A Taxonomy of Computation and Information Architecture
Department of Mathematics,
UCLA, Los Angeles, USA
Chalmers University of Technology
and University of Gothenburg
Nowadays computation is typically understood through the Turing
machine model, in the fields of computability, computational
complexity and even as a basis for present-day computer
hardware and software architectures. Those are technologies
designed in the first place to process data. Being description of
data manipulation, Turing model of computation presents only
one aspect of computation in the real world – an abstraction of the
execution of an algorithm. However, several other possible
aspects of computation, even those existing in today’s
applications, are left outside, thus adequate models in distributed,
self-organized, resource-aware, adaptive, learning computation
systems are needed. This paper presents taxonomy of existing
models of computation. It is connected to more general notion of
natural computation, intrinsic to physical systems, and particularly
cognitive computation in cognitive systems. We see Turing model
of computation as a basic mechanism which can be used to build
more complex computational architectures, that in combination
with interaction with the environment (learning) give advanced
information-processing behaviors in cognitive systems.
Computation, Information Processing, Algorithms, Turing
Machine, Computing Architecture, Computing Taxonomy,
Future progress of new computational devices capable of dealing
with problems of big data, internet of things, semantic web,
cognitive robotics and neuroinformatics depends on the adequate
models of computation. In this article we first present the current
state of the art through systematization of existing models and
mechanisms, and outline basic structural framework of
computation. We argue that defining computation as information
processing, and given that there is no information without
(physical) representation, the dynamics of information on the
fundamental level is physical/ intrinsic/ natural computation. As a
special case, intrinsic computation is used for designed
computation in computing machinery. Intrinsic natural
computation occurs on variety of levels of physical processes,
containing the levels of computation of living organisms
(including highly intelligent animals) as well as designed
The present article offers taxonomy of current models of
computation and indicates future paths for the advancement of the
field; both by the development of new computational models and
by learning from nature how to better compute using different
mechanisms of intrinsic computation.
The question “What is computation?” is answered differently
by different researchers (cf., for example, –). Some did this
in an informal setting based on computational and research
practice, as well as on philosophical and methodological
considerations. Others strived to build exact mathematical models
to comprehensively describe computation. When the Turing
machine (or Logical Computing Machine as Turing originally
named his logical device) was constructed and accepted as a
universal computational model, it was considered as the complete
and exact definition of computation (Church-Turing thesis).
However, the absolute nature of the Turing machine was
disproved by adopting a more general definition of algorithm .
In the spirit of broadening of the concept of computation, the
following definition was proposed: “For a process to qualify as
computation a model must exist such as algorithm, network
topology, physical process or in general any mechanism which
ensures definability of its behavior.” 
Nevertheless, in spite of all efforts, the conception of
computation remains too vague and ambiguous. This vagueness of
the foundations of computing has resulted in a variety of
approaches, including approaches that contradict each other. For
instance,  writes “to compute is to execute an algorithm.”
Active proponents of the Church-Turing Thesis, such as 
claim computation is bounded by what Turing machines are doing
(that is compute mathematical functions). For them the problem of
defining computation was solved long ago with the Turing
machine model. At the same time, Wegner and Goldin insist that
computation is an essentially broader concept than algorithm 
and propose interactive view of computing. While  argues that
computation is symbol manipulation, thus disregarding analog
computers computing over continuous signals, neuroscientists on
the contrary study sub-symbolic computation in neurons. 
Abramsky summarizes the process of successive changing of
computing models as follows:
“Traditionally, the dynamics of computing systems, their
unfolding behavior in space and time has been a mere means to
the end of computing the function which specifies the algorithmic
problem which the system is solving. In much of contemporary
computing, the situation is reversed: the purpose of the computing
system is to exhibit certain behaviour. (…) We need a theory of
the dynamics of informatic processes, of interaction, and
information flow, as a basis for answering such fundamental
questions as: What is computed? What is a process? What are the
analogues to Turing completeness and universality when we are
concerned with processes and their behaviors, rather than the
functions which they compute? 
Abramsky emphasizes that there is the need for second-generation
models of computation, and in particular process models. The first
generation models of computation originated from problems of
formalization of mathematics and logic, while processes or agents,
interaction, and information flow are results of recent
developments of computers and computing. In the second-
generation models of computation, previous isolated systems are
replaced by processes or agents for which the interactions with
each other and with the environment are fundamental. Hewitt too
advocates agent-type, Actor model of computation  which is
suitable for modeling of physical (intrinsic) computation.
