On the Complexity of Determinations
Classical complexity theory measures the cost of computing a function, but many computational tasks require committing to one valid output among several. We introduce determination depth -- the minimum number of sequential layers of irrevocable commi…
Authors: Joseph M. Hellerstein
On the Complexity of Determinations Joseph M. Hellerstein UC Berkeley and Amazon W eb Services Abstract Classical complexity theory measures the cost of computing a function, but many computational tasks require commiing to one valid output among se veral. W e introduce determination depth —the minimum number of sequential layers of irrevocable commitments needed to select a single valid output—and show it is an orthogonal axis measuring the cost of commiing. W e exhibit relational tasks whose commitments are constant-time table lookups yet require exponential parallel width to compensate for any reduction in depth. A conser vation law couples determination and computational depth: enriching the commitment basis trades layers for step complexity , but the total sequential cost is bounded below . For circuit-encoded specications, the resulting depth hierarchy captures the polynomial hierarchy ( Σ 𝑃 2 𝑘 -complete for each xed 𝑘 , PSP ACE-complete for unbounded 𝑘 ). Determination depth and computational depth are orthogonal: neither subsumes the other . Stable matching witnesses the independence sharply—nding a stable matching is in P, yet ev ery nite determination depth arises as the rotation-poset height of some instance . In the distributed seing, the framework r ecovers the Halpern–Moses common-knowledge impossibility . 1 Introduction Classical complexity theory measures the cost of computing a function —producing the unique correct output. But many tasks require resolving a r elation : choosing one admissible outcome from a set, as when a text generator selects one coherent continuation among many , or a distributed system must agree on one valid output among several. W e show that this distinction creates a new , orthogonal axis of complexity . Classical hard functions are computationally expensive but hav e no cost on this new axis. Howe ver , there exist relational tasks in which every step is a constant-time table lookup, yet any strategy using fewer than 𝑘 sequential layers requires e xponential width (e xponentially many parallel candidates), even with unbounded parallelism within each candidate. e boleneck is not computation, nor communication, but irrev ocable commitment . is cost is invisible to classical complexity measures—circuit depth, communication complexity , adaptive queries—be cause they track the cost of computing or distributing an answer , not the cost of commiing to one. W e call this choice process a determination : a collection of irrevocable commitments that narrow the admissible set until a single outcome remains. Commitments whose order ae cts which outcomes survive must be se quenced into layers; those whose order does not maer can be applied in parallel. e determination depth of a spe cication is the minimum number of layers neede d to resolve it—the intrinsic cost of commiing. W e establish three main results. (1) Exp onential separation (Se ction 3 ): the depth–width tradeo is exponential, generalizing pointer chasing [ N W93 , MYZ25 ] from functions to relations. (2) Conser vation law (Section 4 ): enriching the commitment basis trades determination layers for computational depth within layers, but the total sequential cost is bounde d below—choices can be bundle d into richer steps, but the burden does not disappear . e bound is tight. Under a commutativ e basis in the online seing, a supporting oracle characterization shows that determination depth equals the minimum number of observations of 1 the growing histor y , regardless of computational power between observations. (3) Polynomial-hierarchy characterization (Section 5 ): for circuit-encoded sp ecications, “is depth ≤ 𝑘 ?” is Σ 𝑃 2 𝑘 -complete for each xed 𝑘 and PSP ACE-complete for unb ounded 𝑘 , certifying determination depth as a semantic measure well-calibrated against the classical hierarchy . e exponential separation construction is autoregressive: each commitment is a single irrevocable choice, and the depth- 𝑘 lower bound says that 𝑘 sequential choices ar e unavoidable regardless of parallelism. is connects determination depth to chain-of-thought reasoning, wher e each token of sequential infer ence is a commitment layer ( App endix D .3 ). Stable matching provides a complementary structural connection: every nite determination depth arises as the rotation-poset height of some instance ( Appendix D.1 ). e app endices apply the framework to several classical seings, decomp osing known costs into intrinsic determination depth and circumstantial mo del assumptions, and in sev eral cases rev ealing new structure that e xisting models do not capture . Each determination lay er carries a concrete physical cost: a coordination round in distribute d systems, a token of infer ence in language models, a round of play in a game. Existing models can hide this cost by assuming away commitments, or inate it by charging for operations that resolve no ambiguity . e appendices exhibit both phenomena. BSP round complexity over charges by counting communica- tion rounds that involve no commitment, and under charges by assuming xed memb ership. In extensive- form games, determination depth measur es the game’s strategic depth —the number of moves wher e a player must break a tie among equally-optimal options—which can b e strictly less than the game-tree depth; one player can inuence the other’s strategic depth, creating a new dimension of strategic interaction ( Appendix D .2 ). In the distributed seing, the framework recovers the Halpern–Moses imp ossibility of asyn- chronous common knowledge [ HM90 ], with the conservation law and exponential separation providing quantitative predictions beyond the binary threshold. 2 Framework W e model both online and oine seings via histories —partially ordered sets of events that represent everything that has occurred up to a given point. Denition 1 (History) . A histor y is a partially order ed set 𝐻 = ( 𝐸 , →) of events, where → is interpreted as causal precedence [ Lam78 ]. Events ar e of two kinds: environment events (inputs, messages, computation steps), which are given, and commitment events ( dened b elow), which ar e chosen. e results below are parametric in a class H of histories under consideration. Denition 2 (History Extension) . For histories 𝐻 1 = ( 𝐸 1 , → 1 ) and 𝐻 2 = ( 𝐸 2 , → 2 ) in H , we write 𝐻 1 ⊑ 𝐻 2 if 𝐸 1 ⊆ 𝐸 2 and 𝐻 1 is a downward-closed sub-poset of 𝐻 2 (i.e., whenev er 𝑒 ∈ 𝐸 1 and 𝑒 ′ → 𝑒 in 𝐻 2 , then 𝑒 ′ ∈ 𝐸 1 ). 𝐻 2 extends 𝐻 1 by adding later events; ⊑ is a prex order repr esenting information growth. A specication is online if H contains histories with proper extensions (the future is uncertain); it is oine if every history in H is maximal under ⊑ (no further ev ents will occur). e framework handles both uniformly; Example 1 below illustrates the online case. An outcome is an element of a set 𝑂 of externally observable results. Denition 3 (Sp ecication) . A spe cication is a mapping Spec : H → 2 𝑂 assigning to each histor y a set of admissible outcomes . W e call Spec ( 𝐻 ) the admissible set at 𝐻 . A specication may admit multiple outcomes for the same histor y; resolving this ambiguity is the cost that determination complexity measures. 