A Compositional Model of Consciousness based on Consciousness-Only
aa r X i v : . [ q - b i o . N C ] A ug Article
A Compositional Model of Consciousness based onConsciousness-Only
Camilo Miguel Signorelli ∗ , Quanlong Wang ∗ , Ilyas Khan ∗ Department of Computer Science, University of Oxford Cognitive Neuroimaging Unit, INSERM U992, NeuroSpin Cambridge Quantum Computing Ltd St Edmund’s College, University of Cambridge * Abstract:
Scientific studies of consciousness rely on objects whose existence is independent of anyconsciousness. This theoretical-assumption leads to the "hard problem" of consciousness. We avoidthis problem by assuming consciousness to be fundamental, and the main feature of consciousnessis characterized as being other-dependent. We set up a framework which naturally subsumes theother-dependent feature by defining a compact closed category where morphisms represent consciousprocesses. These morphisms are a composition of a set of generators, each being specified by theirrelations with other generators, and therefore other-dependent. The framework is general enough,i.e. parameters in the morphisms take values in arbitrary commutative semi-rings, from which anyfinitely dimensional system can be dealt with. Our proposal fits well into a compositional model ofconsciousness and is an important step forward that addresses both the hard problem of consciousnessand the combination problem of (proto)-panpsychism.
Keywords:
Consciousness; Conscious Agents; Compositionality; Combination problem; Mathematicsof Conciousness; Monoidal Categories; Panpsychism.
1. Introduction
Despite scientific advances in understanding the objective neural correlates of consciousness [1],science has so far failed in recovering subjective features from objective and measurable correlates ofconsciousness. One example is the unity of consciousness. Current models postpone the explanationof that unity, assuming there will be further developments [2]. In the meantime, they reduce consciousexperience to neural events.In this article, we present an alternative approach: consciousness as a fundamental process ofnature. This strategy addresses reductionism and the hard problem of consciousness. Our approachtakes inspiration from the Yogacara school [3,4], and is also in line with the hypothesis of consciousagents [5] and phenomenology [6,7]. In our framework, a key feature of consciousness is characterisedas "other-dependent nature" , i.e. the nature of existence arising from causes and conditions. Withoutfalling into idealism or dualism, we propose that consciousness should be treated as a primitive process.To model the other-dependent nature, we propose a compositional model for consciousness. Thismodel is based on symmetric monoidal categories (Section 2), also called Process Theory [8,9]. Processtheory is an abstract framework which describes how processes are composed, and thus ontologicallyneutral. It has been widely used in various research fields such as the foundations of physicaltheories [10], quantum theory [11,12], causal models [13,14], relativity [15] and interestingly also naturallanguage [16] and cognition [17,18]. At the core of process theory lies the principle of compositionality.Compositionality describes any unity as a composition, possibly non-trivially, of some basic processes of 20 [8,9]. In this paper, we use a fine-grained version of process theory called ZX-calculus to model Alayaconsciousness (Section 3). In our model, we use generators in the form of basic diagrams. A diagramrepresents processes defined by interdependent relations (Section 4, 4.1, 4.1.1 and 4.1.2), exhibiting theother-dependence feature of consciousness. The framework also comes with a standard interpretationfor each diagram (Section 4.1.3 and 4.1.4), making our theory sound, i.e. without internal contradictions.This makes process theory and our compositional framework suitable for investigating the irreduciblestructural properties of conscious experience [19].This framework may become an important step forward, by mathematizing phenomenology totarget major questions of conscious experience [20,21]. For instance, the unity of consciousness naturallyarises as result of composition, and the combination problem of fundamental experiences is described asan application of our framework (Section 5). This new perspective of scientific models of consciousnessinvokes pure mathematical entities, avoiding ontological claims, without the need for any physicalrealization (Section 6).
2. Category Theory and Process Theory
In this section, we briefly introduce the basic notions of Category theory [22], process theory [9] andgraphical calculus [23].
Category
A category C consists of: • a class of objects ob ( C ) ; • for each pair of objects A , B , a set C ( A , B ) of morphisms from A to B ; • for each triple of objects A , B , C , a composition map C ( B , C ) × C ( A , B ) −→ C ( A , C )( g , f ) g ◦ f ; • for each object A , an identity morphism 1 A ∈ C ( A , A ) ,satisfying the following axioms: • associativity: for any f ∈ C ( A , B ) , g ∈ C ( B , C ) , h ∈ C ( C , D ) , there holds ( h ◦ g ) ◦ f = h ◦ ( g ◦ f ) ; • identity law: for any f ∈ C ( A , B ) , 1 B ◦ f = f = f ◦ A .A morphism f ∈ C ( A , B ) is an isomorphism if there exists a morphism g ∈ C ( B , A ) such that g ◦ f = A and f ◦ g = B . A product category A × B can be defined componentwise by two categories A and B . Functor
Given categories C and D , a functor F : C −→ D consists of: • a mapping ob ( C ) −→ ob ( D ) A F ( A ) ; • for each pair of objects A , B of C , a map C ( A , B ) −→ D ( F ( A ) , F ( B )) f F ( f ) , of 20 satisfying the following axioms: • preserving composition: for any morphisms f ∈ C ( A , B ) , g ∈ C ( B , C ) , there holds F ( g ◦ f ) = F ( g ) ◦ F ( f ) ; • preserving identity: for any object A of C , F ( A ) = F ( A ) .A functor F : C −→ D is faithful (full) if for each pair of objects A , B of C , the map C ( A , B ) −→ D ( F ( A ) , F ( B )) f F ( f ) is injective (surjective).A bifunctor (also called binary functor) is just a functor whose domain is the product of twocategories. Natural transformation
Let F , G : C −→ D be two functors. A natural transformation τ : F → G is a family ( τ A : F ( A ) −→ G ( A )) A ∈ C of morphisms in D such that the following square commutes: F ( A ) τ A F ( f ) G ( A ) F ( B ) τ B G ( f ) G ( B ) for all morphisms f ∈ C ( A , B ) . A natural isomorphism is a natural transformation where each of the τ A is an isomorphism. Strict monoidal category
