PossibilitySpace
The definition PossibilitySpace(c) constructs the set of all configurations reachable from a given configuration c through finite sequences of possibility transitions. Researchers working on modal extensions of Recognition Science would cite this when formalizing reachability in configuration space. The definition proceeds by direct set comprehension over paths of length n+1 satisfying the step-wise Possibility predicate.
claimLet $c$ be a configuration. The possibility space of $c$ is the set of all configurations $y$ for which there exists a natural number $n$ and a function path : Fin$(n+1)$ to Config such that path$(0) = c$, path$(n) = y$, and for each $i < n$, path$(i+1)$ belongs to the possibility set of path$(i)$.
background
In the Modal.Possibility module, a configuration is a structure with positive real value, time coordinate in ticks, and log-bound constraint ensuring physical scale. The Possibility relation defines single-step reachability: y is possible from c when y.time equals c.time plus 8 and the J-cost transition plus stasis condition is non-increasing. This definition builds the transitive closure of that relation, enabling modal statements about reachability. Upstream arithmetic results supply the natural-number inequalities needed to index finite paths correctly.
proof idea
The definition is given directly as a set comprehension. It quantifies over n and a path function from Fin(n+1) to Config, enforcing the start and end conditions along with the inductive step using the Possibility predicate at each intermediate point. No tactics are involved; it is a pure definitional expansion.
why it matters in Recognition Science
This definition supports the theorem identity_in_all_possibility_spaces, which shows that the identity configuration lies in every possibility space. It fills the modal logic layer of the Recognition Science framework by formalizing reachability, connecting to the forcing chain and J-cost structures. It touches on the development of possibility operators in the modal extension.
scope and limits
- Does not enumerate explicit paths for computation.
- Does not extend to infinite sequences of configurations.
- Does not impose additional constraints from the full LedgerState.
- Does not address multi-agent or game-theoretic extensions.
formal statement (Lean)
372def PossibilitySpace (c : Config) : Set Config :=
proof body
Definition body.
373 {y : Config | ∃ n : ℕ, ∃ path : Fin (n+1) → Config,
374 path ⟨0, Nat.zero_lt_succ n⟩ = c ∧
375 path ⟨n, Nat.lt_succ_self n⟩ = y ∧
376 ∀ i : Fin n, path ⟨i.val + 1, Nat.succ_lt_succ i.isLt⟩ ∈
377 Possibility (path ⟨i.val, Nat.lt_of_lt_of_le i.isLt (Nat.le_succ n)⟩)}
378
379/-- The identity is in every possibility space. -/