2 Denition 4 (Commitment) . A commitment 𝜑 is a constraint on outcomes. W e write 𝐻 · 𝜑 for the history obtained by adding a commitment event 𝜑 aer all maximal events of 𝐻 (i.e., 𝑒 → 𝜑 for every 𝑒 with no successor in 𝐻 ), giving 𝐻 ⊑ 𝐻 · 𝜑 . Adding a commitment can only narrow the admissible set ( shrinkage ): Spec ( 𝐻 · 𝜑 ) ⊆ Sp ec ( 𝐻 ) . In the distributed seing, each agent’s commitments succeed only its own lo cal maximal events, not the global set; Appendix B develops this extension fully . Each commitment event 𝜑 in a history carries its context: the pr ex of the histor y up to 𝜑 determines the admissible set at which 𝜑 takes eect. Along any chain of extensions, environment events may enlarge or shrink the admissible set, but each commitment event can only shrink it ( shrinkage condition). A relational specication cannot be resolved to a function by environment events alone; commitments ar e required, and the cost of those commitments is what determination depth measures (Section 2.1 ). Denition 5 (Determination) . e determination 𝐷 ( 𝐻 ) of a histor y 𝐻 over a commitment basis Φ is the subsequence 𝜑 1 · · · · · 𝜑 𝑚 of commitment events in 𝐻 , listed in causal order . Its cost is cost ( 𝐷 ( 𝐻 ) ) ≜ 𝑚 . e commitment events in any single-agent history are totally ordered ( each follows all maximal events of its prex), so the determination is a well-dened sequence. Writing 𝐻 𝑗 for the prex of 𝐻 through 𝜑 𝑗 , shrinkage accumulates: Spec ( 𝐻 𝑗 ) ⊆ Sp ec ( 𝐻 𝑗 − 1 ) for each 𝑗 . A determination is valid if each 𝜑 𝑗 preserves nonemptiness: Spec ( 𝐻 ′ ) ≠ ∅ for every 𝐻 ′ ⊒ 𝐻 𝑗 . A spe cication is resolv ed if | Spe c ( 𝐻 ) | = 1 for all 𝐻 ; a history resolves Spe c if its determination leaves | Sp ec ( 𝐻 ) | = 1. In the oine seing, environment events precede all commitment events, so the determination is a single consecutive block. In the online seing, environment events may interleave with commitments, breaking the determination into separate runs; the eect of a commitment dep ends on which environment events hav e occurred, so the online seing requires an adaptive strategy (formalized in Section A ). ree forces on a history . A history evolves under thr ee distinct forces: the environment extends the history with new events (input arrivals, message deliveries); commitments extend the history with events that irrevocably narro w the admissible set; and a determination strategy selects which commitment events to append, based on the history obser ved so far . A determination strategy is a rule 𝜎 : H → Φ ∗ mapping each observable histor y to a sequence of commitments from the basis. It is a semantic object: it spe cies which commitments are made and when in histor y , but says nothing ab out how they are computed. A protocol, algorithm, or agent is a realization of a determination strategy; the complexity results in this paper are stated at the strategy level. Commutation and depth. Commitments 𝜑 and 𝜓 commute with respect to Spec at 𝐻 if the order of extension does not maer: Spec ( 𝐻 · 𝜑 · 𝜓 ) = Spec ( 𝐻 · 𝜓 · 𝜑 ) . Commutation is relative to a spe cication and may change as the histor y grows: 𝜑 and 𝜓 may commute at 𝐻 but not at an extension 𝐻 ′ ⊒ 𝐻 , or vice versa. is histor y-dependence does not arise in classical trace theor y , where independence is static, and it is what makes non-commuting commitments impose irreducible sequencing. (A history records one linearization of the commitments; commutation ensures the choice of linearization is immaterial.) e depth of a determination is the total numb er of commuting layers across all runs of consecutiv e commitments, where commitments within each layer commute pairwise with respect to the spe cication at that point in the history . e full algebraic development appears in Section A . Denition 6 (Determination cost and determination depth) . A determination strategy 𝜎 resolves Spec if, for every history 𝐻 arising under 𝜎 , | Spe c ( 𝐻 ) | = 1 once all of 𝜎 ’s commitments have been applie d. Dene Cost Φ ( Spe c ) ≜ min 𝜎 max 𝐻 cost ( 𝐷 ( 𝐻 ) ) and Depth Φ ( Spe c ) ≜ min 𝜎 max 𝐻 depth ( 𝐷 ( 𝐻 ) ) , wher e 𝜎 ranges over strategies that resolve Spec and 𝐻 ranges over histories arising under 𝜎 ; or ∞ if no resolving strategy exists. 3 In the oine seing (a single history with no environment extensions), the max is trivial and the denition reduces to a min over determinations. Cost Φ ( Spe c ) is the determination cost and Depth Φ ( Spe c ) the determination depth . Both are properties of the specication, not of any particular strategy—the strategy is the witness, just as a circuit witnesses the depth of a function—and they are the semantic analogues of circuit size and circuit depth: cost counts commitments, depth counts sequential layers. In the online seing, the max over 𝐻 gives the environment adversarial control bounded by the history class H ; this is a worst-case measure, and the r esults in this paper focus on this seing. 2.1 Commitment bases Determination depth depends on the commitment basis Φ . W e begin with the canonical minimal basis and then identify three properties it satises that generalize to richer seings. Denition 7 (Atomic basis) . For a history 𝐻 and outcome 𝑜 , the atomic commitment excluding 𝑜 at 𝐻 is the event 𝜑 that, when appende d to 𝐻 , excludes exactly 𝑜 : Spec ( 𝐻 · 𝜑 ) = Spec ( 𝐻 ) \ { 𝑜 } , and 𝑜 ∉ Spec ( 𝐻 ′ ) for every 𝐻 ′ ⊒ 𝐻 · 𝜑 . It has no eect when appended to a history incomparable to 𝐻 . e atomic basis contains one such commitment per pair ( 𝐻 , 𝑜 ) . Atomic commitments are the nest-grained irrevocable commitments: each permanently excludes a single outcome at a single histor y . e permanence condition ( 𝑜 never reappears at any extension) is what makes the exclusion irrev ocable; it is satised by all spe cications studied in this paper . In online seings, depth under the atomic basis arises from forward validity constraints : two atomic commitments made at histor y 𝐻 cannot b e applied in the same layer if their joint ee ct empties the admissible set at an extension of 𝐻 , as in the following example: Example 1 (ree-valued consensus) . Let 𝑂 = { 𝑎, 𝑏 , 𝑐 } with Spec ( 𝐻 0 ) = { 𝑎, 𝑏 , 𝑐 } , and suppose the environ- ment may extend 𝐻 0 to a histor y 𝐸 ⊒ 𝐻 0 with Spec ( 𝐸 ) = { 𝑎, 𝑏 } . e rened admissible sets Spec ( 𝐻 0 · 𝜑 ¬ 𝑎 ) and Spec ( 𝐸 · 𝜑 ¬ 𝑎 ) are both non-empty ( { 𝑏 , 𝑐 } and { 𝑏 } respectively), so 𝜑 ¬ 𝑎 is valid at 𝐻 0 . But the joint rene- ment Spec ( 𝐸 · 𝜑 ¬ 𝑎 · 𝜑 ¬ 𝑏 ) is empty: { 𝑎, 𝑏 } \ { 𝑎, 𝑏 } = ∅ . Symmetrically , if the environment may also extend 𝐻 0 to 𝐸 ′ with Spec ( 𝐸 ′ ) = { 𝑏 , 𝑐 } and to 𝐸 ′′ with Spec ( 𝐸 ′′ ) = { 𝑎, 𝑐 } , then no pair of atomic commitments at 𝐻 0 is jointly valid, so resolution requires two layers: commit, observe the environment, then commit again. Denition 8 (Intrinsic determination depth) . e intrinsic determination depth of a spe cication Spec is Depth Φ ( Spe c ) where Φ is the atomic basis. Intrinsic depth measures the irreducible layering for ced by forward validity constraints alone—the “bare ” sequential cost before any computational or architectural assumptions are introduced. In the oine seing, forward validity constraints vanish and intrinsic depth collapses; the conser vation law (eorem 3 ) shows that the sequential cost reappears as computation within layers. Properties of the atomic basis. e atomic basis satises three properties that we now name for use in the general theory . Denition 9 (Pointwise basis) . A commitment 𝜑 is pointwise if it acts as an outcome-by-outcome lter: whether 𝑜 ∈ Spec ( 𝐻 · 𝜑 ) depends only on 𝑜 and 𝐻 , not on the rest of Spec ( 𝐻 ) . A p ointwise basis consists entirely of pointwise commitments. A commitment whose eect on one outcome depends on the presence of another is non-pointwise . 