Towards a formalization of budgets

We go into the need for, and the requirements on, a formal theory of budgets. We present a simple algebraic theory of rational budgets, i.e., budgets in which amounts of money are specified by functions on the rational numbers. This theory is based on the tuplix calculus. We go into the importance of using totalized models for the rational numbers. We present a case study on the educational budget of a university department offering master programs.

💡 Research Summary

The paper “Towards a Formalization of Budgets” argues that contemporary budget design and financial accounting have become increasingly specialized and embedded within complex IT systems, yet lack a rigorous theoretical foundation. To address this gap, the authors propose an algebraic framework that treats budgets as mathematical objects built from rational numbers. Central to their approach is the use of a totalized field of rational numbers—referred to as a “meadow”—in which division by zero is defined (typically yielding zero). This choice eliminates the complications of partial functions and enables straightforward type‑checking and error detection within budget expressions.

Budgets are modeled as “tuplixes,” a data structure consisting of attribute‑value pairs. An attribute denotes a transaction channel (e.g., a specific cost centre), while the associated rational value represents the monetary amount: positive values indicate payments, negative values indicate receipts. The composition operator ⊕ (denoted in the paper as “”) combines entries and zero‑tests in a commutative, associative manner. Two distinguished constants are introduced: ε (the empty budget) acting as the identity element, and δ (the null budget) which annihilates any composition containing it, thereby representing an inconsistent or erroneous budget. Identical attributes can be merged by adding their amounts, which naturally captures internal reconciliation of payments and receipts.

Conditional constraints are expressed through “zero‑tests” γ(p). If the rational argument p evaluates to zero, the test disappears; otherwise the entire composition collapses to δ. Equality and inequality constraints are encoded by suitable combinations of γ and arithmetic operations (e.g., γ(p‑q) for p = q, γ(|q‑p| ‑ (q‑p)) for p ≤ q). The authors also provide axioms that allow multiple tests to be combined into a single test, simplifying complex constraint handling.

A key operation is encapsulation, written ∂H, which targets a set H of attributes. It sums all entries with attributes in H; if the total is zero, those entries are removed, otherwise the whole budget becomes δ. This formalizes the accounting notion of “netting” or “settlement” across channels. Encapsulation distributes over unions of attribute sets (∂{H∪H’} = ∂H ∘ ∂{H’}), enabling stepwise resolution of multi‑channel settlements.

The theoretical machinery is illustrated with a case study of a university department that runs three MSc programmes (A, B, C). Each programme has its own budget, but they share certain resources (e.g., teaching facilities) modelled via a common channel b. The authors define budget P for programme A (receipt of 30 on channel a, payments of 10 and 20 on b) and budget Q for programme B (receipt of 30 on b, payment of 30 on c). By composing P and Q and then encapsulating channel b (∂_{b}), the b‑related entries cancel, leaving a simplified budget that only contains the net flows on channels a and c. This demonstrates how the algebra can automatically reconcile inter‑programme cost allocations and enforce constraints.

Beyond the case study, the paper draws parallels between budget design and software development. It notes that, unlike software where dynamic testing and static analysis tools are mature, budget validation often relies on ad‑hoc static checks and post‑hoc evaluations. The proposed algebra provides a formal basis for static verification, simulation, and potentially automated tooling, thereby reducing uncertainty and error propagation in financial planning.

In summary, the authors present a concise yet powerful algebraic theory for rational budgets, grounded in totalized rational arithmetic and the tuplix calculus. By formalizing entries, tests, composition, and encapsulation, they enable precise specification, constraint checking, and automated reconciliation of budgets. The framework promises to bridge the gap between informal financial practice and rigorous computer‑science‑inspired modeling, offering a foundation for future tool development and deeper theoretical exploration of budgeting, service‑level agreements, and related aspects of service‑oriented computing.