermann, Chair Jaroslav Pokorny, Co-... more ... Page 8. VIII Organization s s Program Committee Jiˇrı Wiedermann, Chair Jaroslav Pokorny, Co-chair Julius ˇStuller, Co-chair Gerard Tel, Co-chair Bernd Amann Grigoris Antoniou Zohra ... 379 Reliable Broadcasting Without Collision Detection Jaroslaw Kutylowski, Filip Zagorski ...
International Journal of Foundations of Computer Science, 2015
In the late nineteen sixties it was observed that the r.e. languages form an infinite proper hier... more In the late nineteen sixties it was observed that the r.e. languages form an infinite proper hierarchy [Formula: see text] based on the size of the Turing machines that accept them. We examine the fundamental position of the finite languages and their complements in the hierarchy. We show that for every finite language L one has that L, [Formula: see text] for some [Formula: see text] where m is the length of the longest word in L, c is the cardinality of L, and [Formula: see text]. If [Formula: see text], then [Formula: see text] for some [Formula: see text]. We also prove that for every n, there is a finite language Ln with [Formula: see text] such that [Formula: see text] but Ln, [Formula: see text] for some [Formula: see text]. Several further results are shown that how the hierarchy can be separated by increasing chains of finite languages. The proofs make use of several auxiliary results for Turing machines with advice.
Representation and Reality in Humans, Other Living Organisms and Intelligent Machines, 2017
Cognitive processes are often modelled in computational terms. Can this still be done if only min... more Cognitive processes are often modelled in computational terms. Can this still be done if only minimal assumptions are made about any sort of representation of reality? Is there a purely knowledge-based theory of computation that explains the key phenomena which are deemed to be computational in both living and artificial systems as understood today? We argue that this can be done by means of techniques inspired by the modelling of dynamical systems. In this setting, computations are defined as curves in suitable metaspaces and knowledge is generated by virtue of the operation of the underlying mechanism, whatever it is. Desirable properties such as compositionality will be shown to fit naturally. The framework also enables one to formally characterize the computational behaviour of both knowledge generation and knowledge recognition. The approach may be used in identifying when processes or systems can be viewed as being computational in general. Several further questions pertaining to the philosophy of computing are considered.
ABSTRACT We study a versatile model of evolving interactive computing: lineages of automata. A li... more ABSTRACT We study a versatile model of evolving interactive computing: lineages of automata. A lineage consists of a sequence of interactive finite automata, with a mechanism of passing information from each automaton to its immediate successor. Lineages enable a definition of a suitable complexity measure for evolving systems. We show several complexity results, including a hierarchy result.
Recent research in theoretical physics on 'Malament-Hogarth space-times' indicates that so-called... more Recent research in theoretical physics on 'Malament-Hogarth space-times' indicates that so-called relativistic computers can be conceived that can carry out certain classically undecidable queries in finite time. We observe that the relativistic Turing machines which model these computations recognize precisely the Δ 2 −sets of the Arithmetical Hierarchy. In a complexity-theoretic analysis, we show that the (infinite) computations of S(n)-space bounded relativistic Turing machines are equivalent to (finite) computations of Turing machines that use a S(n)-bounded advice f , where f itself is computable by a S(n)-space bounded relativistic Turing machine. This bounds the power of polynomial-space bounded relativistic Turing machines by TM/poly. We also show that S(n)-space bounded relativistic Turing machines can be limited to one or two relativistic phases of computing.
... The previous results can be seen as applications of computability theory to artificial life s... more ... The previous results can be seen as applications of computability theory to artificial life systems. The main result explaining the emergence of the super-Turing computing potential within the respective systems certainly justifies the approach and points to the increasing role that ...