A strict monoidal category consists of: • a category C ; • a unit object I ∈ ob ( C ) ; • a bifunctor − ⊗ − : C × C −→ C ,satisfying • associativity: for each triple of objects A , B , C of C , A ⊗ ( B ⊗ C ) = ( A ⊗ B ) ⊗ C ; for each triple ofmorphisms f , g , h of C , f ⊗ ( g ⊗ h ) = ( f ⊗ g ) ⊗ h ; • unit law: for each object A of C , A ⊗ I = A = I ⊗ A ; for each morphism f of C , f ⊗ I = f = I ⊗ f . Strict symmetric monoidal category
A strict monoidal category C is symmetric if it is equipped with a natural isomorphism σ A , B : A ⊗ B → B ⊗ A of 20 for all objects A , B , C of C satisfying: σ B , A ◦ σ A , B = A ⊗ B , σ A , I = A , ( B ⊗ σ A , C ) ◦ ( σ A , B ⊗ C ) = σ A , B ⊗ C . Strict monoidal functor
Given two strict monoidal categories C and D , a strict monoidal functor F : C −→ D is a functor F : C −→ D such that F ( A ) ⊗ F ( B ) = F ( A ⊗ B ) , F ( f ) ⊗ F ( g ) = F ( f ⊗ g ) , F ( I C ) = I D , for any objects A , B of C , and any morphisms f ∈ C ( A , A ) , g ∈ C ( B , B ) .A strict symmetric monoidal functor F is a strict monoidal functor that preserves symmetricalstructures, i.e., F ( σ A , B ) = σ F ( A ) , F ( B ) . The definition of a general (non-strict) symmetric monoidal functorcan be found in [22]. Strict compact closed category
A strict compact closed category is a strict symmetric monoidal category C such that for each object A of C , there exists a object A ∗ and two morphisms ǫ A : A ⊗ A ∗ → I , η A : I → A ∗ ⊗ A satisfying: ( ǫ A ⊗ A ) ◦ ( A ⊗ η A ) = A , ( ∗ A ⊗ ǫ A ) ◦ ( η A ⊗ ∗ A ) = ∗ A .A strict compact closed category is called self-dual if A = A ∗ for each object A [12]. Process theory is an abstract description of how things have happened, be they mental or physicaland regardless of their nature. In common with all theories, process theory has its own assumptions,albeit with the advantage that it’s major feature is that it contains minimal assumptions.We first assume an event to have occurred. i.e., a change from something typed as A to somethingtyped as B . This is called a process and denoted as a box: fBA Second, we assume that it is impossible that all the things being considered, happenedsimultaneously and thereafter ceased. So there must be processes, say g and f , that happen sequentially: gfABCf happens after g can be seen as a single process from type C to type B , which is denoted by f ◦ g : C → B . This means processes admit sequential composition . As such, three things happening insequence is seen as one process without any ambiguity, i.e., the sequential composition of processes is of 20 associative: ( f ◦ g ) ◦ h = f ◦ ( g ◦ h ) . We also assume that for each type A , there exists a process called theidentity 1 A , which does nothing at all to A . This is depicted as a straight line: A As a consequence, given a process f : A → B , we have 1 B ◦ f = f = f ◦ A .Third, we assume that there should be different "things" happening simultaneously. Two processes f and g that happen simultaneously are described as: f gCDBA If we view two types, say A and C , as a single type which we denote as A ⊗ C , then the simultaneousprocesses f and g can be seen as a single process from type A ⊗ C to type B ⊗ D which we denote as f ⊗ g : A ⊗ C → B ⊗ D . So we have a parallel composition of processes. The above depiction of f ⊗ g is asymmetric: f on the left while g on the right. This is due to the limitation of a planar drawing. Twoprocesses that occur simultaneously should be placed in a symmetric way, which means that if we swaptheir positions, they should be essentially the same where all the types should match. This can be realisedby adding a swap process BA AB such that = f fg gA A AB B BC CCDD D With these basic assumptions, processes can be organised into what is called a process theory in theframework of a strict symmetric monoidal category (SMC). A much more detailed description of processtheory can be found in [12].Furthermore, in this paper, we also consider the origin of space and time as part of our framework.Intuitively, time emerges from sequentially happened processes, and space is a form which displayssimultaneously happened processes. Similar to the theory of relativity where space and time are aunified entity, here we assume that space and time are related to each other in the sense that sequentialcomposition and parallel composition are convertible. This is realized by adding the compact structureto the process theory, then we have: f = f ggABC AB B ∗ C ∗ C Mathematically speaking, we now have a compact closed category. of 20
Since process theory focuses on the processes instead of the objects, they provide aphilosophical advantage: process theories emphasise transformations, avoiding any ontological claimor "substance-like" description.
In general process theory, most of the boxes (processes) are unspecified in the sense that whatis inside a box is unknown, whereas we need to know more details about their interactions in someapplications. In other words, we need a fine-grained version of process theory. The typical way toderive such a version is to generate all the processes by a set of basic processes called generators ,while specifying those generators in terms of equations of processes composed of generators. Below,we illustrate this idea by a typical example called ZX-calculus.ZX-calculus is a process theory invented by Bob Coecke and Ross Duncan as a graphical languagefor a pair of complementary quantum processes (represented by two diagrams called green spider andred spider respectively) [23]. All the processes in ZX-calculus are diagrams composed sequentially or inparallel, either of green spiders with phase parameters, red spiders with phase parameters, straight lines,swaps, caps or cups. These generators satisfy a set of diagrammatic equations called rewriting rules :one can rewrite each diagram into an equivalent one by replacing a part of the diagram which is on oneside of an equation with the diagram on the other side of the equation. All the ZX diagrams modulo and the rewriting rules form a self-dual compact closed category [23]. To guarantee that there are noconflicts in this rewriting system, ZX-calculus needs a property called soundness : there exists a standardinterpretation from the category of ZX diagrams to the category of matrices, i.e., a symmetric monoidalfunctor between them [23].
3. Why use a compositional approach based on consciousness-only
In this section, we motivate and explain the concepts of consciousness as fundamental and also thestructure for consciousness given by the Yogacara School.