Existence of various types and kinds of computation, as well as a
variety of approaches to the concept of computation, shows
remarkable complexity that makes communication of results and
ideas increasingly difficult. Our aim is to explicate present
diversity and so call attention to the necessity of common
understanding: different models of computation may have their
specific uses and applications. It is just necessary to understand
their mutual relationships and assumptions under which they
In this paper, we present methodological analysis of the concept
of computation before and after electronic computers and the
emergence of computer science, demonstrating that history brings
us to the conclusion that efforts in building such definitions by
traditional approaches would be inefficient. An effective
methodology is to find essential features of computation with the
goal to explicate its nature and to build adequate models for
research and technology. In addition, we perform structural
analysis of the concept of computation through explication of
various structures intrinsically related to computation, and in
particular cognitive computation.
One important question related to computation (information
processing) architecture is if cognition can be understood as
computation in the sense of Turing-Church thesis or if cognitive
computation needs additional constructs in case of self-managing
(autonomous) distributed computing systems such as human
brain. Cognition is taken to be the ability to process information,
apply knowledge, and act accordingly, with intent, and it s
accomplishment through various processes that monitor and
control a system and its environment. “Cognition is associated
with a sense of “self” (the observer) and the systems with which it
interacts (the environment or the “observed”). Cognition
extensively uses time and history in executing and regulating
tasks that constitute a cognitive process. In ,  Mikkilineni
argues that “cognition requires more than mere book-keeping
provided by the Turing machines and certain aspects of cognition
such as self-identity, self-description, self-monitoring and self-
management can be implemented using parallel extensions to
current serial von-Neumann stored program control (SPC).”
The paper is organized in the following way. In Section 2 we
develop computational taxonomy, which allows us to extract basic
characteristics of computation and distinguish fundamental types
of computation. The suggested system of computational classes
allows us to reflect natural structures in the set of computational
processes. In Section 3 we study the structural context and
architecture of computation, illuminating the computational triad
and several other structures intrinsically related to computation. In
Section 4 we present the development of computational models in
natural computing (computing nature). Finally, we summarize and
present open questions in Section 5.
2. COMPUTATIONAL TAXONOMY
In his famous work  Turing presents his fundamental model,
which later was called the Turing machine model of computation.
Turing writes: “We may now construct a machine to do the work
of this computer.” Here a computer is a person performing
mechanical procedure executing an algorithm. Thus, algorithms
were the first models of computation. Algorithm was the practice
of algebra in the 18th century. In the 19th century, the term came to
mean any process of systematic calculation. In the 20th century,
Encyclopedia Britannica described algorithm as a systematic
mathematical procedure that produces – in a finite number of
steps – the answer to a question or the solution of a problem.
It is common that we talk about computation as if it would be a
uniquely defined concept. However some think of computation as
algorithm, others as symbol manipulation, while yet others may
have in mind a more general phenomenon of information
Currently there are so many kinds of concepts of computation and
its implementations that it is useful to make classification and
systematization so to better understand their mutual relationships.
In what follows we will present several main ttaxonomies of
computation: existential/substantial, organizational, temporal,
representational, data-oriented, operational, process-oriented and
2.1 Existential taxonomy
On the different levels of organisation of objects/agents (subjects)
(physical, structural, interpretant) we find different types of
computational processes. According to Burgin,  p. 93, reality
in the most abstract way can ben represented by the existential
triad (physical, structural, mental1) which corresponds to Plato’s
triad (material, ideas/forms, mental). This can also be related to
the Peirce's triad of (object, sign, interpretant).
The existential/substantial classification of computation based on
the existential triad  defines the following types:
1. Physical or embodied (object) computations
2. Abstract or structural (sign) computations
3. Mental or cognitive (interpretant) computations
The existential types from this taxonomy have definite subtypes.
The following are types of embodied computations, where each
next level emerges as a consequence of previous ones.
