Change and future models

Expressing deterministic changes in generic computer systems would be a huge undertaking unless one restricted ambitions to general features and trends. Dynamical systems are difficult to trace, even in the simplest of cases, so one cannot expect to get very far without making significant simplifications. The aim of considering a dynamical theory is thus to characterize the significant trends of change which might occur, owing to idealized influences. A full discussion of this topic is beyond the scope of the present paper, however based on the axioms and deliberations presented here, it is possible to outline the way forward in studying them.

The expression of strategies in the previous section is too general to be useful for a fully general, dynamical theory. Taking account of every strategic detail would be a vast undertaking. Instead, one can analyze the development at the level of a generic computer system undergoing generic changes as a matter of principle. The purpose of such a vague preliminary investigation is to elucidate the relationship between the system administration game and the lattice description of the ideal state, presented in section 8.

Once a strategy mixture has been decided, one must address the fact that, in real-world games, the speed of information is finite. It will take a finite amount of time for a response to develop after a strategy is implemented. Moves and counter-moves do not follow a rigid time-plan as in games like chess. This kind of delay leads to races and duels for superiority between competing players. Delay is the province of linear response theory.

The aim, then, is to express the causal structure of system development in the foregoing mathematical language. In order to reduce the dynamical game to algebra we must express each of these in terms of basic primitives. Causality is about relating actions to outcomes, or changes of state $\delta S$. A general action A(t) is built up from a number of primitive action-types T_a (called the generators for the action transformation) in a linear combination

$$A(t) = \sum_i\; a_i(t)\,T_i$$ (57)

where a_i(t) are functions of time (not necessarily differentiable, often step-like) and i takes values which number the full spectrum of primitive actions. The Tⁱ are orthogonal vectors or matrices (indices suppressed), one for each primitive action type, which span an abstract vector space. This vector space is the chequerboard on which the game takes place.

Each complete action A(t), results in a change in the state of the system, which may be denoted $\delta S$. An action can also be a causal chain of sub-actions, characterizing a sequence of changes in the state. This type of causal relationship is summarized by a Green function, propagator, or response-function formulation[15,16]:

$$\delta S(t) = \int dt' G(t,t')\,A(t'),$$ (58)

where G(t,t') is the two-point response function, as yet unspecified. If the rules of the game are independent of time, then G(t,t')=G(t-t'); if the rules change over time, then G(t,t')=G(t-t',t+t'). In this language of dynamical systems, an action plays the role of a source or driving force for the system. The equation above may be inverted to provide an inhomogeneous differential equation for the changing state of the system. If one formally introduces a differential operator D_t which is the inverse of the response function:

$$\int dt'\; D_t\; G(t,t') = 1,$$ (59)

then the differential equation may be written, schematically:

D_t S(t) = A(t), (60)

where, as ad hoc an example one might have,

$$D_t \equiv \frac{d^2}{dt^2} + i\gamma \frac{d}{dt} + \omega_0^2,$$ (61)

for an approximately periodic system which degrades over time, like a damped harmonic oscillator. Each action A(t) thus leads to a response or change of state; this in turn implies that the state of the system must be a linear combination of the same action types:

$$S(t) = \sum_i s_i(t)\, T_i.$$ (62)

The state is thus defined on the same lattice, or chequerboard as the actions themselves. Differential (difference) characterizations of state have been studied in ref. [7]; this type of description is interesting, since it leads often to rich dynamics. Alternating periods of change and stability (riffles and pools in the flow of the system) might be best described by a difference representation.

Returning to the idea of the contest as a game, one writes a strategy as a statistical mixture of actions (i.e. moves in the game) A(t), applied over an interval of time. This stochastic mixture specifies the boundary conditions under which the actions are applied. It may be formed as a linear combination of basic actions A_n, with probability weights w_n:

$$J(t) = \sum_n w_n A_n(t) = \sum_i p_i T_i,$$ (63)

The strategy vector J_i is the vector of probabilities for each primitive action, given the chosen mixture of full actions A_nfor J. In other words, J_i are the components of the decomposition of the strategy J(t) on the space of primitive actions.

$$p_i = \sum_n w_n.$$ (64)

It is easy to normalize these so as to be actual probabilities which sum to unity

$$\sum_i p_i \equiv 1.$$ (65)

Notice that the specific representation of basis generators T_i does not affect the strategy vector, since it only serves to label the lattice-work of independent actions. There is no unique labelling. The components with respect to the basis must be related by a response function $\Pi_{ij}(t,t')$

$$\delta S_i = \int dt' \; \Pi_{ij}(t,t')J_j(t').$$ (66)

The matrix value distribution is related to the pay-off matrix, and a basis of so-called ladder operators, also called creation and annihilation operator. The represent do-action/undo-action operations of the users and system administrator.

$$\Pi(t,t') \sim \pi_{ij}\; \otimes\; \langle \vec d \vert\hat S_+(t) \hat S_-(t') \vert\vec d \rangle$$ (67)

where $\hat S_\pm$ are operators which annihilate a configuration state at t' and create a new configuration at time t. This is the generic mechanism by which the system develops. This form of description might seem unnecessarily formal, but it is actually highly useful, since the continuous generalization of this kind of dynamical system has been widely studied in statistical field theory. By picking out universal features of statistical models and restricting the scope of the computer system, there is a real chance of being able to build toy models which have qualitative, predictive power. However, this is no trivial undertaking and will be considered in a later paper[17].

Since the actions which configure a computer system form a lattice, and these primitive action types do not necessarily commute with one another, one concludes that a suitable idealization of the system administration's stochastic dynamics is found in non-Abelian, statistical field theories. This line of study would be suitable for modelling resource availabilities for large numbers of users, in which all users behave approximately equally on average (like an ideal gas). This approach promises therefore to be relevant to the problem of anomaly detection[8] and will be returned to in later work.