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The notion of information transfer remains cloudy while it is used interchangeably to refer to both concepts. Our thesis in this paper is that the concepts of predictive transfer and causal effect are quite distinct: we aim to clarify them and describe the manner in which they should be considered separately. We argue ... |
For this comparison, we examine Cellular Automata (CAs) (e.g. see Wolfram ( 2002 ): discrete dynamical lattice systems involving an array of cells which synchronously update their states as a homogeneous deterministic function of the states of their local neighbors. In particular we focus on Elementary CAs (ECAs), whic... |
In particular, we examine the transfer entropy and information flow measures on a local scale in space and time in ECAs, in order to provide an explicit comparison between the two. This is the first presentation and examination of the local information flow. We demonstrate that transfer entropy as predictive transfer i... |
On the basis of these results, we suggest that information flow should be used first wherever possible in order to establish the set of causal information contributors for a given destination variable. Subsequently, transfer entropy may be used to quantify the information transfer from these causal sources to the desti... |
II Predictive information transfer II.1 Transfer entropy Schreiber presented transfer entropy as a measure for information transfer Schreiber ( 2000 in order to address deficiencies in the previous de facto measure, mutual information, the use of which was criticized in this context as a symmetric measure of statically... |
[EQUATION] where [MATH] is a time index, [MATH] refers to the [MATH] states of [MATH] up to and including [MATH] , and [MATH] is the state transition tuple [MATH] . It can be viewed as a conditional mutual information, casting it as the average information in the source about the next state of the destination that was ... |
II.2 Local transfer entropy The transfer entropy is an average (or expectation value ) of a local transfer entropy Lizier et al. ( 2008a at each observation [MATH] , i.e. [MATH] where: |
[EQUATION] For lattice systems such as CAs with spatially-ordered agents, the local transfer entropy to agent [MATH] from [MATH] at time [MATH] is represented as: |
[EQUATION] The transfer entropy [MATH] to agent [MATH] from [MATH] at time [MATH] is illustrated in Fig. 1(a) [MATH] is defined for every spatiotemporal destination [MATH] , for every information channel or direction ; sensible values for [MATH] correspond to causal information sources, i.e. for CAs, sources within the... |
The transfer entropy may also be conditioned on other possible causal information sources, to eliminate their influence from being attributed to the source in question |
Schreiber ( 2000 In general, this means conditioning on all sources [MATH] in [MATH] ’s set of causal information contributors [MATH] (except for [MATH] ) with joint state [MATH] , giving the local complete transfer entropy Lizier et al. ( 2008a |
[EQUATION] For CAs this means conditioning on other sources [MATH] within the range [MATH] of the destination to obtain Lizier et al. ( 2008a |
[EQUATION] In deterministic systems (e.g. CAs), complete conditioning renders [MATH] because the source can only add information about the outcome of the destination. Calculations conditioned on no other information contributors (as in Eq. ( )) are labeled as apparent transfer entropy. |
Finally, note that the information (or local entropy) [MATH] required to predict the next state of a destination can be decomposed as a sum of Lizier et al. ( 2008b |
the information gained from the past of the destination (i.e. the mutual information between the past [MATH] and next state [MATH] , known as the active information storage [MATH] ); plus |
the information gained from each causal source considered (in arbitrary order) in the context of that past, incrementally conditioning each contribution on the previously considered sources. |
For example, in ECAs we have: [EQUATION] In this way, the different forms of the transfer entropy as information transfer can be seen to characterize important components of the total information at the destination. |
III Causal effect It is well-recognized that measurement of causal effect necessitates some type of perturbation or intervention of the source so as to detect the effect of the intervention on the destination (e.g. see Pearl ( 2000 ); Hope and Korb ( 2005 ). Attempting to infer causality without doing so leaves one mea... |
III.1 Information flow Following Pearl’s probabilistic formulation of causal Bayesian networks Pearl ( 2000 , Ay and Polani Ay and Polani ( 2008 consider how to measure causal information flow via interventional conditional probabilities. For instance, an interventional conditional probability [MATH] considers the dist... |
In a similar fashion to the definition of transfer entropy as the deviation of a destination from stochastic independence on the source in the content of the destination’s past, Ay and Polani propose the measure information flow as the deviation of the destination [MATH] from causal independence on the source [MATH] |
imposing another set of nodes [MATH] . Mathematically, this is written as: [EQUATION] with [MATH] representing the tuple [MATH] and the modified interventional distribution defined as: |
[EQUATION] The value of the measure is dependent on the choice of the set of nodes [MATH] . To obtain the direct causal information flow from [MATH] to [MATH] we must either include all possible other sources in [MATH] or at least include enough sources to block all non-immediate directed paths from [MATH] to [MATH] |
