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Computational Modelling, Explicit Mathematical Treatments, and Scientific Explanation

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Computational Modelling, Explicit Mathematical Treatments, and Scientific Explanation Computational modelling, explicit mathematical treatments, and scientific explanation John Bryden and Jason Noble School of Computing, University of Leeds, LS2 9JT, UK Abstract findings can be applied more generally. Biological systems are made up of many different subsystems at different levels. A computer simulation model, can produce some interesting and surprising results which one would not expect from ini- Alife models often reside in the interface from one level to tial analysis of the algorithm and data. We question however, the next and can become extremely complex, especially as whether the description of such a computer simulation mod- entities from any level can interact with entities from other elling procedure (data + algorithm + results) can constitute levels. an explanation as to why the algorithm produces such an ef- fect. Specifically, in the field of theoretical biology, can such The discipline of computer simulation modelling allows a procedure constitute real scientific explanation of biological modellers previously unheard-of freedom to build and un- phenomena? We compare computer simulation modelling to derstand systems of many interacting parts. This new ex- explicit mathematical treatment concluding that there are fun- pressive freedom appears to have the potential to become damental differences between the two. Since computer simu- the new modelling paradigm in science, perhaps overriding lations can model systems that mathematical models can not, we look at ways of improving explanatory power of com- traditional techniques which use explicit mathematical treat- puter simulations through empirical style study and mecha- ments. However, this freedom does not come without a cost: nistic decomposition. as more and more detail is added computer simulation mod- els can quickly become unwieldy and too complicated to un- Introduction derstand. It seems possible that computer simulation modelling could How then can computer models contribute to the task of become the new modelling paradigm in biology. As mod- producing scientifically acceptable explanations? The use ellers build transparent, tractable, computer simulation mod- of a complex yet poorly understood model may be accept- els their relaxed assumptions will, in comparison with tra- able as some sort of loose analogy. However, Di Paolo et al. ditional explicit mathematical treatments, make for con- (2000) have argued that without a proper understanding of siderably more realistic models that are close to the data. the internal workings of a computer simulation model, it can The ‘Virtual Biology Laboratory’ is proposed (Kitano et al., be impossible to say whether such a model makes a valuable 1997) where a cycle is applied through comparing computer contribution to the scientific problem it is addressing. They models with empirical evidence: the results from each pro- describe such problematic models as ‘opaque thought exper- cedure inspiring the direction of the other. Animals, such iments’, arguing the need for explanations of the phenomena as C. elegans, have been well studied using computational modelled. They suggest that modellers should use an ‘ex- models, e.g., (Bryden and Cohen, 2004). Indeed the forma- perimental phase’ in which manipulations are made to the tion of a complete model of the organism has been identi- computer model, the results of these manipulations hope- fied as a potential grand challenge for computing research fully generating insights into the workings of the system. (Harel, 2002). However, a full exploration of the rela- Once the internal mechanisms are understood, the transpar- tionship between mathematical and computational models ent model can then not only give new insights into the sys- in biology has not yet been achieved. Questions remain: tem being modelled but can also become a powerful predic- for instance, whether both forms of modelling can peace- tive tool. fully coexist, whether mathematical models should aspire We question whether a computer simulation model can, in to the complexity of computational models, and conversely and of itself, constitute a scientific explanation. For exam- whether computational models can ever be as precise as a ple, one might produce a model in which individual organ- mathematical treatment. isms are explicitly represented and a particular population- In this paper we are mainly concerned with the scientific level phenomenon appears to emerge. But this does not con- modelling of biological systems, however we hope that the stitute an explanation of how entities from one level of a bio- logical hierarchy produce interesting phenomena at another Real World level. Di Paolo et al. (2000) argue that some explanation is Empirical Scientific Corpus required above a basic description of the model and the sys- science tem it represents. In this paper we look further into what an adequate explanation of a model’s mechanisms should en- Conceptual Model tail. We will compare the account that we construct with the more basic position, sometimes seen in the artificial life lit- Results Assumptions erature, that a bare-bones description of a biological system with a computer model that qualitatively produces similar behaviour—with little or no extra analysis or explanation— Working Model can constitute a scientific explanation of some phenomena. Given the above picture we must also consider the tradi- Modelling Tools tional methodology of explicit mathematical treatment. By Figure 1: The cycle of enquiry in scientific modelling within explicit mathematical treatment we mean a model which is the context of scientific investigation. complete and contains no implicit steps, the steps can be logical statements and do not need to be formally written using mathematical symbols. While computer simulation matical treatments reside in the working model area of the models are fundamentally mathematical constructions, they, framework. We take a working model to be a determinis- in the way they are reported, contain implicit mathematical tic and completely specified model of a system. (Whereas steps rather than the explicit steps used by formal mathe- a conceptual model may remain vague in places, a working matical models. An explicit mathematical treatment takes model must be completely fleshed out.) Logical processes logical axioms and specifies a number of clear explicit steps are applied to the axioms and the results of this process are that deductively generate some result. In this paper we com- recorded. Logical processes can include mathematical equa- pare this traditional treatment with the new computational tions, logical deductions and computations. Working mod- approach. els produce results which are used to refine and update the Firstly we set the context, we look at a framework for conceptual model. scientific modelling. Then, by looking at two examples of Before we specifically look at the sorts of results that can a similar system, we identify some properties that charac- be generated by explicit mathematical treatments or com- terise an explicit mathematical treatment and which a com- puter simulation models, we discuss the types of assump- puter simulation is unlikely to share. Having established that tions that can be used to generate a working model. An as- explicit mathematical treatment is the ultimate goal of any sumption is essentially an abstraction from a more complex modelling enterprise, we look at how computer simulation system. There will be many abstractions from the real world models do indeed still have value. We look at how com- in the conceptual model (tested by empirical science) and plex and unwieldy computer simulations may be simplified it will normally be necessary to make further abstractions to more easily generate explicit mathematical treatments— for ease of modelling. One of the main benefits of com- proposing that this can be done by decomposition into sim- puter simulation modelling (Di Paolo et al., 2000) is that pler systems. Finally we set out, in an order of merit, the assumptions can be very easily added to or removed from various different modelling approaches discussed. models to see if they are significant or important. Explicit mathematical treatments tend to be more fixed in their as- A framework for scientific modelling sumptions. The types of abstractions used by either explicit To understand how modelling is important and relevant conceptual models or computer simulation models can be within scientific investigation, we present a framework for distinguished into two groups, reductionist and analogous scientific investigation with the scientific modelling cycle abstractions. We take inspiration for this distinction from highlighted. Figure 1 presents a diagram of the framework. Bedau’s discussion of ‘unrealistic’ models (Bedau, 1999). The primary focus of scientific investigation is the build- In order to highlight the important differences between ing of a good conceptual model of the real world. Expla- the use of computational and mathematical techniques in nations of the real world reside in the conceptual modelling building a working model, we must first consider the out- area of the framework, these are recorded in the scientific comes of a successful working model for the broader scien- corpus. The basic scientific process involves the submission tific project. The more valuable results generated by a work- of concepts to the twin tests of empirical science and scien- ing model will form some kind of explanation of why some tific modelling. The main focus of the framework,
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