4 Pointwise commitments always commute (each outcome is lter ed independently , so application order is irrelevant). Atomic commitments are pointwise, but the pointwise class is strictly larger: a commitment that excludes multiple outcomes simultaneously (e .g., “keep only tuples with 𝑣 1 = 3”) is pointwise but not atomic. Commutativity can also hold for non-pointwise commitments: Denition 10 (Commutative basis) . A commitment basis Φ is commutative if every pair of commitments in Φ commutes at every histor y for every sp ecication: Spec ( 𝐻 · 𝜑 · 𝜓 ) = Sp ec ( 𝐻 · 𝜓 · 𝜑 ) for all 𝜑 , 𝜓 ∈ Φ , all Spec , and all 𝐻 ∈ H . Denition 11 (Constant-depth basis) . A commitment basis Φ is constant-depth for Spec if each commitment in Φ is computable in 𝑂 ( 1 ) circuit depth from a standard polynomial-size encoding of the current history . e atomic basis is both commutative and constant-depth, making it the canonical minimal basis for the online seing where commitments are made before all information has arrived. Under a constant- depth basis, every unit of determination depth corresponds to one unit of sequential time; under a richer basis, some of that time is absorb ed into the computation ne eded to evaluate each commitment. e pointwise, commutative , and constant-depth properties each appear in multiple results: pointwise bases arise in the exponential separation (Section 3 ), the PH characterization (Section 5 ), and the NP-hardness of computing depth; commutative bases in the oracle characterization (Se ction 4.2 ) and the distributed extension ( App endix B ); and constant-depth bases in the conservation law (Se ction 4.1 ). 3 Exponential Depth– Width Separation W e exhibit a family of constrained generation tasks—modeling autoregressiv e text generation—in which every commitment is a constant-time table lookup, y et any strategy using 𝑑 ′ < 𝑘 layers requires width at least ( 𝑚 / 𝑠 ) 𝑘 − 𝑑 ′ . 3.1 e constrained generation task Fix positive integers 𝑘 (number of positions), 𝑚 (domain size), and 𝑠 (constraint sparsity ), with 1 ≤ 𝑠 ≤ 𝑚 . e task has 𝑘 − 1 links 1 → 2 → · · · → 𝑘 , each constraining a position’s value given its predecessor’s. Denition 12 ( 𝑘 -position constraint chain) . A 𝑘 -position constraint chain over domain [ 𝑚 ] with sparsity 𝑠 is a sequence of constraint functions 𝑃 1 ⊆ [ 𝑚 ] and 𝑃 ℓ ⊆ [ 𝑚 ] × [ 𝑚 ] for ℓ = 2 , . . . , 𝑘 , such that | 𝑃 1 | = 𝑠 and, for each ℓ ≥ 2 and each 𝑎 ∈ [ 𝑚 ] , the successor set 𝑃 ℓ ( 𝑎, ·) ≜ { 𝑏 ∈ [ 𝑚 ] : ( 𝑎, 𝑏 ) ∈ 𝑃 ℓ } satises | 𝑃 ℓ ( 𝑎, ·) | = 𝑠 . Denition 13 ( 𝑘 -position generation task) . Given a 𝑘 -position constraint chain ( 𝑃 1 , . . . , 𝑃 𝑘 ) (encoded as environment events in an oine history), the constrained generation task is to output any admissible tuple in Spec cg ( 𝑃 1 , . . . , 𝑃 𝑘 ) ≜ ( 𝑣 1 , . . . , 𝑣 𝑘 ) ∈ [ 𝑚 ] 𝑘 𝑣 1 ∈ 𝑃 1 and ( 𝑣 ℓ − 1 , 𝑣 ℓ ) ∈ 𝑃 ℓ for ℓ = 2 , . . . , 𝑘 . e specication is relational ( 𝑠 𝑘 admissible tuples when 𝑠 ≥ 2) and models autoregr essive generation: each position is a token, [ 𝑚 ] is the vocabulary , and 𝑃 ℓ encodes local coherence constraints. Commitment basis. For each position ℓ ∈ { 1 , . . . , 𝑘 } and value 𝑣 ∈ [ 𝑚 ] , the commitment 𝜑 ℓ , 𝑣 lters to tuples with 𝑣 ℓ = 𝑣 . ese are pointwise, but the links intr oduce dependency: commiing to 𝑣 ℓ before 𝑣 ℓ − 1 may violate a constraint. Each commitment is a constant-time table lookup. Observation 1 (Sequential strategy) . Commiing to one position per layer in order 𝑣 1 , 𝑣 2 , . . . , 𝑣 𝑘 always produces a valid tuple (each 𝑣 ℓ ∈ 𝑃 ℓ ( 𝑣 ℓ − 1 , · ) exists by sparsity), using 𝑘 layers and width 1. 5 3.2 Strategy Model and Lower Bound W e formalize what a strategy with fewer than 𝑘 layers can do. Denition 14 ( 𝑑 ′ -layer strategy with width 𝑤 ) . A 𝑑 ′ -layer strategy with width 𝑤 partitions the 𝑘 positions into 𝑑 ′ groups 𝑆 1 , . . . , 𝑆 𝑑 ′ and produces a single histor y carrying 𝑤 parallel candidate tuples, each contributing 𝑘 commitments (one per position) distributed across the 𝑑 ′ layers. In layer 𝑟 , for each candidate, the strategy commits to values at all positions in 𝑆 𝑟 ; these values may depend on the full input and on earlier-layer commitments, but not on same-layer or later-layer commitments. e strategy sele cts one of the 𝑤 candidates as its output; it resolves the specication if the output lies in Spec cg . e model is strictly more permissive than autoregr essive generation ( which xes le-to-right order and width 1): it allows arbitrary layer assignments, arbitrar y within-layer parallelism, and 𝑤 independent candidates. e only constraint is that commitments within a layer cannot depend on same-layer outcomes— the dening property of parallel execution—so any lower bound here applies to every parallel strategy that respects this constraint, including beam search, speculative deco ding, and diusion-style renement. Each group 𝑆 𝑟 is a commuting layer in which the 𝑤 candidates each commit to | 𝑆 𝑟 | positions in parallel. When 𝑑 ′ < 𝑘 , some adjacent positions share a layer and cannot obser ve each other’s values; the lower bound counts these blind spots ( uninformed links ) and shows that 𝑤 grows e xponentially in their number . Denition 15 ( Uninformed link) . Given a layer assignment 𝑆 1 , . . . , 𝑆 𝑑 ′ , a link ℓ − 1 → ℓ (for ℓ ∈ { 2 , . . . , 𝑘 } ) is informe d if position ℓ − 1 is assigned to a strictly earlier layer than position ℓ , and uninformed otherwise (i.e., 𝑣 ℓ − 1 is not yet determined when 𝑣 ℓ is commied). Observation 2 (Uninforme d-link count) . Any assignment of 𝑘 positions into 𝑑 ′ < 𝑘 layers leaves at least 𝑘 − 𝑑 ′ uninformed links. Denition 16 (Conditionally 𝛾 -spread distribution) . A distribution D over 𝑘 -position constraint chains is conditionally 𝛾 -spread if for e very ℓ ∈ { 2 , . . . , 𝑘 } , every 𝑎, 𝑏 ∈ [ 𝑚 ] , and every xing of all constraint functions except the row 𝑃 ℓ ( 𝑎, ·) : Pr D [ 𝑏 ∈ 𝑃 ℓ ( 𝑎, ·) | all other constraint data ] ≤ 𝛾 . I.e., membership of any single element in a row has pr obability ≤ 𝛾 , conditioned on the rest of the chain. Denition 17 (Random constraint distribution) . In the random ( 𝑘 , 𝑚, 𝑠 ) -distribution , every r ow ( 𝑃 1 and each 𝑃 ℓ ( 𝑎, ·) ) is an independent uniformly random 𝑠 -subset of [ 𝑚 ] . is distribution is conditionally ( 𝑠 / 𝑚 ) -spread. Over a distribution on inputs, a strategy may resolv e some constraint chains but not others; we measur e the probability of resolution. eorem 1 (Exp onential depth–width separation) . Let D be a conditionally 𝛾 -spread distribution over 𝑘 -position constraint chains with 𝛾 < 1 . (a) A sequential strategy with 𝑘 layers and width 1 resolves Spec cg with probability 1 . (b) For any 𝑑 ′ < 𝑘 , any deterministic 𝑑 ′ -layer strategy with width 𝑤 resolves Spec cg with probability at most 𝑤 · 𝛾 𝑘 − 𝑑 ′ . (c) Consequently , achieving constant resolution probability with 𝑑 ′ < 𝑘 layers requires width 𝑤 ≥ ( 1 / 𝛾 ) 𝑘 − 𝑑 ′ . For the