1.1. Physical computations
1.2. Chemical computations
1.3. Biological computations
1.4. Cognitive computations
1 Here mental denotes that which is of or relating to the mind. The
mind is a traditional and vague concept and in scientific terms it
corresponds to an emergent process that emerges from
cognition. Unlike mind, which is ascribed primarily to humans,
cognition can be attributed to both animals and machines and is
therefore a more useful concept in the context of computational
With respects to the structures (objects) that are processed by
computations, it is possible to discern the following types:
2.1 Subsymbolic computations - data/signal processing
2.2 Symbolic computations - data structures processing
2.3 Hybrid/mixed subsymbolic and symbolic computations.
There are connections between the above types. For instance 
differentiates between verbal (linguistic) (symbolic and
subsymbolic) and non-verbal (non-linguistic) (symbolic and
subsymbolic) mental (cognitive) processes. He suggests that the
principle of object formation may be an example of the transition
from a stream of massively parallel subsymbolic micro-functional
events to symbol-type, serial processing through subsymbolic
integration. Even  suggests similar connection between
symbolic and subsymbolic (connectionist) computations.
Mental (cognitive, interpretive) computations can be observed at
the following levels:
3.1 Individual (computational network of the brain)
3.2 Group (computational networks of individuals)
3.3 Social (computational networks of groups)
A taxonomy of computation in relation to cognition is provided by
Fresco , who notes that, depending on the specific kind of
“information” used, information-processing accounts of
computation may differ greatly. Four main types of information in
this context are given:
3.a Syntactic (Shannon) information
3.b Algorithmic information
3.c Semantic information
3.d Instructional information
2.2 Organizational taxonomy
Organization of computation can be characterized as:
1. Centralized computations where computation is controlled
by a single algorithm.
2. Distributed computations where there are separate algorithms
that control computation in some neighbourhood that is
represented by a node in the computational network.
3. Clustered computations where there are separate algorithms
that control computation in clusters of neighbourhoods.
Turing machines, partial recursive functions and limit Turing
machines are models of centralized computations. Neural
networks, Petri nets and Cellular automata are models of
distributed computations. Grid automata in which some nodes
represent networks with the centralized control and the World
Wide Web are systems that perform clustered computations .
Grid automaton is the most advanced abstract model of distributed
computing systems, which performs concurrent computations,
while being a physical distributed computing system.
2.3 Temporal taxonomy
With respect to temporal characteristics, computations can be:
1. Sequential computations, which are performed in linear time.
2. Parallel or branching computations, in which separate steps
(operations) are synchronized in time.
3. Concurrent computations, which do not demand
synchronization in time.
While parallel computation is completely synchronized, branching
computation is not completely synchronized because separate
branches acquire their own time and become synchronized only in
Classical models of computation, such as the classical Turing
machine or partial recursive functions, perform only sequential
computations. Models that appeared later, such as Turing
machines with several heads and tapes or cellular automata,
provide means for parallel computations. There are also various
models for concurrent computations, according to .
In this context, the most advanced device model is grid
automaton, while the most advanced operational model, which
also is a process model, is the EAP (event-action-process) model.
All these models form three classes:
3.1 Device models (Petri nets, Kahn process networks,
dataflow process networks, discrete event simulators, grid
automata, the Linda model and the Actors model).
3.2 Operational models (ACP (algebra of communicating
processes), VCR (extended view-centric reasoning),
EVCR (view-centric reasoning) and ESP (event-signal-
3.3 Process models (CSP (communicating sequential
processes) model and CCS (composite current source
2.4 Representational taxonomy
With respect to the data representation on which computations are
performed, there are following types of computation:
1. Discrete computations, which include interval computations.
2. Continuous computations, which include fuzzy continuous
3. Hybrid/mixed computations, which include discrete and
Digital computing devices and the majority of computational
models, such as finite automata, Turing machines, recursive
functions, inductive Turing machines, and cellular automata,
perform discrete computations.
Examples of continuous computations are given by abstract
models, such as general dynamical systems  and hybrid
systems , and special computing devices, such as the
differential analyzer .
Hybrid/Mixed computations include piecewise continuous
computations, combining both discrete computation and
continuous computation. Examples of mixed computations are
given by neural networks , finite dimensional machines and
general machines of .
It is possible to refine the representational taxonomy in the
following way, obtaining three additional classifications.
2.5 Data-based taxonomy
With respect to the data and the domain of computation, the
following possibility exist:
1. The domain of computation is discrete and data are finite.
For instance, data are words in some alphabet.
2. The domain of computation is discrete but data are infinite.
For instance, data are ω-words in some alphabet. This
includes interval computations because real numbers
traditionally are represented as ω-words.
3. The domain of computation is continuous.
2.6 Operational taxonomy
With respect to operations in computation, the following
taxonomy can be found:
1. Operations in computation are discrete and they transform
discrete data elements. For instance, addition or
multiplication of whole numbers.