In any attempt to model consciousness, we expect to fulfill at least three theoretical requirements.First, one would like a theory with a basic and minimum set of assumptions. Process theory is such aframework. Symmetric monoidal categories start from a minimum and specific intuitive form to dealwith compositions, sequential and parallel, between different mathematical categories and structures(section 2.2). As introduced in section 2, symmetric monoidal categories define process theories, wherethe morphisms of the category are treated as processes or transformations.Second, one would expect those minimum assumptions to be explicit. In other words, we need tomodel the nature of consciousness from explicit, primitive and axiomatic principles. Process theory inparticular, and category theory in general, provides us with an exceptionally well suited mathematicalframework for such axiomatic purposes. Since assumptions in process theory are minimal, any extrastructure needs to be explicitly added and have explicit mathematical meaning.Third, one would like to recover important properties of consciousness from those basic and explicitaxioms. Specifically the unity of consciousness. According to the phenomenology of consciousness,one of the most salient features of conscious experience is its unity [24,25]. Importantly, in processtheory, unity is formed by sequential and parallel compositions. Under those operations, the concept Modulo means using an equivalent relation. of 20 of compositionality defines the whole as compositions of the parts. These parts however, are nottrivial decompositions, they contain in themselves the very properties that define the whole (in ourcase, processes compound other processes). Parts and the whole are therefore defined together.Compositionality is thus a middle ground between reductionism and holism. Due to this foundationalaspect, compositionality is a convenient way to target the unity of consciousness (section 5).
At the heart of a general theory of consciousness there always lies the mind-body problem: howphysical processes (physical properties, neural events and the body) are related to a conscious subjectiveexperience (mental properties, qualia)? [24,26,27]. Answers to this problem diverge into two main paths:dualism and physicalism (sometimes also called materialism). Dualism holds the view that the mentaland the physical are both real and neither can be reduced to the other. The main difficulty of dualism is theproblem of interaction: if the mental and the physical are radically different kinds of things, i.e differentfrom each other, how could they interact with each other while still keeping a unified picture of a creaturepossessed of both a mind and a body [28]? On the other hand, physicalism assumes that everything isphysical, and that mental states are just physical states. Physicalism has two main problems. The firstone is the hard problem of consciousness: why and how does experience arise from a physical basis?[24,26,27]. The second problem of physicalism comes from its basic assumption of objectivity: there existphysical objects whose existence is independent of any consciousness. However, there is an epistemicissue here. Essentially, "our knowledge is limited to the realm of our own subjective impressions, allowingus no knowledge of objective reality in and of itself" [7,29]. This means that consciousness-independentobjectivity is always an assumption that can never be verified.To deal with those issues, we remove the assumption of objectivity, we take consciousness asfundamental and work on the basis that all physical phenomena arise from consciousness. In otherwords, we assume that all primary objects are indeed conscious-dependent. These fundamental andinterdependent interactions form a process theory for consciousness (section 2 and 4).This specific conception of consciousness as fundamental differs from other Western philosophiesthat also consider consciousness as fundamental. Some examples are (proto)-panpsychism and idealism.The former convey the combination problem [30] and the later the dual version of the hard problem ofconsciousness: how do physical phenomenon arise from a subjective basis? One concrete example is therecent conscious agent model [5,31], where the world consists of conscious agents and their experiences.The conscious agent model focuses on the computational properties of consciousness [5] and approachthe mind-body relationship considering the fundamental agent independent, i.e. existing by itself.In view of these, to realise the principle of consciousness as fundamental, we are inspired by theEastern philosophy known as Yogacara. In our model, consciousness as fundamental becomes an axiomthat is equally as valid, but which is more promising at filling "missing gaps", than a model where matterand objectivity are seen as fundamental.
Yogacara (Sanskrit for
Yoga Practice ), also called Vijnanavada (
Doctrine of Consciousness ) orVijnaptimatra (
Consciousness Only ), is one of the two main branches of Mahayana (
Great Vehicle )Buddhism (the other being Madhyamaka,
Middle way ). The key feature of the Yogacara philosophy isconsciousness-only which works on the basis that there is nothing outside of consciousness.To understand the idea of consciousness-only, we should understand another concept fromYogacara, namely Trisvabh¯ava or the three natures. Trisvabh¯ava is the premise that all the possibleforms of existence are divided into three types: i) Parikalpita-svabh¯ava, the fully conceptualized of 20 nature, ii) Paratantra-svabh¯ava, the other-dependent nature, and iii) Parinis.panna-svabh¯ava, the perfect-accomplished-real nature. As explained by [4]: "The first nature is the nature of existence producedfrom attachment to imaginatively constructed discrimination. The second nature is the nature ofexistence arising from causes and conditions. The third nature is the nature of existence being perfectlyaccomplished (real)", which is "the ultimate reality, something that never changes". It is actually "theperfect, complete, real nature of all dharmas" [32].These three natures are inseparable from the mind (translated from the Sanskrit word
Citta ) and itsattributes (
Citta-Caittas ). This is clearly stated in Cheng Weishi Lun [33], a representative work of theYogacara School in China and translated to English by [32] and [34], where consciousness is actually ofthe second nature of existence: the other-dependent nature. In the following, this "other dependence" istaken as the main feature of consciousness processes, unlike the common view of fundamental physicalparticles, whose existences are identified by their own properties like mass, spin and charge, thusindependent of others.The concept of "mind" in the Yogacara School has a rich structure. It is divided into eight types ofconsciousnesses: the first seven consciousnesses—the five sense-consciousnesses (eye or visual, ear orauditory, nose or olfactory, tongue or gustatory, body or tactile consciousnesses), mental consciousness (the sixth consciousness), manas consciousness (the seventh or thought-centre consciousness), andthe eighth consciousness— alaya consciousness (storehouse consciousness). Among them the Alayaconsciousness is of particular note in that the “act of perception of the eighth consciousness is extremelysubtle, and therefore difficult to perceive. Indeed the Alaya is described as incomprehensible becauseit’s internal object (the Bijas (seeds) and the sense-organs held by it) is extremely subtle while its externalobject (the receptacle-world) is immeasurable in its magnitude” [34]. These eight consciousnesses are notindependent of each other. "...the Alaya consciousness and the first seven consciousnesses generate eachin a steady process and are reciprocally cause and effect. [32]". As a feature of the Yogocara School, "inthe Three Worlds (Dhatus in Sanskrit) there is nothing but mind" [34], which means consciousness-onlyin the world.Each type of consciousness is capable of being transformed (parinama in Sanskrit) into twodivisions: the perceived division (nimittabhaga in Sanskrit) and the perceiving division (darsanabhagain Sanskrit), and the function of the latter is to perceive the former. The phenomenon of the physicalworld and the body which we feel everyday comes from the perceived division of Alaya consciousness:"it transforms internally into seeds and the body provided with organs, and externally into the worldreceptacle. These things that are its transformations become its own object of perception (dlanzbana)"[32]. The receptacle-world and the Body as part of the perceived division of Alaya consciousness shouldnot be thought of as the physical world and the physical body that we feel in our normal lives, but as beingrelated in that the appearance of the latter is based on the existence of the former. As a consequence, theobjectivity of the world comes from the same structure shared by different sentient beings in the perceiveddivision of their Alaya consciousnesses. Furthermore, we note that the sixth consciousness (mentalconsciousness) is close to modern notions of awareness. Perceptual objects in mental consciousness areknown as the inner or the sixth guna, which are composed of impressions of colours, shapes, sounds,smells, tastes, and touches.