In MANETs and similar link-free networks of communicating objects there is no central authority f... more In MANETs and similar link-free networks of communicating objects there is no central authority for naming and connection management. Autoconfiguration of network nodes is therefore desirable and, building on approaches in IPv6, a number of 'zero configuration' networking protocols have been proposed for this case. Typically these protocols do not easily scale and have difficulty with network partitioning and merging. In this paper we propose a number of novel, decentralized techniques for name resolution in zero configuration protocols that are more flexible and yet lead to name extensions of smallest possible length, assuming that objects sufficiently mix within their ranges. Finally, the techniques are converted into a fully decentralized, scalable autoconfiguration protocol for use in ad hoc networks with directional antennas. The protocol is 'lazy' in the sense that name resolution is postponed until the moment that conflicts prevent the correct functioning of the communication structure.
The ÿrst attempts concerning formalization of the notion of fuzzy algorithms in terms of Turing m... more The ÿrst attempts concerning formalization of the notion of fuzzy algorithms in terms of Turing machines are dated in late 1960s when this notion was introduced by Zadeh. Recently, it has been observed that corresponding so-called classical fuzzy Turing machines can solve undecidable problems. In this paper we will give exact recursion-theoretical characterization of the computational power of this kind of fuzzy Turing machines. Namely, we will show that fuzzy languages accepted by these machines with a computable t-norm correspond exactly to the union 0 1 ∪ 0 1 of recursively enumerable languages and their complements. Moreover, we will show that the class of polynomially time-bounded computations of such machines coincides with the union NP ∪ co-NP of complexity classes from the ÿrst level of the polynomial hierarchy.
ABSTRACT We develop a model of computation as an unbounded process, measuring complexity by the n... more ABSTRACT We develop a model of computation as an unbounded process, measuring complexity by the number of observed behavioural changes during the computation. In a natural way, the model brings effective unbounded computation up to the second level of the Arithmetical Hierarchy, unifying several earlier concepts like trial-and-error predicates and relativistic computing. The roots of the model can be traced back to the circular a-machines already distinguished by Turing in 1936. The model allows one to introduce nondeterministic unbounded computations and to formulate an analogue of the PP-versus-NPNP question. We show that under reasonable assumptions, the resource-bounded versions of deterministic and nondeterministic unbounded computation have equal computational power but that in general, the corresponding complexity classes are different (Pmind⊊NPmind)(Pmind⊊NPmind).
We present a new model of parallel computation called the "array processing machine" or APM (for ... more We present a new model of parallel computation called the "array processing machine" or APM (for short). The APM was designed to closely model the architecture of existing vector-and array processors, and to provide a suitable unifying framework for the complexity theory of parallel combinatorial and numerical algorithms. It is shown that every problem that is solvable in polynomial space on an ordinary, sequential random access machine can be solved in parallel polynomial time on an APM (and vice versa). The relationship to other models of parallel computation is discussed.
The computational potential of artificial living systems can be studied without knowing the algor... more The computational potential of artificial living systems can be studied without knowing the algorithms that govern their behavior. Modeling single organisms by means of socalled cognitive transducers, we will estimate the computational power of AL systems by viewing them as conglomerates of such organisms. We describe a scenario in which an artificial living (AL) system is involved in a potentially infinite, unpredictable interaction with an active or passive environment, to which it can react by learning and adjusting its behaviour. By making use of sequences of cognitive transducers one can also model the evolution of AL systems caused by 'architectural' changes. Among the examples are 'communities of agents', i.e. by communities of mobile, interactive cognitive transducers. Most AL systems show the emergence of a computational power that is not present at the level of the individual organisms. Indeed, in all but trivial cases the resulting systems possess a super-Turing computing power. This means that the systems cannot be simulated by traditional computational models like Turing machines and may in principle solve noncomputable tasks. The results are derived using non-uniform complexity theory.
... Page 8. VIII Organization s s Program Committee Jiˇrı Wiedermann, Chair Jaroslav Pokorny, Co-... more ... Page 8. VIII Organization s s Program Committee Jiˇrı Wiedermann, Chair Jaroslav Pokorny, Co-chair Julius ˇStuller, Co-chair Gerard Tel, Co-chair Bernd Amann Grigoris Antoniou Zohra ... 379 Reliable Broadcasting Without Collision Detection Jaroslaw Kutylowski, Filip Zagorski ...
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