2. Operations in computation are discrete but they transform
(operate with) continuous sets. For instance, addition or
multiplication of all real numbers or of real functions.
3. Operations in computation are continuous. For instance,
integration of real functions.
2.7 Process-oriented taxonomy
1. The process of computation is discrete, i.e. it consists of
separate steps in the discrete domain, and it transforms
discrete data elements. For instance, computation of a Turing
machine or a finite automaton.
2. The process of computation is discrete but it employs
continuous operations. An example is given by analogue
computations  as quoted in  p. 122.
3. The process of computation is continuous but it employs
discrete operations. For instance, computation of a limit
Turing machine  p. 140.
2.8 Computation levels taxonomy
In  three generality levels of computations are introduced:
1. At the top and most abstract/general level, computation is
perceived as any transformation of information and/or
2. At the middle level, where computation is distinguished as a
discretized process of transformation of information and/or
3. At the bottom, least general level, computation is defined as
a discretized process of symbolic transformation of
information and/or symbolic information representation.
There are also spatial levels or scales of computations. Even
though the highest levels subsume the lower ones, computation is
performed/interpreted at a given level, as follows:
1. Macro-level includes computations performed by mechanical
calculators as well as electromechanical devices.
2. Micro-level includes computations performed by integrated
3. Nano-level includes computations performed by fundamental
parts that are not bigger than a few nano meters.
4. Molecular level includes computations performed by
5. Quantum level includes computations performed by atoms
and subatomic particles.
At present there are no commercially available nano-computers,
molecular or quantum computers, but they are being developed.
3. STRUCTURAL FRAMEWORK AND
ARCHITECTURE OF COMPUTATION
The earliest idea of computation was application of an algorithm.
The Turing machine model of computation is equivalent to an
algorithm. Thus, the first and most commonly encountered
computational structure is the computational dyad 
(computation, algorithm). The computational dyad reflects the
existing duality between computations and algorithms. In 1971
Dijkstra in his A Short Introduction to the Art of Programming
 defined an algorithm as a static description of computation,
which is a dynamic state sequence induced in a machine by the
algorithm. Later a more systemic explication of the duality
between computations and algorithms was elaborated. Namely,
computation is a process of information transformation, which is
organized and controlled by an algorithm, while an algorithm is a
system of rules for a computation . In this context, an algorithm
is a compressed informational/structural representation of a
process. A computer program is an algorithm written in
(represented by) a programming language.
Thus, an algorithm is an abstract structure and it is possible to
realize the same algorithm as different programs (in different
programming languages). It is important to understand the
difference between algorithm and its representation or
embodiment. An algorithm is an abstract structure, which can be
represented in a multiplicity of ways: as a computer program, a
control schema, a graph, a system of cell states in the memory of a
computer, a mathematical system, such as an abstract finite
automaton, etc. Interestingly, many people think that neural
networks perform computations without algorithms. However,
this is not true as neural networks algorithms have representations
even though very different from traditional representations of
algorithms as systems of rules/instructions. The neural networks
algorithms are represented by neuron weights and connections
between neurons. This is similar to hardware
representation/realization of algorithms in computers (analog
The computational dyad is a simplification and incomplete
because there is always a system that uses algorithms to organize
and control computation. This observation suggests that the
computational dyad has to be extended to the triangular basic
computational triad (algorithm, computation, device/agent) that
reflects that specifies computation is performed by a device, such
as a computer, or by an agent, such as a human being or bacteria.
Functioning of the device or work of the agent that/who performs
computation is directed or controlled by an algorithm embodied in
a program, plan, scenario or hardware. Consequently, the process
of computation is also directed/controlled by an algorithm.
Note that the computing device can be either a physical device,
such as a computer, or an abstract device, such as a Turing
machine, or a programmed (virtual or simulated) device when a
program simulates some physical or abstract device. For instance,
neural networks and Turing machines are usually simulated by
programs on conventional computers. A Java virtual machine can
be run on different operating systems and is processor- and
operating system- independent. Besides, with respect to
architecture, it can be an embracing device, in which computation
is embodied and exists as a process, or an external device, which
organize and control computation as an external process.
The basic computational triad reflects the structure of the world
represented by the existential triad .