With consciousness as fundamental, we now compare western idealism with eastern Yogacaraphilosophy. Yogacara philosophy is fundmentally different from idealism. A notable first difference isthe richer structure of consciousness. Yogacara identifies eight different types of consciousness and theirrelationships. Idealism and other types of monism do not have this complex structure. Secondly, the of 20 interdependence between the three natures of existence, and specifically between types of consciousness.The eight consciousnesses are reciprocally a cause and effect of the others [32], while idealism in generaldoes not present these reciprocal cause and effect interactions. The third difference corresponds tothe concept of Alaya consciousness itself. The subtle nature of Alaya consciousness in addition to theperceived and the perceiving division is absent in philosophies such as idealism. The world arises fromthe perceived and the mind from perceiving transformations of Alaya consciousness. In other words,they share similar structures, but they are not reduced to each other, as would happen in idealism ormaterialism. A final main difference is the third nature (perfect-accomplished-real nature) which is thereal nature of each consciousness process in Yogacara philosophy. The feature of this real nature is thatit is unchangeable and unconditional, never affecting nor being affected by anything. On the other hand,it makes the existence of any changeable thing possible - things can not exist if they have no real nature,and can not change if they have self-identities. This idea does not exist in western idealism.
4. Compositional Model for Consciousness-Only
After the discussion in section 3, we now provide a compositional model of consciousness basedon the Yogacara philosophy of consciousness-only. The full enterprise means to use process theory andmodel the eight types of consciousness and their relations. In this paper, we first focus on the modelof two important types, Alaya consciousness and mental consciousness. We leave the modelling of themanas consciousness and the five sense-consciousnesses for future work.
The first step is to show how to model Alaya consciousness. In order for this, we need tomake explicit the key features of Alaya consciousness. The first feature of Alaya consciousness isother-dependence, which means each process of Alaya consciousness is dependent on other processes.The general process theory can not display the other-dependence feature because most of its processes arenot specified (see section 2.3). So we need a fine-grained version of process theory which has generatorsspecified by explicit rewriting rules. The second feature of Alaya consciousness is its deepness andsubtleness. To realise this feature we request that each process in the chosen process theory has no explicitmeaning in consciousness and any parameter appeared in the theory is not a concrete number. The thirdfeature of Alaya consciousness is that the structure of the physical world is included in its perceiveddivision. Since quantum theory is a fundamental formalism for the physical world, we would expect thefine-grained process theory to be quantum-related and has space and time arising from.Based on the requirements for a fine-grained process theory that are noted above, we introducea formalism called qufinite ZX ∆ -calculus, which is a generalisation of the normal ZX-calculus [23]regarding the following aspects: 1) a labelled triangle symbol is introduced as a new generator, that’swhy there is a ∆ in the name of the generalised ZX-calculus, 2) all the qudit ZX-calculus (ZX-calculus forqudits– quantum versions of d-ary digits) are unified in a single framework, 3) the parameters (phases)of normal ZX-calculus are generalised from complex numbers to elements of an arbitrary commutativesemiring.We claim that the qufinite ZX ∆ -calculus meet all the requirements of a desired fine-grained processtheory for Alaya consciousness. First, all the processes in the qufinite ZX ∆ -calculus are either generatorsthemselves which are specified by relations with other diagrams or are composed of generators, soother-dependence is realised. Second, all the processes in the qufinite ZX ∆ -calculus are just diagramswithout explicit meaning, and parameters are just general elements of an arbitrary commutative semiring.Therefore deepness and subtleness are embodied. Finally, the qufinite ZX ∆ -calculus is naturallyquantum-related and has the compact structure which relates space and time. We give the details below of the qufinite ZX ∆ -calculus: generators, rewriting rules and its standardinterpretation. Throughout this section, N = {
0, 1, 2, · · · } is the set of natural numbers, 2 ≤ d ∈ N , ⊕ isthe modulo d addition, S is an arbitrary commutative semiring [35]. All the diagrams are read from topto bottom as in previous sections.4.1.1. Generators of Qufinite ZX ∆ -calculusWe give the generators of the qufinite ZX ∆ -calculus in Table 1. mn −→ α d ...... mn ...... dd d j d s ts s ssst ts s st t Table 1.
Generators of qufinite ZX ∆ -calculus, where m , n ∈ N ; −→ α d = ( a , · · · , a d − ) ; a i ∈ S ; i ∈ { · · · , d − } ; j ∈ {
0, 1, · · · , d − } ; s , t ∈ N \{ } . Remark 1.
Each input or output of a generator is labeled by a positive integer. For simplicity, the first fourgenerators have each of their inputs and outputs labelled by d, and we just give one label to a wire.
For simplicity, we use the following conventions: −→ d ... d ...: = ...... d : = d j d j d : = d j d j d k : = −−→ e d − k and ε : ··· · · ·· ·· ··· ··· · : = where −→ d = d − z }| { ( · · · , 1 ) ; j ∈ {
0, 1, · · · , d − } ; k ∈ { · · · , d − } ; −−→ e d − k = d − z }| { ( · · · , 1 | {z } d − k , · · · , 0 ) ; ε represents an empty diagram. ZX ∆ -calculusWe provide rewriting rules for qufinite ZX ∆ -calculus in Figure 1 and Figure 2. These rules specifythe generators as listed in Table 1. Concretely, it means that two or more generators define each other. Forexample, the green dot d is specified by the rule d = ddd in the way that it is the only greenspider which has no input and one output and can be copied by the red spider d . Moreover, thered spider d is also specified by the effects in the green dot d .... = ... −→ β d ... ... −−→ α d β d ... −−→ α d β d ... ...... = −→ α d ... = = d d d = dd d = d i d j d i ⊕ j = ...... ... d ... = ... ... d ...... d d ... d ...... ... d = = dddd d ddd = ddd = ddd m m ...... d = dd j d j d j · = · −→ α d ·· ··· ··· d ·· ·· ·· = d = d dd j d j d j Figure 1.