Ay and Polani ( 2008 . The minimum to satisfy this is the set of all direct causal sources of [MATH] excluding [MATH] , including any past states of [MATH] that are direct causal sources. For computing direct information flow across one cell to the right in ECAs (see Fig. 1(c) ) where [MATH] and [MATH] , this means [MA... |
[EQUATION] Establishing the value of [MATH] requires determination of the underlying interventional conditional probabilities. By definition these may be gleaned by observing the results of intervening in the system, however this is not possible in many cases. |
One alternative is to use detailed knowledge of the dynamics, in particular the structure of the causal links and possibly the underlying rules of the causal interactions. This also is often not available in many cases, and indeed is often the very goal for which one turned to such analysis in the first place. Regardle... |
Furthermore, certain cases exist where one can construct these value from observational probabilities only Ay and Polani ( 2008 . For example, the “back-door adjustment” (Section 3.3.1 of Pearl ( 2000 suggests that where a set of nodes [MATH] satisfies the “back-door criteria” relative to [MATH] , i.e. that: |
1. “no node in [MATH] is a descendant of” [MATH] , and 2. [MATH] blocks every path between” [MATH] and [MATH] that contains a directed causal link into [MATH] |
then the interventional conditional probability [MATH] is given by: [EQUATION] The back-door adjustment could be applied to [MATH] in ECAs in Fig. 1(c) with the set of nodes satisfying the back-door criteria marked there as [MATH] ; for [MATH] the set [MATH] would be used. In general, note that the back-door adjustment... |
III.2 Local information flow We can define a local information flow [EQUATION] in a similar manner to the localization performed for the transfer entropy. The meaning of the local information flow is slightly different however. Certainly, it is an attribution of local causal effect of [MATH] on [MATH] were [MATH] impos... |
For lattice systems such as CAs, we use the notation [MATH] to denote the local information flow into agent [MATH] from the source agent [MATH] at time step [MATH] (i.e. flow across [MATH] cells to the right), giving: |
[EQUATION] [EQUATION] with [MATH] defined in 11 IV Application to Cellular Automata Here, we apply the local transfer entropy and local information flow to the raw states of ECA rule 54 in Fig. . This rule exhibits a (spatially and temporally) periodic background domain, with gliders traveling across the domain and col... |
Focusing on transfer and flow one step to the right per unit time step, we measure the average transfer values being [MATH] and [MATH] bits for apparent and complete transfer entropy respectively, and the information flow at [MATH] bits. Much more insight is provided by examining the local values of each measure howeve... |
IV.1 Background domains are highly causal As an extension of the example of coupled Markov chains in Ay and Polani ( 2008 to more complex dynamics, we first look at the background domain region of the CA where each cell executes a periodic sequence of states. The four time step period of the (longest) sequences is long... |
Lizier et al. ( 2008a , while the local information flow [MATH] in Fig. 2(b) measures a periodic pattern of causal effect at similar levels to those in the glider/blinker regions. |
Both results are correct, but from different perspectives . From a computational perspective, the cells in the domain region are executing information storage processes – their futures are (almost) completely predictable from their pasts Lizier et al. ( 2008a . Note that to achieve these long periods, some of this info... |
IV.2 Gliders distinguished as emergent information transfer We then examine the measurements at the gliders, the emergent structures which propagate against the background domain. Here we see that the local transfer entropies [MATH] and [MATH] measure strong information transfer in the direction of glider motion in Fig... |
Lizier et al. ( 2008a , while the local information flow [MATH] in Fig. 2(b) measures similar levels of causal effect to those in the background domain. |
Again, both results are correct from different perspectives . The cell states in the glider region provide strong predictive information about the next states in the direction of glider motion: this is why gliders have long been said to transfer information about the dynamics in one part of the CA to another (as quanti... |
It is possible that a macroscopic formulation of the information flow might distinguish gliders as highly causal macroscopic structures, but certainly (when applied to the same source and destination pair as transfer entropy) as a directional measure of direct local causal effect it does not distinguish these emergent ... |
IV.3 Information transfer to be measured from causal sources only Fig. 2(f) measures the local apparent transfer entropy [MATH] for two steps to the right per unit time step. This profile is fairly similar to that produced for one step to the right per unit time step. However, this measurement is for superluminal trans... |
To check the correctness of the information flow measure, we apply it here assuming the CA is of neighborhood-5 (i.e. two causal contributors on either side of the destination). As expected, the local information flow profile computes no causal effect across two cells to the right per unit time step (not shown). Import... |