4. NATURAL COMPUTATION/
In previous sections we presented concept of computation and its
models and developed computational taxonomies in the context of
information processing. Adopting the approach of structural
realism we provided structural framework of computation based
on Burgin’s existential triad (physical, structural, mental/
cognitive) of the world . We have chosen triadic structures as
fundamental elements in our study of structures, based on the
classical triad of Plato – (matter, structure, mind), Peirce's triad of
(object, sign, interpretant) and Poppers structural triad (physical
world, knowledge, mentality).
In what follows, we will concentrate on the first two elements of
the existential triad – physical and structural in their cognitive
aspect, both of them addressed through the idea of computing
nature (natural computing),  .
According to the Handbook of Natural Computing  natural
computing is “the field of research that investigates both human-
designed computing inspired by nature and computing taking
place in nature.” In particular, it addresses: intrinsic computation
performed by natural materials and computational nature of
processes taking place in nature. Natural computing comprises,
among others, areas of cellular automata and neural computation,
evolutionary computation, molecular, quantum and organic
computing, biocomputing, nature-inspired algorithms.
Knowledge in natural computing is generated bi-directionally,
through the interaction between computer science and natural
sciences. As the natural sciences are promptly absorbing concepts,
tools and methodologies of information processing, computer
science on its side is broadening the notion of computation,
recognizing information processing found in nature as special type
of computation – intrinsic, natural computation.  
This development in understanding of computation led Denning to
argue that computer science today is a natural science  .
In his “Simulating Physics with Computers” Feynman argued that
the specific forms of computation are best carried out by the
physical substrate we are trying to describe . Even though
DNA computation can be emulated in silico, the efficiency of
computing with DNA substrate instead of its digital representation
is higher by many orders of magnitude . Similar view of
physical (material) computation is presented in  where it is
argued for clear benefits of intrinsic (material) computation, as in
the Brook’s famous observation that “the world is its own best
model” . Cooper addresses the question of the relationship
between abstract mathematical models and physical computation
that underlies every real-world computation, suggesting that we
are under the spell of the “mathematician bias” and arguing for
the return to embodied computation  –showing how
nature can help us to compute.
4.1 Intrinsic vs. designed computation
We are used to quick increase of computational power, memory
and usability of our computers, but the limit of miniaturization is
approaching as we are getting close to quantum dimensions of
hardware components. We are also facing the explosive increase
in the amounts of data, “big data” and the emergence of the
“internet of things” where computers are becoming an integral
part of practically all devices and indeed of our physical
environment. All of this poses huge demands on effective
computation techniques. Ever since the time of Turing, one of the
ideals was intelligent computing, which would besides mechanical
symbol manipulation include even intelligent problem solving.
That would help us manage complexity and vast amounts of data
that have to be processed, often in real time. In that direction
currently, there is a development of cognitive computing –
 aimed towards human-level abilities of machines that
process/organize/ and even understand information.
At the same time the development of computational models of
human brain has for a goal to reveal the exact mechanisms of
human brain function (https://www.humanbrainproject.eu) that
will help us understand not only how humans actually perform
information processing when they follow an algorithm, but also
how humans create algorithms or models in a recursive cascades
of self-reflective computational processes from physical substrate
to information/patterns to mental states through cognitive
computing. Those new developments can be seen as a part of the
research within the field of natural computing, where natural
system performing computation is embodied and embedded
The new understanding of computation as complex distributed
concurrent computational system of systems with inspiration in
natural information processing allows among others learning
about nondeterministic complex computational systems, such as
living organisms, with self-* properties (self-organization, self-
configuration, self-optimization, self-healing, self-protection, self-
explanation, and self-awareness). Natural computation has a
potential to provide a basis for a unified understanding of
phenomena of embodied cognition, intelligence and knowledge
Recently, a focus issue of the journal Chaos was dedicated to the
intrinsic and designed computation under the title “Information
Processing in Dynamical Systems—Beyond the Digital
Hegemony” addressing challenges of intrinsic computing in
dynamical systems as complementary to designed computing in
digital systems. 
This relates to the view of a brain as a dynamical system
processing information (computing) at different levels of
organization – from molecular/electro-chemical, cellular
processes (DNA, protein networks), through neural circuits,
cortical through columns (morphologically distinct regions of the
brain processing and exchanging information), and finally through
the level of whole-brain information integration that is considered
to provide the function of consciousness, . Cellular and
whole-brain levels of computation correspond to the cognition
level of cells and the brain. In a biological sense, cognition is the
property of an autopoietic system, with self-production, self-
organisation and closure and in structural coupling with the
At this point, there are several avenues of the future development
of computing both in the setting of physical devices and of
theoretical models. Natural computation promises new and
broader ways of understanding of computational processes that
can be found intrinsically in various physical systems. In
connection to that, the structure and its particular case -
architecture of computational processes becomes increasingly
important, especially in the case of cognitive computing.
