> For the complete documentation index, see [llms.txt](https://novusedge0.gitbook.io/docket-docs/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://novusedge0.gitbook.io/docket-docs/design-and-research/outcome-formalism.md). # Outcome formalism September 2026. These working notes define a formalism for a proposed paper. They identify prior art the paper must cite and claims that remain unresolved. ## Core objects | Symbol | Meaning | | ------ | ------------------------------------- | | `S` | Set of steps. | | `O` | Set of outcomes. | | `Γ` | Set of claims. | | `C` | Set of finite chains. | | `R` | Relation between chains and outcomes. | | `ι` | Outcomes a chain aimed at. | A chain is a finite sequence of steps: $$ C \subseteq S^\* $$ The realization relation connects chains to outcomes. The pair `(c, o) ∈ R` means that chain `c` realizes outcome `o`: $$ R \subseteq C \times O $$ For a chain `c` and an outcome `o`, define: $$ R(c) = {o \in O : (c,o) \in R} $$ $$ R^{-1}(o) = {c \in C : (c,o) \in R} $$ `R` is a relation. `R(c)` is the set of outcomes realized by one chain, and `R⁻¹(o)` is the set of chains that realize one outcome. ## Outcome taxonomy The proposed distinctions describe the cardinality of `R` and one labeling function. They do not add independent structural axes. | Distinction | Condition | Interpretation | | ------------- | ----------------- | --------------------------------------------------------- | | Many-to-one | \` | R⁻¹(o) | | One-to-many | \` | R(c) | | Intentional | `R(c) ∩ ι(c)` | Realized outcomes the chain aimed at. | | Unintentional | `R(c) \ ι(c)` | Realized outcomes outside the chain's aim. | | Direct | `R(c)` | Outcomes directly realized by the chain. | | Indirect | `Cl(R(c)) \ R(c)` | Consequences added by the specified consequence operator. | Define intent as the outcomes a chain aimed at: $$ ι : C \to 2^O $$ Then the intentional and unintentional sets are: $$ \operatorname{Intentional}(c) = R(c) \cap ι(c) $$ $$ \operatorname{Unintentional}(c) = R(c) \setminus ι(c) $$ For every exhaustive and disjoint binary partition of `R(c)`, choosing the intended class as `ι(c)` reconstructs both classes. The result does not derive `ι` from `R`. Direct and indirect outcomes require a separately specified consequence operator: $$ Cl : 2^O \to 2^O $$ Generic closure laws do not determine the meaning of the indirect set. Extensivity, monotonicity, idempotence, and preservation of the empty set are insufficient. Specify `Cl` through a fixed logical, causal, or transition relation, and require a consequence path from a direct outcome to each indirect outcome. ## Claims and justification Each step has supporting claims. Each claim can have several alternative justification sets. Every inner set is one complete support for the claim: $$ j : \Gamma \to 2^{2^\Gamma} $$ The claim classifications are: * A claim `γ` is **atomic** when `j(γ) = ∅`. * A claim `γ` is **reasoned** when it has at least one non-empty justification set. These definitions leave the empty-support case $j(\gamma) = {\varnothing}$ unclassified. This structure extends a truth maintenance system with alternative support (Doyle 1979; de Kleer 1986). When an atomic premise becomes unavailable, propagate that change through the support relation. Retain a reasoned claim while one complete justification set survives. Retain other atomic claims until the work withdraws them explicitly. The regress problem remains. Every reasoned claim needs support, and that support needs support. The chain terminates at atomic claims, loops, or continues without end. Treating atomic claims as self-evident is foundationalism. It is an assumption that the formalism does not establish, and the paper should state it as one. ## Decomposition Let `k` index the steps that require reasoning, and let `d` be a decomposition limit: $$ k \subseteq {1, \ldots, n} $$ Each step indexed by `k` expands into at most `d` sub-steps. The user's intent and effort level set `d`, so it is a budget and a practical stopping rule. A decomposed chain has a tree structure. Shared support can turn that tree into a directed acyclic graph. The sequence notation is therefore too weak when `d > 1`: $$ C \subseteq S^\* $$ A value-of-information test gives a stronger stopping rule: expand a step only when a different answer can change the final outcome. ## Sections and parallelism Let `≺` be a strict partial order on steps. Read `sᵢ ≺ sⱼ` as “step `sⱼ` depends on step `sᵢ`.” A section is a contiguous subsequence of a chain. A partition is **parallel-valid** when every pair of section transformations commutes. An absent crossing edge does not establish parallel validity. The dependency relation must include every pair of steps that do not commute. Step-level commutativity is sufficient: if all cross-section step pairs commute, the complete section transformations commute. When transformations do not commute, each section can be locally correct while the composition changes with the schedule. This is compositional incoherence (2605.30335). One result argues against wide parallelism. A single long chain can search an exponentially larger space than the same compute spent on many short parallel chains (2505.21825). The paper must address this result. ## Prior art the paper must cite **Intention.** Ward, MacDermott, Belardinelli, Toni, and Everitt, “The Reasons that Agents Act: Intention and Instrumental Goals” (2402.07221, AAMAS 2024) defines intention formally in structural causal influence models. It separates intended outcomes from foreseen but unintended ones by a counterfactual criterion. Read it in full. If this definition does not mechanically diverge from theirs, rename this section's contribution. Philosophical sources include Anscombe on intention and foresight, Bratman's planning theory, and the doctrine of double effect. “On Automating the Doctrine of Double Effect” (IJCAI 2017) gives a computational treatment. **Side effects.** Impact-measure research provides related prior art for studying the effects of an agent's actions. Compare its definitions directly with the direct/indirect and intentional/unintentional classifications here. See Krakovna et al. on stepwise relative reachability (1806.01186) and future tasks (2010.07877), and Turner et al. on attainable utility preservation (1902.09725). The critique is 2101.12509. Review how these impact measures have already been applied to LLM and tool-using agents before claiming a contribution in that setting. The current source review leaves that question open. **Outcome multiplicity.** Self-consistency (2203.11171) exploits many-to-one as a confidence signal and is the paper a reviewer is likely to raise first. Argumentation theory calls the structure convergent arguments and argument accrual. Classical planning calls it top-k and top-quality planning (2404.01503), with plan equivalence as the formal apparatus. Whether prior work treats multiplicity as a named taxonomic axis remains an open literature question. Any novelty claim requires citing and distinguishing self-consistency. **Adjacent concepts.** Instrumental and terminal goals classify desired outcomes. Direct and indirect outcomes classify realized outcomes. Add a footnote so readers do not conflate them. **Evaluation precedent.** Agent benchmark surveys already separate outcome-level success from step-level process metrics, and some score trajectory harms alongside task success (2507.21504, 2605.16282). This is the applied form of the taxonomy, implemented as two scores rather than stated as a formalism. ## Open problems The formalism leaves these questions open: 1. Answer order is irrelevant when all answer transformations commute. For two transformations, order invariance holds exactly when they commute: $$ f\_2 \circ f\_1 = f\_1 \circ f\_2 $$ Real systems do not always satisfy this condition. Option order changes some benchmark results by up to 75 percent (2308.11483), and premise order changes results on deductive tasks (2502.04134). 2. Deciding that a claim is atomic is itself a reasoning step and can be wrong. There is no ground truth for self-evidence. 3. A visible chain often fails to reflect the computation that produced the answer (2503.08679, 2606.13603). A formalism over stated chains therefore describes what the system reports; the underlying computation remains unverified. ## Verification limits Several citations came from search summaries rather than direct paper reads. Check each citation before it carries weight. Read 2402.07221 directly because the intention-section novelty claim depends on its exact definition. --- # Agent Instructions This documentation is published with GitBook. GitBook is the documentation platform designed so that both humans and AI agents can read, navigate, and reason over technical content effectively. 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