Kairos Language Specification — 2. Types and Layers

Translated from the canonical Japanese chapter spec/10-types.md. The source_sha above records the source revision; a consistency check flags this page when the Japanese original changes.

2.1 The two-layer structure

Kairos consists of two layers (corresponding to SQL’s DDL/DML).

Premise layer — builds calendar systems and calendars. May be multi-line and declarative. The layer that defines what a “month” is and what a “business day” is.
Body layer — weaves schedules. One line, pipe-like, wherever possible. The layer that derives firing instants using the vocabulary already defined.

Both layers share the same stream vocabulary, but their types are distinct. The premise layer absorbs the complexity; the body layer stays thin.

2.2 The three types

The time-stream type (the body layer’s first-class value)

A lazy, infinite, ordered sequence of points on the base Chronos. Equal definitions are equivalent. This is the extension (the generated sequence of points), and the body layer’s only first-class value. Within the body layer the type is kept to this single one.

The premise type (intension — generation rules)

The generation rules themselves — calendar systems, calendars, and the like. If the time-stream type is the extension, the premise type is its intension. There are primitive and derived definitions (§3.6/§3.7); derivation is a premise → premise closure, staying closed within the premise type. Primitive and derived share the same type.

The value type (numbers, booleans, enums, lists, strings, instants)

Values that are not time streams: numbers, booleans, enums (Mon, Preceding, etc.), lists ([…], indexing, and the membership predicate in), strings ("Asia/Tokyo"; the values of TZs and provenance, ADR-32), and instants (the value of a date literal; the point a lambda binds has the same makeup. A bare value binding d0 = 2026-05-15 is also legal = ADR-43/F97 — this removes the duplication of writing the same date twice, in anchor:/from: and in a table). Used in the rules of calendar-system definitions (leap-year tests, etc.; §3.6) and in body-layer arguments (the n of shift(n)). The third type, alongside the time-stream type and the premise type. A list whose elements are instants is promoted to a time-stream constant (table literals, §3.8; ADR-26. The empty list [] is promoted only with a covering: postfix = the empty table, ADR-45).

Evaluating a premise yields a time stream (never the reverse). The two layers are asymmetric, with a single conduit at the boundary where the base coordinates (asof, TZ, granularity scale, data phase; §3.8) fix the premise. The overall picture of the three types and the two layers (the diagram includes the closure of §2.3 and the one-way hierarchies of §2.4):

flowchart TD
  subgraph P["premise layer — intension (generation rules); absorbs the complexity"]
    PRIM["primitive definitions (Gregorian etc.)<br/>build calendar systems from the base"]
    DERIV["derived definitions (fiscal calendars, entity overrides, etc.)<br/>premise → premise closure"]
    DATA["data bindings<br/>(table literals, external; §3.8)"]
    PRIM -- "with overrides (cannot be expanded back into primitives = one-way)" --> DERIV
  end
  P == "evaluation — the single conduit at the boundary (asof, TZ, granularity scale, data phase fix the premise)<br/>never the reverse: a stream cannot yield a premise" ==> ST
  P -. "premise arguments (the axes of on:/unit:/axis:; in-scope calendar:)" .-> CORE
  subgraph B["body layer — extension (sequence of points); kept thin"]
    ST["time-stream type — lazy, infinite, ordered<br/>the body layer's only first-class value"]
    CORE["core family (generators, point transforms, combinators,<br/>filters, windows, selectors, strides)"]
    SUGAR["sugar (monthEnd, businessDays, etc.)"]
    SUGAR -- "erasable by expansion into core (one-way)" --> CORE
    ST -- "input" --> CORE
    CORE -- "output — no escape from the type (I2 closure)" --> ST
  end
  VAL["value type — numbers, booleans, enums, lists, strings, instants"]
  ST -- "binding-name projection name(d) = point → value (§4.9)" --> VAL
  VAL -- "window-instance reference W(v) = value → point sequence (§2.7, ADR-42)" --> ST
  VAL -. "arguments (the n of shift, predicates, label expressions)" .-> CORE
  VAL -. "instant lists promote to constants (§3.8)" .-> DATA

2.3 Closure

Every operator is (stream…, premise) → stream. There is no escape from the type. This is what enables the composition existing languages lacked — “taking the derived dates as a basis and building yet another definition on top of them.” Instead of one monolithic expression, you write a pipe in which each stage takes the previous stage’s output as its input.