4.2 Levels of physical organization in natural
computing as natural information processing
In the section on Taxonomy of computation, under the heading
Hierarchy of computation levels, we mentioned levels and scales
of physical organization at which computation is performed in
designed computation. In the case of cognitive computing, the
following levels of designed, nature-inspired computation are
distinguished: synapses, neurons, microcircuits (modeling brains
cortical columns), long range interconnections (modeling axons
function) and the whole-brain level integration of processes,
representing non-von-Neumann architecture:
“overarching cognitive computing architecture is an on-chip
network of light-weight cores, creating a single integrated system
of hardware and software. This architecture represents a critical
shift away from traditional von Neumann computing to a
potentially more power-efficient architecture that has no set
programming, integrates memory with processor, and mimics the
brain’s event-driven, distributed and parallel processing.”
Mikkillineni ,  describes cognitive architectures based on
his Distributed Intelligent Computing Element (DIME)
computing model, consisting of a “recursive managed distributed
computing network, which can be viewed as an interconnected
group of such specialized Oracle machines” that “provides the
architectural resiliency, which is often associated with cellular
organisms, through auto-failover; auto-scaling; live-migration;
and end-to-end transaction security assurance in a distributed
system.” Mikkillineni argues that “the self-identity and self-
management processes of a DIME network” add the elements of
cognition into Turing machine type computing. This approach
extends Burgin’s view of computing consisting of hardware,
software and infoware  with one more architectural layer
corresponding to cognitive function related to knowledge.
5. CONCLUSIONS AND OPEN
Present account suggests that the concept of computation as
information processing develops together with the constantly
increasing scientific knowledge and tools of analysis . We
present the structural framework of information processing and
computation starting with existential triadic relationships between
(physical, structural, mental/cognitive) and taxonomies of
numerous aspects of computation that follow this basic existential
layered architecture that is paralleled by physico-chemical,
chemo-biological and bio-cognitive levels of information
processing . As there is no information without (physical)
representation , the dynamics of information is implemented
on different levels of granularity by different physical processes,
including the level of computation performed by computing
machines (designed computation), as well as by living organisms
We highlight several topics of importance for the development of
new understanding of computation and its role: natural
computation, interactivity as fundamental for computational
modeling of concurrent information processing systems such as
living organisms and their networks, and the new developments in
modeling needed to support this generalized framework for
cognitive architectures . In such a way, we achieve better
understanding of computation as information processing on
There are still many open problems related to the nature of
information and computation, as well as to their relationships.
How is information dynamics represented in computational
systems, in machines, as well as in living organisms? Are
computers capable of processing only data or information and
knowledge as well, as recent work of Mikkillineni suggests? What
can we know of computational processes in machines and living
organisms and how these processes are related to the
computational architectures? What can we learn from natural
computational processes in cognitive systems that can be useful
for engineered information systems and knowledge management?
 A. M. Turing, “On computable numbers, with an
application to the Entscheidungs problem,” Proc. London
Math. Soc., vol. 42, no. 42, pp. 230–265, 1936.
 A. N. Kolmogorov, “On the Concept of Algorithm,”
Russ. Math. Surv., vol. 8, no. 4, pp. 175–176, 1953.
 B. J. Copeland, “What is computation?,” Synthese, vol.
108, no. 3, pp. 335–359, 1996.
 M. Burgin, Super-Recursive Algorithms. New York:
Springer-Verlag New York Inc., 2005.
 P. Denning, “What is computation?: Editor’s
Introduction,” Ubiquity, no. October, pp. 1–2, 2010.
 M. Burgin and G. Dodig-Crnkovic, “Information and
Computation – Omnipresent and Pervasive,” in
Information and Computation, New
York/London/Singapore: World Scientific Pub Co Inc,
2011, pp. vii –xxxii.
 H. Zenil, A Computable Universe. Understanding
Computation & Exploring Nature As Computation.
Singapore: World Scientific Publishing
Company/Imperial College Press, 2012.
 C. Hewitt, “What is computation? Actor Model versus
Turing’s Model,” in A Computable Universe,
Understanding Computation & Exploring Nature As
Computation, H. Zenil, Ed. World Scientific Publishing
Company/Imperial College Press, 2012.