random ( 𝑘 , 𝑚, 𝑠 ) -distribution, 𝛾 = 𝑠 / 𝑚 , giving width 𝑤 ≥ ( 𝑚 / 𝑠 ) 𝑘 − 𝑑 ′ . Part (a) is Observation 1 . Part (c) follo ws immediately from part (b), which we prove no w: 6 Proof of eorem 1 (b). Fix a deterministic 𝑑 ′ -layer strategy with width 𝑤 and layer assignment 𝑆 1 , . . . , 𝑆 𝑑 ′ . By Observation 2 , at least 𝑡 ≥ 𝑘 − 𝑑 ′ links ℓ 1 , . . . , ℓ 𝑡 are uninformed. Fix a single candidate tuple ( 𝑣 1 , . . . , 𝑣 𝑘 ) and process the uninformed links in order . At link ℓ 𝑗 , condition on all constraint data e xcept the row 𝑃 ℓ 𝑗 ( 𝑣 ∗ ℓ 𝑗 − 1 , · ) and on the outcomes at prior links. Under this conditioning, both 𝑣 ∗ ℓ 𝑗 − 1 (from an earlier layer or the conditioning) and 𝑣 ℓ 𝑗 (from the same or a later layer ) are xed, while the ro w 𝑃 ℓ 𝑗 ( 𝑣 ∗ ℓ 𝑗 − 1 , · ) remains random by the 𝛾 -spread condition. Hence Pr [ ( 𝑣 ∗ ℓ 𝑗 − 1 , 𝑣 ℓ 𝑗 ) ∈ 𝑃 ℓ 𝑗 | conditioning ] ≤ 𝛾 . Multiplying across all 𝑡 ≥ 𝑘 − 𝑑 ′ links gives success probability at most 𝛾 𝑘 − 𝑑 ′ per candidate; a union bound over 𝑤 candidates gives the result. Corollary 1 (Hard instances for any xed strategy) . For every 𝑘 , 𝑚, 𝑠 with 𝑚 ≥ 2 𝑠 , every 𝑑 ′ < 𝑘 , and every (possibly randomized) 𝑑 ′ -layer strategy with width 𝑤 , there exists a constraint chain on which the strategy resolves Spec cg with probability at most 𝑤 · ( 𝑠 / 𝑚 ) 𝑘 − 𝑑 ′ . eorem 1 (b) bounds the expected resolution probability of any deterministic strategy over the random ( 𝑘 , 𝑚, 𝑠 ) -distribution by 𝑤 · ( 𝑠 / 𝑚 ) 𝑘 − 𝑑 ′ , so for each deterministic strategy some chain achie ves this bound. Y ao’s minimax principle [ Y ao77 ] gives the same guarantee for randomized strategies. Orthogonality . Determination depth and computational depth are independent axes: the constrained generation task has determination depth 𝑘 and computational depth 𝑂 ( 1 ) per step, while any computa- tionally hard function has determination depth 0. Within a single problem, however , the two axes interact: Section 4.1 shows that enriching the basis trades determination layers for computational depth within layers, but the total sequential cost is bounded b elow (eorem 3 ). Remark 1 (Approximate determination and the exchange rate) . When a strategy tolerates constraint violations, the expected violation count is at least ( 𝑘 − 𝑑 ′ ) ( 1 − 𝛾 ) by eorem 1 ; for the random ( 𝑘 , 𝑚, 𝑠 ) - distribution this is tight. e parameter 𝛾 thus controls the exchange rate between distributional knowledge and sequential depth. For a distribution that is not 𝛾 -spread—one with exploitable correlations—a strategy can achieve p er-link success probability 𝛾 ′ > 𝛾 . At 𝛾 ′ = 1 (perfe ct prediction), no layers are neede d; at 𝛾 ′ = 𝛾 (no exploitable structure), the full 𝑘 layers are required. Remark 2 (Generality b eyond autoregressive generation) . e strategy model applies to any procedure operating in se quential rounds of parallel renement, including diusion models. A diusion model with 𝑇 denoising steps is a 𝑇 -round strategy; if 𝑇 < 𝑘 , the uninformed-link argument applies and the exponential lower bound holds with 𝑑 ′ = 𝑇 . 3.3 Distributed Extension and Pointer Chasing e constraine d generation task shares the same constraint-chain structure as 𝑘 -step p ointer chasing [ N W93 , PS84 ], diering only in sparsity 𝑠 , and plays the same role for determination complexity that pointer chasing plays for communication complexity . e following tradeo extends the width bound to a distributed seing where communication can substitute for width. Consider a 𝑘 -party communication model in which the constraint chain is distributed: player ℓ holds 𝑃 ℓ privately and sends 𝑏 ℓ bits to a central r eferee, who must output an admissible tuple. eorem 2 (Depth–width–communication tradeo ) . For the random ( 𝑘 , 𝑚, 𝑠 ) -distribution in the 𝑘 -party model, any 𝑑 -round protocol with width 𝑤 and p er-player communication 𝑏 ℓ bits satises log 𝑤 + Í ℓ ∈ 𝑈 𝑏 ℓ ≥ | 𝑈 | · log ( 𝑚 / 𝑠 ) , where 𝑈 is the set of uninformed links ( | 𝑈 | ≥ 𝑘 − 𝑑 ). Seing 𝑏 ℓ = 0 recovers the width bound; seing 𝑤 = 1 gives a communication lower bound. Both are tight. Proof in Appendix C.2 . 7 Observation 3 (Pointer chasing as degenerate determination) . At sparsity 𝑠 = 1 the spe cication is functional (determination depth 0); hardness appears only when constraints are distributed ( 𝑘 -step pointer chasing, Ω ( 𝑛 / 𝑘 ) -bit trade o [ N W93 ]). At 𝑠 ≥ 2 the specication is relational (determination depth 𝑘 ); the exponential width bound applies even to a centralized machine. e transition from polynomial to exponential occurs exactly at 𝑠 = 1 vs. 𝑠 ≥ 2—the boundary b etween functions and relations. 4 Structural Results e exponential separation shows that determination depth is a meaningful complexity measure with concrete consequences. W e now establish structural results that explain why the separation works and what it implies for the general the ory . e results in this section concern the interaction b etween determination depth and computational depth; communication re-enters in the distributed seing ( App endix B ). e conservation law (eorem 3 ) shows that enriching the commitment basis trades determination layers for computational depth within lay ers, but the total sequential cost is bounded below—explaining the tightness witnessed by the constrained generation task. e oracle characterization (Proposition 1 ) shows that under a commutative basis in the online seing, ev en a strategy with access to an unbounded disclosure oracle cannot reduce the number of layers b elow Depth Φ ( Spe c ) . Finally , stable matching (Proposition 2 ) witnesses the orthogonality of the two axes: computational cost is p olynomial regardless of determination depth. 4.1 Conservation of sequential depth A specication may have internal dep endency structure—a chain of commitments where each depends on the previous one ’s outcome—even in the oine seing, where for ward validity constraints are absent. e following theorem says this structure cannot be eliminated. eorem 3 (Conservation law for sequential depth) . Let Φ 0 be a constant-depth basis for Spec (Denition 11 ). For any basis Φ and any valid determination over Φ that resolves Spec in 𝑑 layers, let 𝑐 𝑖 be the circuit depth of layer 𝑖 (wher e the circuit’s input is the curr ent history , which includes all prior commitment events and their eects on the admissible set). en Í 𝑑 𝑖 = 1 ( 1 + 𝑐 𝑖 ) ≥ Depth Φ 0 ( Spe c ) . A richer basis can reduce the number of layers 𝑑 by bundling multiple constant-depth commitments into a single commitment, but evaluating that commitment requires circuit depth 𝑐 𝑖 that accounts for the bundled dependencies. e bound says the total sequential depth of commitment e valuation—summing ( 1 + 𝑐 𝑖 ) over layers—cannot fall below Depth Φ 0 ( Spe c ) . e b ound is not a counting identity: a richer basis could in principle exploit algebraic cancellations to resolve the specication in fewer total sequential steps than the longest dependency chain, and the theorem’s content is that no such shortcut exists. e constrained generation task (Section 3 ) witnesses tightness: for every split of the 𝑘 dependency links, the bound is achieved with equality ( Appendix C.2 ). e proof ( App endix C.1 ) traces the longest path in the Φ 0 -dependency DA G: each pair either crosses a layer boundary (contributing to 𝑑 ) or forces a circuit path within a layer ( contributing to 𝑐 𝑖 ). 