2.4 Two symmetric hierarchies (core/sugar, primitive/derived)

The same pattern — separating complexity from shorthand — appears in both layers.

Body layer: the core family and sugar — the core family (generators, point transforms, combinators, filters, windows, selectors, strides) is minimal and strict. Sugar (monthEnd, businessDays, nextWeekday, etc.) is shorthand that names compositions of it, erasable by expansion into core. The dependency is one-way (sugar → core), so a bug in the sugar layer never propagates into core. Everyday writing stays short with sugar; the semantics stay strict in core.
Premise layer: primitive and derived definitions — primitives (Gregorian, etc.) build calendar systems from the base; derived definitions (fiscal calendars, etc.) are made by overriding an existing premise. A derived definition cannot be expanded into primitives (it is a new rule). The same one-way asymmetry.

2.5 Symbols: one-to-one with three roles

No symbol carries more than one role. Stage connection, namespace reference, and stream union are assigned to separate symbols, one-to-one.

Symbol	Role
`\\|>`	Stage connection (flows the time stream into the next stage; in the premise layer, connects premise → premise)
`.`	Premise qualification (hierarchical namespace reference; `Gregorian.month`)
`\\|`	Combinator (stream union; paired with intersection `&` and difference `\`; §4.5)

2.6 Invariants

Properties the language upholds structurally (details in 10-domain-model.md and 20-adr/).

I1 Fixed base — every element is a point on the single base Chronos. Every calendar system and granularity is a projection of it.
I2 Closure — every operator is (stream…, premise) → stream.
I3 Explicit resolution — any operator that can land on an invalid or absent position takes a roll convention and anchor resolution as mandatory arguments (making silent bugs syntactically impossible).
I4 Window-relative — selectors are always relative to a containing window. A selector without a window is a type error. “The Nth” also depends on the window’s origin (a two-step dependency).
I5 Exhaustiveness verification — partition-type windows are checkable for exhaustiveness over the axis and non-overlap. Interval-sequence types state the meaning of gaps explicitly.
I6 Context flow — TZ, WKST, asof, and out-of-coverage origin (interval annotations) flow as evaluation context and evaluation annotations, not as element data (“origin of emptiness” generalizes to origin of the interval — it attaches to non-empty results too; §4.10, ADR-15 revised/37).
I7 Pure and lazy — over infinite streams, every operator is defined online.
I8 Generator purity — generators depend only on the calendar system, never on calendars (business days, holidays). Calendar dependence is confined to point transforms, filters, and combinators onward (with difference \ as the entry point = the standard bizDay derivation, ADR-35).

2.7 Typing rules for application — position-dependent name interpretation (ADR-42)

The same name receives different interpretations depending on where it appears and on the types of its arguments. One rule governs all of it: narrow the candidate set by the expected type of the occurrence position; if more than one interpretation survives the narrowing, never choose silently = ambiguity error (the same facet as ADR-17 and ADR-35 decision 4). This principle has three concrete faces.

Position	Kind of name	Interpretation	Details
Axis position (`on:`/`unit:`/`axis:`)	premise name	Determining the entity’s identity + rereading it as the standard derivation	§3.9, ADR-35 decision 4
Application, point argument `W(d)`	binding with a label source	Binding-name projection (point → value)	§4.9, ADR-30/34/39
Application, value argument `W(v)`	window binding with a label source	Window-instance reference (value → point sequence) = preimage	§4.9, ADR-42

Dispatch (deciding point versus value) branches uniquely on the type of the argument expression. The type of a lambda variable is fixed by the context of its binding site (the lambda of filter/label: binds a point; the lambda of span binds a window ordinal = a number; value functions get theirs from the typing of the body), and the decision moment is after sugar expansion and after actual-argument binding (the same computation moment as alignment, §4.5) — at that moment the type of the actual argument is always unique, so an ambiguity error can structurally never fire at an application position. A free wrapper (f = v => year(v)) has its type fixed per call (polymorphism is allowed). W(v) is legal in every stream-expecting position (head position, combinator operands, on:/unit:, the S slot of coincides, etc.); a pure value position (x = year(2020) + 1) is a type error (streams cannot mix into value expressions = the type separation of §2.2). An instant is a member of the value type (§2.2, ADR-43), and dispatch reads “point → projection; any non-point value → instance reference” — a date literal and a value variable holding a point both branch to projection (year(2026-05-15) = year(d0) = projection = 2026).