 G. Dodig-Crnkovic, “Significance of Models of
Computation from Turing Model to Natural
Computation,” Minds Mach., vol. 21, no. 2, pp. 301–322,
 L. Fortnow, “The enduring legacy of the Turing
machine,” Comput. J., vol. 55, no. 7, pp. 830–831, 2012.
 D. Goldin, S. Smolka, and P. Wegner, Eds., Interactive
Computation: The New Paradigm. Berlin, Heidelberg:
 J. S. Conery, “Computation is symbol manipulation,”
Comput. J., vol. 55, no. 7, pp. 814–816, 2012.
 D. E. Angelaki, A. G. Shaikh, A. M. Green, and J. D.
Dickman, “Neurons compute internal models of the
physical laws of motion,” Nature, vol. 430, no. 6999, pp.
 S. Abramsky, “Information, Processes and Games,” in
Philosophy of Information, J. Benthem van and P.
Adriaans, Eds. Amsterdam, The Netherlands: North
Holland, 2008, pp. 483–549.
 R. Mikkilineni, Designing a New Class of Distributed
Systems, SpringerBr. Berlin Heidelberg: Springer, 2011.
 R. Mikkilineni, “Going beyond Computation and Its
Limits: Injecting Cognition into Computing,” Appl.
Math., vol. 3, pp. 1826–1835, 2012.
 M. Burgin, Structural Reality. New York: Nova Science
 R. Landauer, “The Physical Nature of Information,”
Phys. Lett. A, vol. 217, p. 188, 1996.
 W. Bucci, Psychoanalysis and Cognitive Science: A
Multiple Code Theory. New York: Guilford Press, 1997.
 A. Clark, Microcognition: Philosophy, Cognitive
Science, and Parallel Distributed Processing.
Cambridge, MA: MIT Press, 1989.
 N. Fresco, Physical Computation and Cognitive Science.
Berlin Heidelberg: Springer Berlin Heidelberg, 2014.
 M. Burgin and M. L. Smith, “Concurrent Composition
and Algebras of Events, Actions, and Processes,”
 O. Bournez, “Achilles and the Tortoise climbing up the
hyper-arithmetical hierarchy,” Theor. Comput. Sci., vol.
210, pp. 210–211, 1977.
 V. Gupta, R. Jagadeesan, and V. A. Saraswat,
“Computing with continuous change,” Sci. Comput.
Program., vol. 30, no. 1–2, pp. 3–49, 1998.
 C. Shannon, “Mathematical Theory of the Differential
Analyzer,” J. Math. Phys., vol. 20, pp. 337–354, 1941.
 C. Moore, “Recursion theory on the reals and
continuous-time computation,” Theor. Comput. Sci., vol.
162, no. 1, pp. 23–44, 1996.
 W. S. McCulloch and W. Pitts, “A logical calculus of the
ideas immanent in nervous activity. 1943.,” Bull. Math.
Biol., vol. 52, no. 1–2, pp. 99–115; discussion 73–97,
 L. Blum, F. Cucker, M. Shub, and S. Smale,
“Complexity and Real Computation: A Manifesto,” Int.
J. Bifurc. Chaos, vol. 6, no. 1, pp. 3–26, 1996.
 M. Burgin and E. Eberbach, “Evolutionary Computation
and Processes of Life,” Ubiquity, no. August, pp. 1–13,
 E. W. Dijkstra, “A Short Introduction to the Art of
Computer Programming,” 1971.
 G. Dodig-Crnkovic and R. Giovagnoli, Computing
Nature. Berlin Heidelberg: Springer, 2013.
 G. Rozenberg, T. Bäck, and J. N. Kok, Eds., Handbook
of Natural Computing. Berlin Heidelberg: Springer,
 G. Rozenberg and L. Kari, “The many facets of natural
computing,” Commun. ACM, vol. 51, pp. 72–83, 2008.
 S. Stepney, S. L. Braunstein, J. A. Clark, A. M. Tyrrell,
A. Adamatzky, R. E. Smith, T. R. Addis, C. G. Johnson,
J. Timmis, P. H. Welch, R. Milner, and D. Partridge,
“Journeys in Non-Classical Computation I: A Grand
Challenge for Computing Research,” Int. J. Parallel
Emerg. Distr. Syst., vol. 20, pp. 5–19, 2005.