Furthermore, measuring the complete transfer entropy [MATH] in this neighborhood results in a zero information transfer profile (not shown), since all the required information to predict the next state of the destination is contained within the interior neighborhood for this deterministic system. This aligns well with ... |
IV.4 Complete transfer entropy as a next best inference for information flow The approximation that the complete transfer entropy provides to the information flow goes beyond similar inference of a lack of influence. Consider the profile of [MATH] in Fig. 2(c) – note how similar it is to the profile of the local inform... |
The complete transfer entropy is therefore a candidate method for inferring causal structure in a multi-variate time series in these appropriate conditions, so long as one understands it is neither a direct nor exact measure of causal effect. |
Importantly, the complete transfer entropy must condition on (at least) the correct neighborhood of causal sources in order to provide best approximation of the information flow. Since this is exactly what is being searched for in this circumstance, one would in fact need to build knowledge of the causal contributors f... |
Importantly also, while the complete transfer entropy can at least function in the absence of observations spanning all possible combinations of the variables, if crucial combinations are not observed it can give quite incorrect inferences here. For example, consider the classical causal example of a short circuit whic... |
Discussion and Conclusion The concepts of information transfer and causal effect have often been confused. In this paper, we have demonstrated the complementary nature of these concepts while emphasizing the distinctions between them. On an information-theoretical basis, information flow quantifies causal effect using ... |
Causal effect is a fundamental micro-level property of a system. Information flow should be used as a primary tool (where possible) to establish the presence of and quantify causal relationships. Where this is not possible (e.g. where one has no ability to intervene in the system, no knowledge of the underlying dynamic... |
Information transfer can then be analyzed in order to gain insight into the emergent computation being carried out by the system. Importantly, the transfer entropy should only be measured for causal information contributors to the destination, otherwise its result cannot be interpreted as information transfer. To do so... |
Acknowledgements. The authors thank Daniel Polani and Nihat Ay for helpful discussions regarding the nature of the information flow measure and in particular how to estimate it from observational data. JL thanks John Mahoney for discussions regarding measuring transfer entropy from non-causal sources, and the Australia... |
# Source: arxiv 0901.0317 # Title: Design of a P System based Artificial Graph Chemistry # Sections: all # Downloaded: 2026-03-03T01:56:10.579906+00:00 |
Design of a P System based Artificial Graph Chemistry Abstract Artificial Chemistries (ACs) are symbolic chemical metaphors for the exploration of Artificial Life, with specific focus on the origin of life. In this work we define a P system based artificial graph chemistry to understand the principles leading to the ev... |
Basic Framework of Artificial Chemistries Aim of this section is to present a brief introduction to artificial chemistries. We will start with a discussion on the epistemological foundations of the area and will illustrate further details using examples relevant to this proposal. The examples are followed by discussion... |
1.1 Introduction It is a long held topic of scientific debate whether there are any biological principles of life and other complex biological phenomena, which are not directly reducible to physical and chemical laws. Living beings, however small and consisting of the same molecular components as non living things, non... |
The direct ways to understand this complex biological phenomena are usually difficult and error prone because living structures are by default complex and hard to manipulate. Even cellular level experiments are difficult to carry out and their simulations are quite cumbersome. |
Artificial life (AL) is a tool to study principles explaining this complex phenomena of life without directly getting involved with the real biological systems. The fundamental assumption here is that principles of life are independent of the medium and carbon based life on earth is just one example of the possible for... |
Living phenomena has several aspects to study, one such is the origin of life or biogenesis . Here the problem is to understand how first primitive form of life such as metabolism and self replicating structures could have come into existence starting from non living chemical compounds. Artificial chemistries (AC) are ... |
An AC has three main components, a set of objects or molecules , a set of reaction rules or collision rules, and a definition of population dynamics |
Objects can be abstract symbols, numbers, lambda expressions, binary strings, character sequences, abstract data structures etc. Reaction rules might be string matching, string concatenation, reduction rules, abstract finite state machines, Turing machines, matrix multiplication, simple arithmetic operation, cellular a... |
A survey on various ACs is given in Ditt01 , which also has some broad classification of ACs based upon the kind of molecular abstractions (explicit or implicit), type of reaction rules (constructive or non constructive), and population dynamics. |
To illustrate, we take examples from two kinds of ACs. One where no spatial structures are considered, that is, all molecules evolve as a whole in a reactor tube and all molecules can interact with each other according to the collision rules. The examples of AlChemy (Section 1.2.1 ) and CHAM/ARMS (Section 1.2.2 ) are o... |