4.2 Oracle characterization e conservation law b ounds the sequential depth of commitment evaluation in both the oine and online seings. What if the strategy itself has unbounded computational power between commitments? In the online seing, this question is non-trivial: Proposition 1 (Oracle p ower does not reduce depth) . Under a commutative basis Φ in the online seing, grant the strategy a free disclosure oracle call at the start of ev ery layer (a function that may inspect the 8 entire history , perform unbounded computation, and return any value). Any determination strategy that resolves Spec still requires at least Depth Φ ( Spe c ) layers. Under a commutative basis, the only constraint on co-applying commitments within a layer is for ward validity : their combined ee ct must preserve nonemptiness of the admissible set at all extensions. For ward validity is a property of the specication, not of the strategy’s knowledge, so no oracle can increase the number of commitments that t in one layer (Example 1 ). Commutativity is essential: without it, non-commuting commitments can be se quenced within a single layer , collapsing multiple algebraic layers. e conservation law and the oracle characterization are complementar y: the former bounds computa- tion within layers ( commitment evaluation), the laer bounds the number of layers themselves ( strategy computation between layers cannot reduce it). In the distribute d seing, the oracle characterization ser ves as a single-agent baseline: App endix B shows that asynchronous multi-agent systems cannot always achieve this baseline, recovering the Halp ern–Moses imp ossibility of common knowledge as a special case and providing quantitativ e depth predictions beyond the binary threshold. 4.3 Orthogonality: stable matching Within a single specication, the conser vation law couples determination and computational depth. Across specications, these are independent: neither subsumes the other . Stable matching witnesses this sharply . Given preference lists for two disjoint sets of 𝑛 agents, the spe cication maps the input to the set of all stable matchings [ GS62 ]. e structural theory of stable matchings is organized around rotations — cyclic reassignments that transform one stable matching into an adjacent one in the laice of stable matchings [ IL86 , GI89 ]. Rotations are partially order ed by precedence ( 𝜌 must be applied b efore 𝜌 ′ can be exposed); the resulting rotation poset encodes the determination structure. Proposition 2 (Stable matching depth) . For any stable matching instance , determination depth under the rotation basis equals the height of the rotation poset [ IL86 , GI89 ]. By the Ir ving–Leather realization theorem, every nite poset arises as a rotation poset, so every determination depth is achievable. Finding a stable matching is in P [ GS62 ], yet determination depth can be arbitrarily large: computational cost is polynomial regardless of determination depth. e Gale–Shapley algorithm nds a matching in 𝑂 ( 𝑛 2 ) steps but is not depth-optimal; a strategy that applies all e xposed rotations simultaneously at each layer achiev es depth equal to the poset height, recovering the parallel algorithm of Garg [ Gar20 ]. Proof and full development in Appendix D .1 . 5 Metacomplexity of Determination Depth e preceding sections establish determination depth as a complexity measure with concrete separations and structural laws. W e now ask: how hard is it to compute the determination depth of a given sp ecication? e answer ranges from NP-hard to PSP ACE-complete dep ending on the seing, with the p olynomial hierarchy arising as the exact hierar chy of determination depths. Proposition 3 (NP-hardness in the oine seing) . Computing determination depth is NP-hard in the oine seing: for decision tree synthesis, determination depth equals minimum tree depth, which is NP-hard to compute [ HR76 ]. Proof in Appendix E . Depth here r eects the tree structure: a depth- 𝑑 decision tree has 𝑑 levels of nodes, and each level is one r ound of variable-test commitments. In the online seing, an adversarial environment adds universal quantication—the environment xes some variables, the determiner xes others—and the metacomplexity captures the full polynomial hierarchy . 9 eorem 4 (Determination depth captures the polynomial hierarchy ) . Given a Boolean formula 𝜃 ( 𝑥 1 , . . . , 𝑥 𝑛 ) encoded by a p olynomial-size circuit, the outcome set is 𝑂 = { 0 , 1 } 𝑛 (all variable assignments) and the initial admissible set is Spec ( 𝐻 0 ) = 𝑂 . e commitment basis consists of pointwise lters “set 𝑥 𝑖 = 𝑏 ”; the environment xes some variables adversarially , the determiner xes the rest. Aer all 𝑛 variables are xed, the strategy succeeds i the resulting assignment satises 𝜃 . (i) For each xed 𝑘 , “is determination depth ≤ 𝑘 ?” is Σ 𝑃 2 𝑘 -complete. (ii) For unbounded 𝑘 (given as input), the problem is PSP A CE-complete. Proof sketch. Upper bound. e determiner guesses 𝑘 rounds to control; the environment controls the remaining 𝑛 − 𝑘 . Each round, the controlling player xes one variable. e determiner’s choices are existential quantiers, the environment’s are adversarial (universal). In the worst case these alternate ( ∃∀ ∃∀ · · · ), and consecutive same-player rounds collapse , giving at most 2 𝑘 alternations ( Σ 𝑃 2 𝑘 ); for unbounded 𝑘 , PSP ACE. Hardness. Re duce from Σ 2 𝑘 -QBF for xed 𝑘 , or from TQBF for unbounded 𝑘 . Construct a specication with 2 𝑘 variables: at odd rounds the determiner xes an existential variable, at even rounds the environment xes a universal variable. e admissible set starts as 𝑂 ; each round narrows it by xing one variable. e determiner has a depth- 𝑘 strategy i the corresponding QBF is true . Full proof in Appendix E , where the game is formalized as a QBF instance with variable-xing commitments. e p olynomial hierarchy is thus precisely the hierarchy of determination depths for circuit-encoded specications. (Oine, the metacomplexity is NP-hard but does not capture the full hierarchy; see Proposi- tion 3 .) Where cir cuit depth and communication complexity parameterize the PH thr ough computational resources, determination depth parameterizes it through a semantic one—layers of irre vocable choice. e correspondence is not the contribution; it is a calibration. An alternating Turing machine has a xed quantier prex determined by the program; here the determiner optimizes which variables to control and in which order , so the quantier schedule is itself part of the problem. More fundamentally , the pap er’s main results—the e xponential separation (eorem 1 ) and the conservation law (eorem 3 )—ar e oine results about depth–width and depth–computation tradeos within a single sp ecication, a regime not captured by the quantier alternation that A TMs model. 