Qufinite ZX ∆ -calculus rules I, where −→ α d = ( a , · · · , a d − ) ; −→ β d = ( b , · · · , b d − ) ; −−→ α d β d =( a b , · · · , a d − b d − ) ; a k , b k ∈ S ; k ∈ { · · · , d − } ; j ∈ {
0, 1, · · · , d − } ; m ∈ N . d = dd d d j = d j −→ α d + −→ d −→ α d = d = −→ d d −→ α d d ddd d = d −→ α d −→ α d dd = dd dd d ddddd d ddd d d = dd dd ddd d dd = d d −→ α d −→ β d −→ α d + −→ β d st ts = st st st ts ts = s tst ts stu u = us stutut sts t stst = st Figure 2.
Qufinite ZX ∆ -calculus rules II, where −→ d = d − z }| { ( · · · , 1 ) ; −→ d = d − z }| { ( · · · , 0 ) ; −→ α d =( a , · · · , a d − ) ; −→ β d = ( b , · · · , b d − ) ; a k , b k ∈ S ; k ∈ { · · · , d − } ; j ∈ { · · · , d − } ; s , t , u ∈ N \{ } . In order to form a compact closed category of diagrams, we also need the following structural rules: = ss s s = s s s s = = sss s sss (1) = ...... ...... ... ...=... ... = s s s k t t t l u u s s s k s s s k s s s k t t t l t t t l t t t l u uu uuu ss tt tsts (2)where s t l s t t ...... s k is an arbitrary diagram in the qufinite ZX ∆ -calculus.The first two diagrams in equation (1) mean the cap η s and the cup ǫ s are symmetric, while thelast diagram means the connected cap and cup can be yanked. The first two diagrams of equation (2)mean any diagram could move across a line freely, representing the naturality of the swap morphism.The last diagram of equation (2) means the swap morphism is self-inverse. Note that now we havea self-dual compact structure rather than a general compact structure, which makes representation ofdiagrams much easier.From the rewriting rules noted above, we form a strict self-dual compact closed category Z of ZXdiagrams. The objects of Z are all the positive integers, and the monoidal product on these objects aremultiplication of integer numbers. Denote the set of generators listed in Table 1 as G . Let Z [ G ] be a freemonoidal category generated by G in the following way - i) any two diagrams D and D are placedside-by-side with D on the left of D to form the monoidal product on morphisms D ⊗ D , or ii) theoutputs of D connect with the inputs of D when their types all match to each other to form the sequentialcomposition of morphisms D ◦ D . The empty diagram is a unit of parallel composition and the diagramof a straight line is a unit of the sequential composition. Denote the set of rules listed in Figure 1, Figure2, equations (1) and equations (2) by R . One can check that rewriting one diagram to another diagramaccording to the rules of R is an equivalence relation on diagrams in Z [ G ] . We also call this equivalenceas R , then the quotient category Z = Z [ G ] / R is a strict self-dual compact closed category. The qufinite ZX ∆ -calculus is seen as a graphical calculus based on the category Z .4.1.3. Standard interpretation of qufinite ZX ∆ -calculusTo ensure that qufinite ZX ∆ -calculus is sound, we need to test its rules in a preexisting reliablesystem which we now describe. These interpretations, however, does not represent the explicit meaningin terms of our consciousness processes. They are given here to test soundness.Let Mat S be the category whose objects are non-zero natural numbers and whose morphisms M : m → n are n × m matrices taking values in a given commutative semiring S . The compositionis matrix multiplication, the monoidal product on objects and morphisms are multiplication of naturalnumbers and the Kronecker product of matrices respectively. Then Mat S is a strict self-dual compact closed category. We give a standard interpretation, namely J · K , for the qufinite ZX ∆ -calculus diagrams in Mat S : mn −→ α d ...... }(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)~ = d − ∑ i = a j | i i ⊗ m h i | ⊗ n ; a = a i ∈ S ; mn ...... d }(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)~ = ∑ ≤ i , ··· , i m , j , ··· , j n ≤ d − i + ··· + i m ≡ j + ··· + j n ( mod d ) | i , · · · , i m i h j , · · · , j n | ; uwv d j }(cid:127)~ = d − ∑ i = | i i h i ⊕ j | ; t d | = | i h | + d − ∑ i = ( | i + | i i ) h i | ; uv d }~ = d − ∑ i = | i i h i | ; uv st ts }~ = s − ∑ k = t − ∑ l = | kt + l i h kl | ; uv s st t }~ = st − ∑ k = (cid:12)(cid:12)(cid:12)(cid:12) [ kt ] (cid:29) (cid:12)(cid:12)(cid:12)(cid:12) k − t [ kt ] (cid:29) h k | ; t ··· · · ·· ·· ··· ··· · | = uwwv s t }(cid:127)(cid:127)~ = s − ∑ k = t − ∑ l = | kl i h lk | ; s s s { = s − ∑ i = | i i | i i ; s ss { = s − ∑ i = h i | h i | ; J D ⊗ D K = J D K ⊗ J D K ; J D ◦ D K = J D K ◦ J D K ;where s , t ∈ N \{ } ; h i | = d z }| { ( · · · , 1 | {z } i + , · · · , 0 ) ; | i i = ( d z }| { ( · · · , 1 | {z } i + , · · · , 0 )) T ; i ∈ {
0, 1, · · · , d − } ; and [ r ] is theinteger part of a real number r .One can verify that the qufinite ZX ∆ -calculus is sound in the sense that for any two diagrams D , D ∈ Z , D = D must imply that J D K = J D K . This standard interpretation J · K is actually a strictsymmetric monoidal functor from Z to Mat S .According the standard interpretation, if S is the field of complex numbers, then the green spidercorresponds to the computational basis | i i} d − i = , with d − d j diagram represents the j -th unitary which is also a permutation matrix, with j ranging from 0to d . The triangle diagram labelled with d acts as a successor of phase parameters (adding 1’s to them).The two trapezium diagrams represent unitaries between the Hilbert space of H s ⊗ H t and the Hilbertspace H st , these two diagrams are invertible to each other. Remark 2.