 S. Stepney, S. L. Braunstein, J. A. Clark, A. M. Tyrrell,
A. Adamatzky, R. E. Smith, T. R. Addis, C. G. Johnson,
J. Timmis, P. H. Welch, R. Milner, and D. Partridge,
“Journeys in Non-Classical Computation II: Initial
Journeys and Waypoints,” Int. J. Parallel Emerg. Distr.
Syst., vol. 21, pp. 97–125, 2006.
 J. Crutchfield, W. Ditto, and S. Sinha, “Introduction to
Focus Issue: Intrinsic and Designed Computation:
Information Processing in Dynamical Systems—Beyond
the Digital Hegemony,” Chaos, vol. 20, no. 037101,
 P. Denning, “Computing is a natural science,” Commun.
ACM, vol. 50, no. 7, pp. 13–18, 2007.
 P. Denning and P. Rosenbloom, “The fourth great
domain of science,” ACM Commun., vol. 52, no. 9, pp.
 R. P. Feynman, “Simulating Physics with Computers,”
Int. J. Theor. Phys., vol. 21, no. 6/7, pp. 467–488, 1982.
 M. Nadin, “Anticipatory Computing. From a High-Level
Theory to Hybrid Computing Implementations,” Int. J.
Appl. Res. Inf. Technol. Comput., vol. 1, no. 1, pp. 1–27,
 S. Stepney, “The neglected pillar of material
computation,” Phys. D Nonlinear Phenom., vol. 237, no.
9, pp. 1157–1164, 2008.
 R. Brooks, “Elephants don’t play chess,” Rob. Auton.
Syst., vol. 6, pp. 3–15, 1990.
 S. B. Cooper, B. Löwe, and A. Sorbi, New
Computational Paradigms. Changing Conceptions of
What is Computable. Springer Mathematics of
Computing series, XIII. Springer, 2008.
 S. B. Cooper, “Turing’s Titanic Machine?,” Commun.
ACM, vol. 55, no. 3, pp. 74–83, 2012.
 S. B. Cooper, “How Can Nature Help Us Compute?,” in
SOFSEM 2006: Theory and Practice of Computer
Science - 32nd Conference on Current Trends in Theory
and Practice of Computer Science, 2006, pp. 1–13.
 S. B. Cooper, “The Mathematician’s Bias - and the
Return to Embodied Computation,” in A Computable
Universe: Understanding and Exploring Nature as
Computation, H. Zenil, Ed. World Scientific Pub Co Inc,
 Y. Wang, “On Abstract Intelligence: Toward a Unifying
Theory of Natural, Artificial, Machinable, and
Computational Intelligence,” Int. J. Softw. Sci. Comput.
Intell., vol. 1, no. 1, pp. 1–17, 2009.
 Y. Wang, “Toward a Formal Knowledge System Theory
and Its Cognitive Informatics Foundations,” in
Transactions on Computational Science V, vol. 5540, M.
L. Gavrilova, C. J. K. Tan, Y. Wang, and K. C. C. Chan,
Eds. Berlin, Heidelberg: Springer Berlin Heidelberg,
2009, pp. 1–19.
 Y. Wang, “On Contemporary Denotational Mathematics
for Computational Intelligence,” Trans. Comput. Sci.,
vol. 2, pp. 6–29, 2008.
 Y. Wang, “The Theoretical Framework of Cognitive
Informatics,” Int’l J. Cogn. Informatics Nat. Intell., vol.
1, no. 1, pp. 1–27, 2007.
 G. Dodig-Crnkovic and V. Mueller, “A Dialogue
Concerning Two World Systems: Info-Computational vs.
Mechanistic,” World Scientific Pub Co Inc, Singapore,
 G. Tononi, “The Integrated Information Theory of
Consciousness: An Updated Account,” Arch. Ital. Biol.,
vol. 150, no. 2/3, pp. 290–326, 2012.
 H. Maturana, “Autopoiesis, Structural Coupling and
Cognition: A history of these and other notions in the
biology of cognition,” Cybern. Hum. Knowing, vol. 9,
no. 3–4, pp. 5–34, 2002.
 G. Dodig-Crnkovic, “The Development of Models of
Computation with Advances in Technology and Natural
Sciences,” in Symposium of The British Society for the
Study of Artificial Intelligence, AISB 2013, 2013.
 G. Dodig-Crnkovic, “Why we need info-computational
constructivism,” Constr. Found., vol. 9, no. 2, pp. 246–