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.
... 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 ...
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).
... 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.
... 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 ...
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).
... 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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