Similar to the situation that ZX and ZW calculus over qubits are isomorphic to the category of matriceswith size powers of [36], we would like to prove in future work that the qufinite ZX ∆ -calculus over semiring S is isomorphic to the category of Mat S (maybe more rules to be added). If this can be done, then the structure of thecategory of diagrams of the qufinite ZX ∆ -calculus is independent of the choice of generators and rules. ZX ∆ -calculus. A general diagramrepresents some sort of conscious process and a diagram with outputs but without inputs will representa state of consciousness. Sequential composition of two diagrams represents two successive consciousprocesses happening one after the another, while parallel composition of two diagrams represents twoconscious processes happening simultaneously.Furthermore, we model the perceived and perceiving division of Alaya consciousness. On the onehand, as we have introduced in section 3.3, the content of the perceived version of Alaya consciousness isthe phenomenon of the physical world and the body which is supposed to have the same mathematicalstructure for all sentient beings in this world. Since each physical object is supposed to be composed ofquantum systems, the perceived version of Alaya consciousness is modelled here by the category FdHilb :the category whose objects are all finite dimensional complex Hilbert spaces and whose morphismsare linear maps between the Hilbert spaces with ordinary composition of linear maps as compositionsof morphisms. The usual Kronecker tensor product is the monoidal tensor, and the field of complexnumbers C (which is a one-dimensional Hilbert space over itself) is the tensor unit. FdHilb is the categoryof quantum processes which composes the physical world.On the other hand, the function of the perceiving division of Alaya consciousness is to perceive theperceived division, which means a perceiving action of the Alaya consciousness. Thus, the perceivingdivision of Alaya consciousness is modelled by a functor from Z to FdHilb . This functor is set up as amodification of the standard interpretation functor J · K , i.e.: just choose a semiring homomorphism f from S to C and let {| i i} d − i = a standard basis of a Hilbert space with dimension d , then replace a i with f ( a i ) in the codomain of the interpretation J · K . One can check that a monoidal functor is obtained in this way,where a semiring homomorphism from S to C is selected. After describing the category for Alaya consciousness, we now consider a model for mentalconsciousness. Consider N -semimodules [35] freely generated by a finite set of perceptions (impressions),either of colours, shapes, sounds, smells, tastes or touch feelings. We call these N -semimodulessingle-type perception semimodules. Let X be the category whose objects are finite tensor products ofsingle-type perception semimodules, and whose morphisms are semimodule homomorphisms betweenthem [35]. Then X forms a symmetric monoidal category [37]. An object of X is called here an experiencespace. We give an example of experience space as follows. An experience space about two shapes ofa square and a triangle is a free N -semimodule with a basis {square, equilateral triangle }. A generalelement in this semimodule is of form m(square)+n(equilateral triangle), which means an impressionwhere there are m squares and n equilateral triangles. Therefore mental consciousness is modelled bythe category X whose objects are explained as experience spaces and whose morphisms are explained asmental consciousness processes which transform from one experience space to another. The reason whywe use the semi-ring N is because we take our experiences as being basically finite.As we described in section 3.3, mental consciousness (or the sixth consciousness) is generatedfrom the alaya consciousness. Since mental consciousness and alaya consciousness are modelled by thecategory X and the category Z respectively, it is natural to model the generation of mental consciousnessas a symmetric monoidal functor from Z to X . First, we set up a functor F from FdHilb N to X , where FdHilb N is the category obtained from FdHilb by restricting the coefficients of complex numbers tonatural numbers. Clearly we can have an interpretation of diagrams of Z in FdHilb N similar to J · K , whichis denoted by J · K N . For each object H n of dimension n , F ( H n ) is a single-type perception semimodulegenerated by n elements { x i } n − i = which has a bijection σ : | i i → x i with an orthonormal basis {| i i} n − i = of H n . Obviously, σ and σ − can be linearly extended to semimodule homomorphisms which will be calledwith the same names. For each linear map f from H m to H n , F ( f ) is the semimodule homomorphism σ ◦ f ◦ σ − . Also we give the morphism F ( J g K N ) : F ( H s ) ⊗ F ( H t ) −→ F ( H st ) x i ⊗ x j x it + j where g is the following generator of the qufinite ZX ∆ -calculus: st ts One can check that F ( J g K N ) is a natural isomorphism and F is a symmetric monoidal functor. Thenthe functor from Z to X is given by the composite functor B = F ◦ J · K N , which is a symmetric monoidalfunctor (SMF) since both components are SMFs.
5. The Unity of Experience
As an application of our model of consciousness, we consider the combination problem on the unityof experience. Our approach is an alternative to conserve the irreducible and fundamental nature ofexperience. It is not, however, the only one. Panpsychism and Panprotopsychism, among others, alsoconsider experience seriously, but assigns a quantifiable character to that experience. According to theseviews, consciousness is present in all fundamental physical entities [38] and the composition of basicblocks of experience creates our conscious experience. Nevertheless, an important question remains:How "microphenomenal seeds of consciousness" constitute macrophenomenal conscious experiencesas we experience them? —the so-called combination problem for Panpsychism and Panprotopsychism[30]. In other words, how these building blocks of experience compound one single unified phenomenalsubjective experience [25]: the phenomenal unity of experience [25,39]. Basically, the dualism betweenmind and matter is now replaced by two modes, micro and macro experience, of the same ontology.