6 Conclusion Complexity theor y has measured the cost of computing a uniquely determined answer . is paper introduces determination depth to measure the cost of commiing to an answer when many ar e admissible—a semantic complexity measure orthogonal to computation. e exponential separation (eorem 1 ) shows this cost is real, the thr ee-way tradeo (eorem 2 ) shows depth, width, and communication are fungible , the PH characterization (eorem 4 ) shows it is well-calibrated against the classical hierarchy . In the distributed seing, the framework recov ers the Halpern–Moses imp ossibility of asynchronous common knowledge as a special case (e orem 5 ), and the conservation law provides quantitative depth–cost predictions beyond the binary threshold. e app endices ground the framework in BSP round comple xity , chain-of-thought reasoning, stable matching, extensive-form games, and distributed graph coloring; App endix F sur veys related work in detail. A numb er of directions remain open: Distributional vs. worst-case hardness. e exponential separation is distributional: for any single xed chain, a strategy with full knowledge resolves in one layer . Characterizing which restrictions on the strategy’s knowledge or power yield worst-case har dness is open. Fault tolerance and randomization. In every deterministic, fault-free multi-party seing we have examined, the spe cication’s semantic structure fully accounts for the known round complexity via determination depth ( App endices D.4 – D .4 ). e scop e of this correspondence is open: fault-tolerant seings 10 and randomized protocols lie outside the current framework, though the Lov ´ asz Local Lemma already shows that randomization can collapse determination depth in some seings ( App endix D .4 ). Reversible commitments. e framework assumes commitments are irre vocable. In many seings— autoregressiv e generation with backtracking, transactional rollback, spe culative execution—commitments can be reversed at a cost. Extending the commitment algebra to include inverses (a group rather than a monoid) and characterizing the resulting depth–cost tradeos is open. Prov enance of determinations. Classical data provenance explains quer y answers via a commutative semiring over monotone derivations [ GKT07 ]. 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IEEE, 1977. 13 A Formal Denitions for Determination Depth is appendix colle cts the formal denitions deferred from Se ction 2 ; no new results are claimed. Fix a commitment basis Φ and a specication Spec throughout. e goal is to characterize the parallelism available in a determination 𝐷 ( 𝐻 ) (the commitment sub- sequence of a history 𝐻 ). In the oine seing, all commitment events are consecutive (no environment events interleav e), so the entire determination can be analyzed as a single sequence: adjacent commitments that commute can b e parallelized into layers, and the depth of 𝐷 ( 𝐻 ) is the minimum numb er of such lay ers. In the online seing, environment events may separate conse cutive commitments, and commutation is only meaningful for commitments that share the same history prex. e layering then applies to each maximal run of consecutive commitments (between successive environment e vents), and the depth of 𝐷 ( 𝐻 ) is the total number of layers across all such runs. Depth Φ ( Spe c ) is dened (Denition 6 ) as the minimum worst-case depth ov er all strategies that resolve Spec . A.1 Commuting Layers and Depth W e extend the · notation to multisets: 𝐻 · 𝐿 is the histor y obtained by appending all commitments in 𝐿 aer 𝐻 (in any order). Denition 18 (Commuting layer) . A nite multiset 𝐿 of commitments from a common basis Φ is a commuting layer for Spec at 𝐻 if applying its commitments aer 𝐻 is order-independent: for any two listings 𝜓 1 , . . . , 𝜓 𝑡 and 𝜓 ′ 1 , . . . , 𝜓 ′ 𝑡 of the elements of 𝐿 (respecting multiplicity), Spec ( 𝐻 · 𝜓 1 · 𝜓 2 · · · 𝜓 𝑡 ) = Spe c ( 𝐻 · 𝜓 ′ 1 · 𝜓 ′ 2 · · · 𝜓 ′ 𝑡 ) . Pairwise commutation (Denition 2 ) at 𝐻 is sucient; every basis used in this paper satises this condition. Denition 19 (Layering and determination depth) . Let 𝐷 ( 𝐻 ) = 𝜑 1 · · · · · 𝜑 𝑚 be the determination of a history 𝐻 over Φ . A run is a maximal subsequence of consecutive commitment events in 𝐻 (with no intervening environment events). A layering of a run 𝑅 = 𝜓 1 · · · · · 𝜓 𝑟 is a partition into nonempty multisets 𝐿 1 , . . . , 𝐿 𝑘 such that each 𝐿 𝑖 is a commuting layer for Spec at the history 𝐻 0 · 𝐿 1 · · · 𝐿 𝑖 − 1 , where 𝐻 0 is the history prex immediately before the run. e depth of a run is the minimum 𝑘 over all its layerings. e depth of 𝐷 ( 𝐻 ) , wrien depth ( 𝐷 ( 𝐻 ) ) , is the sum of the depths of its runs. In the oine seing, the entire determination is a single run. For a single run, we write 𝐿 1 ⊲ 𝐿 2 ⊲ · · · ⊲ 𝐿 𝑘 to denote the layered determination; the ⊲ notation makes the layering (and hence depth) visible. Ever y run admits a depth-minimal layering; in the oine seing (a single run), we call this the layered normal form . Depth is bounded by cost ( depth ( 𝐷 ) ≤ cost ( 𝐷 ) , since layers are nonempty ), but can b e much smaller when most commitments commute. A.2 Dependency Chains and Depth Characterization T o lower-bound depth, we exhibit commitments that cannot share a layer . Denition 20 (Forced dep endency) . Let 𝜑 and 𝜓 be commitment events in a determination 𝐷 ( 𝐻 ) . W e say 𝜓 locally depends on 𝜑 (in 𝐻 ) if, in every layering of 𝐷 ( 𝐻 ) , 𝜑 appears in a strictly earlier layer than 𝜓 . W e say 𝜓 universally depends on 𝜑 if this holds in ev ery resolving history whose determination contains both (use d only in the distributed seing, A ppendix B ). In practice, forced dependency arises when 𝜓 acts as the identity (or violates validity ) until 𝜑 has be en applied. e stable matching application (Section D ) exhibits this paern. 14 Remark 3 (Online depth and forced dependency) . e oine results (Proposition 4 below) use local forced dependency . e distributed results (eorem 5 ) use universal forced dependency . In the online seing, depth may exceed the longest dep endency chain: run b oundaries force additional layers even b etween independent commitments. e oracle characterization (Proposition 1 ) handles the online seing directly . Denition 21 (Dependency chain) . A sequence 𝜑 1 , 𝜑 2 , . . . , 𝜑 𝑘 is a dependency chain of length 𝑘 if each 𝜑 𝑖 + 1 locally depends on 𝜑 𝑖 . Proposition 4 (Depth equals longest dependency chain (oine)) . In the oine seing, Depth Φ ( Spe c ) equals the maximum length of a dependency chain ov er all resolving determinations. In the online seing, the longest chain is a lower bound on depth; run boundaries may force additional layers. Proof. A dependency chain of length 𝑘 requires 𝑘 layers ( each successive commitment must appear in a strictly later layer), so Depth Φ ( Spe c ) ≥ the longest chain in both seings. In the oine seing (a single run), any determination induces a dependency D AG on its commitments (draw an e dge 𝜑 → 𝜓 whenever 𝜓 depends on 𝜑 ). A depth-minimal lay ering corresponds to a minimum-height topological layering of this D A G, whose height e quals the length of the longest chain. B Distributed Determinations and Common Knowledge is appendix extends the oracle characterization (Section 4.2 ) to the distributed seing, connecting determination depth to synchronization and common knowledge. Distributed setting. W e extend the frame work to multiple agents. Fix a nite set of agents { 1 , . . . , 𝑛 } . Each event in a history 𝐻 is associated with an agent; for agent 𝑝 , the projection 𝐻 | 𝑝 is the subgraph of 𝐻 induced on 𝑝 ’s events—local computation steps, message sends, and message receives at 𝑝 —with only the edges of 𝐻 whose both endpoints are events at 𝑝 . (Causal paths through other agents’ e vents ar e not visible in the projection; this is the standard Lamp ort projection.) e partial order on 𝐻 | 𝑝 is the transitive closure of → restricted to 𝑝 ’s events: 𝑒 1 precedes 𝑒 2 in 𝐻 | 𝑝 whenever 𝑒 1 → ∗ 𝑒 2 in 𝐻 and both are events at 𝑝 . e frontier of 𝐻 | 𝑝 is the set of sinks of 𝐻 | 𝑝 (e vents with no successors under → ). Indistinguishability . In the Halp ern–Moses framework [ HM90 ], two global states are indistinguishable to agent 𝑝 if 𝑝 ’s local state is the same in b oth. In our formalism, the analogue of local state is the projection 𝐻 | 𝑝 . T wo global histories 𝐻 , 𝐻 ′ are 𝑝 -indistinguishable , wrien 𝐻 ∼ 𝑝 𝐻 ′ , if 𝐻 | 𝑝 and 𝐻 ′ | 𝑝 are isomorphic as partial orders of typed events—that is, there exists an order-preserving bijection b etween the events of 𝐻 | 𝑝 and 𝐻 ′ | 𝑝 that preserves event types (message contents, commitment values, and local computation steps). Local knowledge and mutual knowledge. A global property 𝑆 (a set of histories) is known by agent 𝑝 at 𝐻 if 𝑆 holds at every histor y 𝑝 -indistinguishable from 𝐻 : 𝐾 𝑝 ( 𝑆 , 𝐻 ) ⇔ ∀ 𝐻 ′ : 𝐻 ∼ 𝑝 𝐻 ′ ⇒ 𝐻 ′ ∈ 𝑆 . 