It seems, during the pre-biotic evolution of life, spatial structures (e.g., membranes etc) emerged starting from the open reactor type system without any spatiality. This spetial structure based classification is one of the main motivations for P system based AC definition, we propose in the next section. |
1.2 Examples Next we illustrate the common design of ACs using examples. Each example is followed by a discussion on the relative strengths and limitations of it w.r.t. real chemistry. |
1.2.1 Algorithmic Chemistry - AlChemy We consider [MATH] expression based AC proposed in Font92 Font94 called AlChemy. Molecules - [MATH] Terms: The object space consists of abstract lambda expressions (also called terms ). These terms are generated as follows: There is an infinite supply of variable names [MATH] [MATH... |
The set of terms, [MATH] , is defined inductively: (1) [MATH] (2) [MATH] (abstraction) (3) [MATH] (application) A variable [MATH] is said to be bound if it occurs inside a sub-term with the form [MATH] , otherwise it is free. The set of free variables in an expression [MATH] is denoted by [MATH] |
Syntactical Transformation: The schemes of transformation are oriented rewrite rules. Structures on the left hand side are replaced by structures on the right hand side. More precisely, |
Substitution (4) [MATH] (5) [MATH] ; if [MATH] (6) [MATH] ; if [MATH] and [MATH] (7) [MATH] Renaming (8) [MATH] Reaction Rules - Function Composition and Normal Form Reduction: The reaction rules in Alchemy consist of application of one lambda term over the other, which is then reduced to a normal form. The choice of l... |
Formally a reaction between molecules [MATH] and [MATH] can be written as a binary operation ( [MATH] ) defined as [EQUATION] Where [MATH] is used from the convention of writing chemical equations to represent that the molecules are present in reactor. |
[MATH] uses some consistent reduction strategy to reduce the term in finitely many steps to a normal form. This choice of finite step normal form reduction actually results in equivalence classes consisting of all the expressions which are functionally same modulo finite execution steps. The choice of [MATH] gives flex... |
Population Dynamics - Stochastic Molecular Collisions: Initially a large pool of random lambda terms of finite lengths is generated. Only those terms, which are in normal form are considered. In each iteration two molecules are chosen at random and one is applied to the other (function composition) according to [MATH] ... |
The relative quantitative dynamics of various molecules is captured in terms of differential equations. Replicator equations of Lotka-Volterra type Font92 JK98 are used to describe the relative concentration of self replicating molecules. |
Discussion: In actual chemistry, especially in case of organic compounds with chains of carbon atoms and possible branching, chemical reactions substitute parts of one molecule with other molecule thus leading to structural rearrangement in the chemical composition of these molecules. This is the main motivation behind... |
With stochastic collision dynamics and choice of reaction type ( [MATH] ), the AlChemy gives rise to some interesting forms of organizations, classified as level-0 level-1 , and level-2 organizations. While level-0 organization consists of only self replicating molecules whose frequencies are modelled using replicator ... |
Though AlChemy captures certain basic aspects of real chemical compounds and their reactions, it has its own limitations. Most important of those is related to the choice of lambda calculus. Even though lambda calculus is computationally universal and has a consistent reduction strategy (i.e., order of reduction steps ... |
Thus the first limitation is the lack of selective substitution, which means, in case of actual chemical reactions, new compounds are formed (with substitution) based on the relative strengths of chemical bonds in reactants and relatively higher stability of the products. On the other hand in case of substitution in la... |
Second limitation is the poor abstraction of structural properties of chemical compounds. The only kind of compounds which might be resembling the lambda terms structurally are those which have long carbon chains with possible branching. Double helix structure of DNA with complementarity is difficult to capture using l... |
, which is so common in living forms as well cannot be captured using lambda terms. The significance of this lack of structural abstraction of geometrical properties is not very clear. |
Since chemical reactions are driven by thermodynamic constraints like rate of collision, pressure etc, and the properties of colliding molecules, they are usually symmetrical in nature. Thus the result of collision of the molecules A and B is same as that of B and A since there is no order on A and B. On the other hand... |
Functional Equivalence - the kind of functional equivalence defined in case of lambda chemistry does not capture the equivalence which life-like forms demonstrate. In case of living structures, it is the interaction which objects have with external environment or other objects that plays important role. This element of... |
Lack of information abstraction - this is true in general for almost all of the proposed ACs. And that is one of the focus of this proposal to understand the role information plays in the emergence of life-like phenomena in ACs. |
1.2.2 The Chemical Abstract Machine The Chemical Abstract Machine (CHAM) was proposed in Berr96 as an abstract formalism for concurrent computation using closely a metaphor of chemical reactions. |