The combination problem has three aspects [30]: structural, subject and quality. Each one of theseaspects leads to a specific sub-problem. On the one hand, the structure of the micro world, mostlyassociated with quantum mechanics, gives the impression of being different from the structure of macroexperiences. This is the structural mismatch problem, which also appears between macro experiencestructure and macro physical structures in the brain [30]. On the other hand, there is the question of howmicro subject and micro qualities combine to give rise to macro subjects and qualities. It seems that nogroup of micro subjects need the existence of a macro subject, and additionally, it is not clear how possiblelimited micro qualities yield to the many macro qualities that can be experienced, including differentcolors, shapes, sounds, smells, and tastes (for detail see [30]). According to Chalmers, a satisfactorysolution of the combination problem must face all these three aspects.Our framework targets all of these aspects of the combination problem. First, the mathematicalstructure of the qufinite ZX ∆ -calculus for Alaya consciousness is a unification of all dimensional quditZX-calculus. If generators are interpreted in Hilbert space, the latest becomes a graphical language forquantum theory. This means that the ZX ∆ -calculus for conscious processes shares a similar structureto quantum theory. This similarity solves the mismatch at the level of micro experience. At the levelof macro experiences we avoid any match or mismatch with macro physical structures because themodel does not reduce experience to neural events (non-isomorphic relationship). Second, the model does not distinguish between subject and quality, everything is a conscious process. Those fundamentalconscious processes of reality, namely the generators of the theory, compound other conscious processesjust by means of connecting them together: via sequential and parallel composition. The result of thosecompositions are other subjective and qualitative processes. New compounded processes depend on thebasic generators, while the generators are interrelated to define themselves. In other words, each processneed other processes to specify itself. If someone insists on generators being matched with subjects oragents, then micro (generators) and macro subjects (composition of generators) necessitate themselves asimposed by the other dependent nature. This deals with the problem of subject composition. An examplefor quality composition in mental consciousness is discussed in the next section. In our framework, unityof consciousness is naturally described as a result of process composition [40]. One application of the above comments is instantiated for the combination of qualitative experiencesat the level of mental consciousness. Since we have modelled mental consciousness as the category X , thecombination of qualitative experiences should be modelled as a morphism within this category. Given anexperience space of rank s (the smallest number of generators) and an experience space of rank t , we claimthat a combination of experiences from these two spaces to an experience space of rank st is modelled bythe morphism F ( J g K N ) as given in section 4.2.Now we show by an example why F ( J g K N ) could model a combination of experiences. Considerthat there is a colour experience space A freely generated by {green, red} and a shape experience space B freely generated by {square, circle}. Then F ( J g K N ) is seen as a combination scheme to gain an experiencespace C of shapes with colour freely generated by {green square, green circle, red square, red circle}: F ( J g K N ) : A ⊗ B −→ C green ⊗ square green squaregreen ⊗ circle green circlered ⊗ square red squarered ⊗ circle red circle where J g K N : H ⊗ H −→ H | i 7→ | i| i 7→ | i| i 7→ | i| i 7→ | i Here two combined experiences presented at the same time are modelled by the superposition of thetwo experiences. For example, a green square and red circle that show up in our mind simultaneouslyare represented as green ⊗ square + red ⊗ circle . One can then check that the morphism F ( J g K N ) is theabstract mechanism that realises the combination: given green square and red circle simultaneously, agreen square and a red circle is obtained simultaneously via F ( J g K N ) ; the other cases are similar. One maywonder that whether the morphism F ( J g K N ) is just a renaming of the basis. In general, any isomorphismcan be seen as a renaming of a basis, however, as we pointed out in section 4.2, F ( J g K N ) is a naturalisomorphism, thus mathematically more complex than just a renaming of basis.
6. Conclusions
In approaching the problem of consciousness through the framework of qufinite ZX ∆ -calculus, weavoided reductionism in tackling the “hard problem” described above.Our framework is based on arbitrary commutative semirings as a compositional model ofconsciousness, with the emphasis on its potential use for the mathematical and structural studiesof consciousness [19–21]. We utilise generators and processes as abstract mathematical structures,resembling quantum theory. The philosophy that underlies our approach is taken from the Yogacaraschool of Buddhism which assumes that consciousness is fundamental and which characterizes the mainfeature of consciousness as other-dependence.A positive consequence of this approach is that the structure is close, but not the same, as quantumtheory, and if we restrict our semiring to the field of complex numbers, adding the standard interpretationof the diagrams in matrices, we get to finite-dimensional quantum theory. Therefore, the qufinite ZX ∆ -calculus is a unification, in this respect, of all finite dimensional qudit ZX-calcului, which aregraphical languages for quantum theory when interpreted in Hilbert space.In a future work, we expect to generalise the qufinite ZX ∆ -calculus to the infinite dimensional case,from which standard quantum mechanics might be recovered. It is to be noted that we have not recoveredstandard quantum mechanics. To do so would mean generalising our model in order to derive theSchrödinger equation. This is important because once subjectivity is taken as fundamental, a new inverseproblem comes into play. Namely, how do objective phenomena such as quantum physics or relativityarise from subjective experiences?The aim of models such as the conscious agent model is to recover fundamental physics from theagent’s interactions, as for instance in quantum mechanics [31]. It is not clear that current versions ofthe conscious agent model are capable of recovering the entire objective realm (see objections and repliessection in [31]). In our framework part of the reconstruction goal pursued by the conscious agent modelis achieved for free, and without overhead, invoking only phenomenal aspects. In doing so, our approachto consciousness processes and quantum theory share a similar mathematical structure. We are hopefulthat due to its other-dependent feature, and sufficient generality, our framework may pave the way forfurther research on the scientific study of consciousness.In following works, we also expect the extension of the model to, inter alia, fivesense-consciousnesses and manas consciousness, to consider infinite diagrams for Alaya consciousnessand infinite dimensional Hilbert spaces for its perceived division. This mean adding more structure formental consciousness, allowing us to compare our approach to other models of qualia space.We close by remarking that a process theory for consciousness is not only about modellingconsciousness with any type of mathematics, but about modelling consciousness with category theory ina graphical form, i.e. axiomatic mathematics. This form of mathematics explicitly introduces structures,assumptions and axioms. We believe this approach is better suited to describing the conscious experienceas fundamental. Author Contributions:
Conceptualization, CMS and QW; investigation CMS, QW and IK; writing-original draftpreparation, CMS; writing-review and editing, CMS, QW and IK; visualization, CMS and QW.
Funding:
CMS is funded by Comisión Nacional de Investigación Ciencia y Tecnología (CONICYT)through Programa Formacion de Capital Avanzado (PFCHA), Doctoral scholarship Becas Chile: CONICYTPFCHA/DOCTORADO BECAS CHILE/2016 - 72170507. QW is supported by AFOSR grant FA2386-18-1-4028.
Acknowledgments:
The authors appreciate valuable feedback and discussions from Bob Coecke, KonstantinosMeichanetzidis and Robert Prentner. The authors would also like to thank the anonymous reviewers for their prettyhelpful comments.
Conflicts of Interest:
The authors declare no conflict of interest.
References
1. Seth, A.K. Consciousness: The last 50 years (and the next).
Brain and Neuroscience Advances , , 239821281881601. doi:10.1177/2398212818816019.2. Crick, F.; Koch, C. Consciousness and neuroscience. Cerebral cortex , , 97–1007.doi:10.1093/cercor/8.2.97.3. Lusthaus, D. Buddhist Phenomenology , first ed.; Routledge Curzon, 2002; p. 632. doi:10.4324/9781315870687.4. Makeham, J. Introduction. In
Transforming Consciousness: Yogacara Thought in Modern China ; Makeham, J., Ed.;Oxford University Press, 2014. doi:10.1093/acprof:oso/9780199358120.001.0001.5. Fields, C.; Hoffman, D.D.; Prakash, C.; Singh, M. Conscious agent networks: Formal analysis and applicationto cognition.