𝑆 is known to ev eryone at 𝐻 if 𝐾 𝑝 ( 𝑆 , 𝐻 ) for every 𝑝 ; write 𝐸 ( 𝑆 , 𝐻 ) for this. Dene 𝑘 -th order mutual knowledge inductively: 𝐸 0 ( 𝑆 , 𝐻 ) holds i 𝐻 ∈ 𝑆 ; 𝐸 𝑘 + 1 ( 𝑆 , 𝐻 ) holds i for ev ery agent 𝑝 and every 𝐻 ′ with 𝐻 ∼ 𝑝 𝐻 ′ , 𝐸 𝑘 ( 𝑆 , 𝐻 ′ ) holds. Common knowledge of 𝑆 at 𝐻 means 𝐸 𝑘 ( 𝑆 , 𝐻 ) for all 𝑘 ≥ 0. Synchronization p oints. W e assume each agent’s projection 𝐻 | 𝑝 is a chain (a total order on 𝑝 ’s events); this holds whenever each agent executes sequentially , as in standard distributed computing models. A determination strategy may invoke multiple synchr onization points in sequence. e 𝑗 -th synchro- nization point is a set of distinguished events { 𝑒 ∗ 𝑗 , 1 , . . . , 𝑒 ∗ 𝑗 , 𝑛 } , one p er agent, satisfying three conditions: (i) 𝑒 ∗ 𝑗 ,𝑝 is in the projection of 𝑝 for each 𝑝 ; (ii) the events form a consistent cut : no 𝑒 ∗ 𝑗 ,𝑝 causally follows any 15 𝑒 ∗ 𝑗 , 𝑞 for 𝑝 ≠ 𝑞 (so the global history truncated to the cut is well-dened); and (iii) the admissible set at the cut is the same as seen by each agent— Spec ( 𝐻 ≤ 𝑗 ) is independent of which agent’s persp ective is used, where 𝐻 ≤ 𝑗 is the global history through the cut. At a synchronization point, ev ery agent sees the same admissible set (condition (iii)), and each agent knows the synchronization occurred ( condition (i): 𝑒 ∗ 𝑗 ,𝑝 is in its projection). Lemma 1 (Synchronization establishes common kno wledge) . Assume the synchronization protocol ( con- ditions (i)–(iii)) is common knowledge among all agents. en at each synchronization point, the current admissible set is common knowledge. Proof. Condition (iii) gives 𝐸 ( 𝑆 , 𝐻 ) ; common knowledge of the protocol iterates this to all levels. Connection to the oracle framework. e oracle characterization (Section 4.2 ) assumes a single global oracle that sees the entire history . In the distributed seing, each agent has access only to a local oracle that sees its projection 𝐻 | 𝑝 . A synchronization round ( conditions (i)–(iii)) is precisely the mechanism that elevates local oracles to global ones: aer synchronization, every agent’s local state includes the same admissible set (by condition (iii) and Lemma 1 ), so a lo cal oracle can determine the admissible set without global access. Between synchr onization points, local oracles are strictly weaker—each sees only its own projection. Proposition 5 (Determination depth equals synchronization points) . In the online seing under a com- mutative basis Φ with 𝑛 ≥ 2 agents, if every layer b oundary involves a cross-agent forced dependency (Denition 20 ), then Depth Φ ( Spe c ) equals the minimum number of synchronization p oints ne eded to resolve Spec . When some layers inv olve only local dependencies, the minimum number of synchronization points may be smaller than Depth Φ ( Spe c ) , but Depth Φ ( Spe c ) synchronization points always suce. Proof. Lower b ound. Between tw o consecutive synchronization points, each agent acts on its projection 𝐻 | 𝑝 (its local events, r eceived messages, and environment inputs). Within a single layer , an agent can safely apply its own commitments without synchronization: by denition, the commitments in a layer commute and are jointly valid, so each agent’s share can be applied independently . e constraint arises at layer boundaries. By the cross-agent assumption, each layer boundary has a commitment in layer 𝑖 + 1 at some agent 𝑞 that universally depends (Denition 20 ) on a commitment in layer 𝑖 at a dierent agent 𝑝 . Without a synchronization point, 𝑞 cannot verify that 𝑝 has completed its layer- 𝑖 commitment: 𝑞 ’s projection may be consistent with histories in which 𝑝 has not yet acted. Applying a layer- ( 𝑖 + 1 ) commitment in such a history risks invalidity . A synchronization point establishes common knowledge that layer 𝑖 is complete, enabling all agents to pr ocee d to layer 𝑖 + 1. Hence each inter-synchr onization interval accomplishes at most one layer , and at least Depth Φ ( Spe c ) synchronization points are needed. Upper bound. A synchronous protocol in which all agents exchange pr ojections at each synchronization point can simulate the oracle: the combined projections reconstruct the global history , and the strategy selects the next layer’s commitments. At the synchronization point, every agent knows the global histor y (from the exchange), kno ws that every other agent knows it (the exchange was simultaneous), and so on at all levels—establishing common kno wledge of the current state. Hence Depth Φ ( Spe c ) synchronization points suce. e following theorem recovers the Halp ern–Moses impossibility of asynchronous common knowl- edge [ HM90 ] as a consequence of the determination framework, via an independent proof that uses only histories, projections, and universal forced dependency . eorem 5 (A synchronous impossibility via determination) . Let H async be an asynchronous history class: for every pair of agents 𝑝 , 𝑞 and every history 𝐻 ∈ H async containing an ev ent 𝑒 at 𝑝 that is not in the projection 16 of 𝑞 , there exists 𝐻 ′ ∈ H async with 𝐻 ∼ 𝑞 𝐻 ′ in which 𝑒 has not occurred. If Spec has a cross-agent univ ersal forced dependency—a commitment 𝜑 at some agent 𝑝 and a commitment 𝜓 at a distinct agent 𝑞 such that 𝜓 universally depends on 𝜑 (Denition 20 )—then no determination strategy over a commutative basis can resolve Spec over H async . Proof. Let 𝜑 at agent 𝑝 and 𝜓 at agent 𝑞 ≠ 𝑝 be a cross-agent universal for ced dependency . Consider any history 𝐻 in which 𝜑 has been applied at 𝑝 . Since 𝜑 is an event at 𝑝 and is not in the pr ojection of 𝑞 , the asynchronous condition guarantees a history 𝐻 ′ ∈ H async with 𝐻 ∼ 𝑞 𝐻 ′ in which 𝜑 has not occurred. At 𝐻 ′ , 𝜓 is invalid: by universal forced dependency , 𝜓 acts as the identity or violates validity without the prior eect of 𝜑 . Since 𝑞 cannot distinguish 𝐻 from 𝐻 ′ , any deterministic strategy that applies 𝜓 at 𝐻 also applies it at 𝐻 ′ , producing an invalid determination. Hence no strategy can apply both 𝜑 and 𝜓 without a synchronization point between them, and Spec is not resolvable over H async . eorem 5 —like Halp ern-Moses—is a qualitative threshold: synchronization is needed or not. Determination depth renes this to a cost measure: Proposition 5 says at most 𝑘 synchronization points are needed for depth 𝑘 (e xactly 𝑘 when every layer boundary involves a cross-agent dependency ), and the conservation law (eorem 3 ) ensures the total se quential cost cannot be eliminated by enriching the basis. More broadly , the determination