There are two description levels. On the upper level, CHAM abstractly defines a syntactic framework and a simple set of structural behavior laws. An actual machine is defined by adding a specific syntax for molecule and a set of transformation rules that specify how to produce new molecules from old ones. |
Molecules are terms of some algebra . A general membrane construct transforms a solution into a single molecule, and an associated general airlock construct makes the membrane somewhat porous to permit communication between an encapsulated solution and its environment. The generic reactions laws specify how reactions d... |
cooling rules . The presence of membrane type structure gives universal computational power to the model. Dynamics of CHAM goes like this - on each iteration a CHAM may perform an arbitrary number of transformations in parallel, provided that no molecule is used more than once to match the left side of a reaction law. ... |
Sujuki and Tanaka used CHAM to model chemical systems by defining an ordered abstract rewriting system on multiset called chemical ARMS Sujuki01 Molecules are the abstract symbols . The reaction rules are multiset rewriting rules . The reactor is represented by a multiset of symbols with a set of input strings. An opti... |
The qualitative dynamics of ARMS is investigated by generating rewriting rules randomly. This led them to derive a formal criteria for the emergence of cycles Sujuki96 in terms of an order parameter, which is roughly the relation of the number of heating rules to the number of cooling rules |
Sujuki98 . For small and large values of this order parameter, the dynamics remains simple, i.e., the rewriting system terminates and no cycles appear. For intermediate values, cycles emerge. |
Discussion: Although CHAM was not defined as an AC, it is quite close to actual cellular chemistry in some aspects. The presence of membrane structure gives rise to important resemblance with cellular reactions mediated by membranes. Another significant property of CHAM model is that it is very general hence provides f... |
The main limitation of CHAM model is that the allowed abstract terms of algebra are not adequate to capture the structural properties of real chemical compounds, as discussed in case of AlChemy. |
Second limitation comes due to nature of rewriting rules, they are actually grammar rules rather than being close to the chemical reactions. Because of this problem with multiset rewriting, in ARMS analysis is done by randomly generating these rewriting laws, and it is not clear whether chemical reactions where molecul... |
1.2.3 Artificial Chemistry on a Planar Graph This model of AC was proposed in Piet01 , where an AC is embedded in a planar triangular graph. Molecules are placed on the vertices of the undirected graph and interact with each other only via the edges. The planar triangular graph can be manipulated by adding and deleting... |
Molecules and Reactions: There is an (infinite) set of potential molecules [MATH] and a reaction mechanism which computes the reaction product for two colliding molecules [MATH] There may be an arbitrary number of products for each such collision. Molecules are built from different types of substrate of elements called... |
Dynamics: At every step they pick two neighboring molecules [MATH] and apply the first [MATH] to the second [MATH] creating a (multi )set of new molecules. These product molecules are randomly inserted in the two faces next to the link between [MATH] and [MATH] |
[MATH] is replaced with first molecule after the reaction (the result of the combinator reduction) and [MATH] is finally deleted. Molecules cannot change their positions in the graph. |
In this system, it is observed that clusters of molecules which do not interact with the neighboring molecules arise. The clusters can be regarded as membranes when they divide the graph into different regions. There also arises a cell organization, that is, a subgraph that can maintain the membranes by themselves. |
Discussion: As noted in Ditt01 , the presence of spatial topology gives rise to certain phenomena which is not possible to emerge easily in cases where there is no spatial topology present in the model. For example in the case of this planar graph based AC, an emergence of membrane type structure is something which is ... |
On the other hand, the choice of planar graph based topology is not something usually present in cellular structures neither it can be a simplified spatial structure for initial chemical environment responsible for emergence of life. Absence of abstraction of geometrical or structural properties is yet another problem. |
1.3 More Discussion on Artificial Chemistries ACs are basically motivated by and developed to understand the pre-biotic evolution or the problem of origin of life, which is still an open problem despite lots of advancements in molecular biology Smith99 Dev00 Dys99 . The problem of pre-biotic evolution differs significa... |
Smith93 or neutral theory of random drifts Kimu83 , etc can be used to explain the emergence of higher and more complex forms of life. Still the emergence of this genetic material which is so fundamental for the proper functioning of even the simplest forms of life is what makes the problem of pre-biotic evolution so d... |
Therefore the kind of problems mainly of focus in ACs and in this proposal are the search for principles governing the emergence of life-like forms from non life-like structures in AC systems. This also involves proper level of abstraction from real chemistry without loosing generality. |
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