Cognitive Systems Research , , 186–213. doi:10.1016/j.cogsys.2017.10.003.6. Evan Thompson. Mind in Life ; Harvard University Press, 2007.7. Varela, F.J. Neurophenomenology: A Methodological Remedy for the Hard Problem.
Journal of ConsciousnessStudies , , 330–349.8. Coecke, B. An Alternative Gospel of Structure: Order, Composition, Processes. In Quantum Physics andLinguistics:A Compositional, Diagrammatic Discourse ; Heunen, C.; Sadrzadeh, M.; E. Grefenstette., Eds.; OxfordUniversity Press, 2013; [1307.4038]. doi:10.1093/acprof:oso/9780199646296.003.0001.9. Coecke, B.; Duncan, R.; Kissinger, A.; Wang, Q. Generalised Compositional Theories and DiagrammaticReasoning. In
Quantum Theory: Informational Foundations and Foils. Fundamental Theories of Physics. ; Chiribella,G.; Spekkens, R., Eds.; Springer, 2016; Vol. 181, pp. 309–366, [1506.03632]. doi:10.1007/978-94-017-7303-4_10.10. Coecke, B., Ed.
New Structures for Physics , lectures n ed.; Springer Berlin Heidelberg, 2011; p. 1034.doi:10.1007/978-3-642-12821-9.11. Abramsky, S.; Coecke, B. A categorical semantics of quantum protocols. 19th Annual IEEE Symposium onLogic in Computer Science (LICS’04), 2004, pp. 415–425.12. Coecke, B.; Kissinger, A.
Picturing Quantum Processes. A first Course in Diagrammatic reasoning ; CambridgeUniversity Press: Cambridge, UK, 2017. doi:10.1017/9781316219317.13. Kissinger, A.; Uijlen, S. A categorical semantics for causal structure. 32nd Annual ACM/IEEE Symposiumon Logic in Computer Science (LICS), 2017, pp. 1–12. doi:10.1109/LICS.2017.8005095.14. Pinzani, N.; Gogioso, S.; Coecke, B. Categorical Semantics for Time Travel . [1902.00032].15. Kissinger, A.; Hoban, M.; Coecke, B. Equivalence of relativistic causal structure and process terminality .[1708.04118].16. Coecke, B.; Sadrzadeh, M.; Clark, S. Mathematical Foundations for a Compositional Distributional Model ofMeaning.
Linguistic Analysis , , 345–384, [arXiv:1003.4394v1].17. Bolt, J.; Coecke, B.; Genovese, F.; Lewis, M.; Marsden, D.; Piedeleu, R. Interacting Conceptual Spaces I :Grammatical Composition of Concepts. ArXiv , [arXiv:1703.08314].18. Signorelli, C.M.; Dundar-Coecke, S.; Wang, V.; Coecke, B. Cognitive Structures of Space-Time.
Frontiers inPsychology , p. (Submitted).19. Prentner, R. Consciousness and topologically structured phenomenal spaces.
Consciousness and Cognition , , 25–38. doi:10.1016/j.concog.2019.02.002.20. Yoshimi, J. Mathematizing phenomenology. Phenomenology and the Cognitive Sciences , , 271–291.doi:10.1007/s11097-007-9052-4.21. Tsuchiya, N.; Saigo, H. Applying Yoneda’s lemma to consciousness research: categories of level and contentsof consciousness. Preprint . doi:10.31219/osf.io/68nhy.22. Mac Lane, S.
Categories for the Working Mathematician , second ed.; Springer New York, 1978.doi:10.1007/978-1-4757-4721-8.23. Coecke, B.; Duncan, R. Interacting quantum observables: Categorical algebra and diagrammatics.
New Journalof Physics , , [arXiv:0906.4725v3]. doi:10.1088/1367-2630/13/4/043016.24. Searle, J.R. Consciousness. Annual Review of Neuroscience , , 557–578.doi:10.1146/annurev.neuro.23.1.557.
25. Bayne, T.; Chalmers, D.J. What is the unity of consciousness?
The Unity of Consciousness: Binding, Integration,and Dissociation , pp. 1–41. doi:10.1093/acprof:oso/9780198508571.003.0002.26. Thomas Nagel. What is it like to be a bat?
The Philosophical Review , , 435–450.27. Chalmers, D. The puzzle of conscious experience. Scientific American , , 80–86.28. Howard, R. Dualism, 2017.29. Mulder, D.H. Objectivity.30. Chalmers, D.J. The Combination Problem for Panpsychism. Panpsychism; Brüntrup, G.; Jaskolla, L., Eds.Oxford University Press, 2016, pp. 179–214. doi:10.1093/acprof:oso/9780199359943.003.0008.31. Hoffman, D.D.; Prakash, C. Objects of consciousness. Frontiers in Psychology , , 1–22.doi:10.3389/fpsyg.2014.00577.32. Xuanzang.; Cook, F.H.; Vasubandhu. Three Texts on Consciousness Only ; Numata Center for BuddhistTranslation and Research: Berkeley, 1999.33. Xuanzang. Cheng Weishi Lun. In no. 1585 of Taisho shinshu daizokyo ; Vol. 31.34. Xuanzang, Tat Wei.; Vasubandhu.
Cheng Wei Shi Lun; The Doctrine of Mere-Consciousness. ; Ch’eng Wei-shihLun Publication Committee: Hong Kong, 1973.35. Golan, J.S.
Semirings and their Applications ; Springer Netherlands, 1999. doi:10.1007/978-94-015-9333-5.36. Hadzihasanovic, A.; Ng, K.F.; Wang, Q. Two complete axiomatisations of pure-state qubitquantum computing.
Proceedings - Symposium on Logic in Computer Science , pp. 502–511.doi:10.1145/3209108.3209128.37. Heunen, C. An embedding theorem for hilbert categories.
Theory and Applications of Categories , , 321–344.38. Chalmers, D.J. Panpsychism and Panprotopsychism. Amherst Lecture in Philosophy , .39. Revonsuo, A.; Newman, J. Binding and consciousness. Consciousness and cognition , , 123–127.doi:10.1006/ccog.1999.0393.40. Signorelli, C.M.; Wang, Q.; Coecke, B. Reasoning about conscious experience with axiomatic mathematics. (inprogress)2020