framework applies beyond the distributed seing—to single-agent oine problems, combinatorial structures, and autoregr essive generation—making the distributed recovery one instantiation of a domain-independent measure. Corollary 2 (Common knowledge assumptions collapse depth) . If common knowledge of a property 𝑃 is assume d as part of the mo del (i.e., 𝑃 holds at every histor y in H and this is common knowledge among all agents), and establishing 𝑃 through synchronization w ould require 𝑗 layers, then the assumption r educes Depth Φ ( Spe c ) by 𝑗 . is unies the diagnostic observations: • BSP assumes common kno wledge of membership (the process set is xed and globally known), saving 1 layer (Section D .4 ). • In extensive-form games , a dominant-strategy me chanism makes every node subgame-trivial, reducing the strategic depth to 0 (Section D.2 ). • LOCAL assumes common knowledge of unique identiers, collapsing determination depth to 1 (Section D.4 ). In each case, the standard model silently enriches the commitment basis by assuming common knowledge of a property whose establishment would otherwise cost determination layers. C Proofs for Conser vation and Tightness C.1 Conservation of sequential depth (eorem 3 ) Let Φ 0 be a constant-depth basis for Spec (Denition 11 ), and let 𝐺 Φ 0 be its forced-dep endency DA G (Denition 20 ). Each edge 𝜑 → 𝜓 in 𝐺 Φ 0 has a concrete witness exclusion : a triple ( 𝐻 , 𝐻 · 𝜑 , 𝑜 ) such that excluding outcome 𝑜 at 𝐻 is invalid, but excluding 𝑜 at 𝐻 · 𝜑 is valid—the dependency is witnessed by a specic outcome that cannot be excluded until 𝜑 has taken eect. Any basis Φ that resolves Spec must also exclude each witness outcome 𝑜 ; the conser vation law measures how Φ pays for these exclusions. Observation 4 (Semantic dependency implies circuit dependency) . Let 𝜑 𝑗 and 𝜑 𝑗 + 1 be commitments from a constant-depth basis Φ 0 such that 𝜑 𝑗 + 1 is forced to depend on 𝜑 𝑗 . If a circuit 𝐶 computes a valid layer of a determination that achieves a witness exclusion of both 𝜑 𝑗 and 𝜑 𝑗 + 1 , then 𝐶 contains a directed path from the gates computing 𝜑 𝑗 ’s witness exclusion to the gates computing 𝜑 𝑗 + 1 ’s witness exclusion. 17 Proof. By forced dependency , there exist instances where 𝜑 𝑗 + 1 ’s exclusion is invalid unless 𝜑 𝑗 ’s exclusion has already occurred. If 𝐶 contained no path from 𝜑 𝑗 ’s gates to 𝜑 𝑗 + 1 ’s, then 𝜑 𝑗 + 1 ’s output would be independent of 𝜑 𝑗 ’s, and on such an instance the circuit would pr oduce an invalid exclusion—a contradiction. eorem 3 (Conservation law , restated) . For any basis Φ and any valid determination over Φ that resolves Spec in 𝑑 layers with per-layer circuit depths 𝑐 1 , . . . , 𝑐 𝑑 , Í 𝑑 𝑖 = 1 ( 1 + 𝑐 𝑖 ) ≥ 𝑑 ∗ , where 𝑑 ∗ = Depth Φ 0 ( Spe c ) is the determination depth under a constant-depth basis Φ 0 . Proof of eorem 3 . Let 𝜑 1 → · · · → 𝜑 𝑝 be a longest path in 𝐺 Φ 0 , so 𝑝 = 𝑑 ∗ (Corollary 4 ). Each edge 𝜑 𝑗 → 𝜑 𝑗 + 1 has a witness exclusion ( 𝐻 𝑗 , 𝐻 𝑗 · 𝜑 𝑗 , 𝑜 𝑗 ) : outcome 𝑜 𝑗 cannot be excluded at 𝐻 𝑗 until 𝜑 𝑗 has taken eect. Fix any determination 𝐷 under basis Φ with 𝑑 layers and per-layer circuit depths 𝑐 1 , . . . , 𝑐 𝑑 . Since 𝐷 resolves Spec , each 𝑜 𝑗 is eventually excluded; let 𝜆 ( 𝑗 ) be the Φ -layer in which 𝑜 𝑗 is rst excluded. e richer basis Φ may exclude multiple witness outcomes within a single lay er . For each consecutive pair 𝜑 𝑗 → 𝜑 𝑗 + 1 in the path ( 𝑗 = 1 , . . . , 𝑝 − 1), exactly one of the following holds. Case 1: 𝜆 ( 𝑗 ) < 𝜆 ( 𝑗 + 1 ) . e predecessor’s ee ct is achieved in a strictly earlier Φ -layer . is pair contributes at least 1 to 𝑑 . Case 2: 𝜆 ( 𝑗 ) = 𝜆 ( 𝑗 + 1 ) = 𝑖 . Both ee cts ar e achiev ed in the same Φ -layer 𝑖 . By Obser vation 4 , the circuit computing Φ -layer 𝑖 ’s commitments contains a directed path from the gates for 𝜑 𝑗 ’s exclusion to those for 𝜑 𝑗 + 1 ’s, contributing at least 1 to 𝑐 𝑖 . Case 3: 𝜆 ( 𝑗 ) > 𝜆 ( 𝑗 + 1 ) . e successor’s eect is achieved in an earlier Φ -layer than the predecessor’s. en 𝜑 𝑗 + 1 ’s exclusion is valid without 𝜑 𝑗 ’s prior exclusion, contradicting forced dep endency . Hence Case 3 does not arise. Each of the 𝑝 − 1 conse cutive pairs contributes at least 1 to either 𝑑 (Case 1: a layer boundar y) or some 𝑐 𝑖 (Case 2: a circuit path within a layer), and each layer visite d by the path contributes its 1 term. Hence Í 𝑑 𝑖 = 1 ( 1 + 𝑐 𝑖 ) ≥ 𝑝 = 𝑑 ∗ . Constrained generation instantiation. e conser vation law applies directly to the constrained gen- eration task (Se ction 3.1 ). e outcome space is a product 𝑂 = [ 𝑚 ] 𝑘 , and the constant-depth basis Φ 0 is the coordinate basis: each commitment “x 𝑣 ℓ = 𝑣 ” selects a value for one position, which is computable in 𝑂 ( 1 ) circuit depth. e dependency DA G 𝐺 Φ 0 is the chain 1 → 2 → · · · → 𝑘 , since the feasible values at position ℓ depend on the value chosen at position ℓ − 1 (via the successor set 𝑃 ℓ ( 𝑣 ℓ − 1 , · ) ). In Case 2 of the conservation-law proof, this dep endency becomes a literal circuit data path: the sub-circuit computing 𝑣 ℓ 𝑗 + 1 must have 𝑣 ℓ 𝑗 on its input path, because the successor sets 𝑃 ℓ 𝑗 + 1 ( 𝑎, ·) dier across predecessor values 𝑎 . Case 3 cannot arise for the same reason: 𝑣 ℓ 𝑗 + 1 ’s feasibility depends on 𝑣 ℓ 𝑗 ’s value, so the successor’s exclusion cannot be valid before the predecessor’s. e constrained generation task is oine : the constraint chain is fully given, so there is one complete history with no extensions. is rules out branch-sp ecic conditioning—the strategy cannot make dierent choices for dierent futures, because there are no branches. e following remark shows why this maers. Remark 4 (Branch-specic commitments can bypass conser vation) . In the online seing, a basis that allows branch-specic commitments—whose eects depend on which future history materializes—can bypass the conservation bound. In the three-valued consensus of Example 1 (an online specication), Depth Φ ( Spe c ) = 2 (under the atomic basis), but a single non-pointwise commitment can cho ose dierently per branch (keep 𝑏 at 𝐻 1 , 𝑐 at 𝐻 2 , 𝑐 at 𝐻 3 ), resolving the specication in one layer with 𝑂 ( 1 ) circuit depth. e oine product-space seing of the constrained generation task rules out this escap e: all commitments are applie d aer the history is xe d, and the coordinate basis is commutative, so no branch-specic conditioning is possible. 18 C.2 Tightness of the conservation law and the three-way tradeo W e rst prov e the three-way tradeo, then the conser vation tightness. eorem 2 (Depth–width–communication tradeo, restated) . For the random ( 𝑘 , 𝑚, 𝑠 ) -distribution in the 𝑘 -party model, any 𝑑 -round protocol with width 𝑤 and per-player communication 𝑏 ℓ bits satises log 𝑤 + Í ℓ ∈ 𝑈 𝑏 ℓ ≥ | 𝑈 | · log ( 𝑚 / 𝑠 ) , where 𝑈 is the set of uninforme d links ( | 𝑈 | ≥ 𝑘 − 𝑑 ). Proof of eorem 2 . Fix a 𝑑 -round protocol with width 𝑤 and layer assignment 𝑆 1 , . . . , 𝑆 𝑑 . Let 𝑈 b e the set of uninformed links. Index the uninformed links as ℓ 1 , . . . , ℓ 𝑡 ; Observation 2 tells us 𝑡 ≥ 𝑘 − 𝑑 . Message-decoding lemma. Let 𝑅 ⊆ [ 𝑚 ] be a uniformly random 𝑠 -element subset, let 𝑀 = 𝑀 ( 𝑅 ) be a 𝑏 -bit message (a deterministic function of 𝑅 ), and let 𝑔 be any deco der that outputs an element 𝑔 ( 𝑀 ) ∈ [ 𝑚 ] . en Pr [ 𝑔 ( 𝑀 ) ∈ 𝑅 ] ≤ min ( 1 , 2 𝑏 · 𝑠 / 𝑚 ) . Proof: Count pairs ( 𝑅 , 𝜇 ) with 𝜇 = 𝑀 ( 𝑅 ) and 𝑔 ( 𝜇 ) ∈ 𝑅 . For each of the at most 2 𝑏 message values 𝜇 , the decoder outputs a xe d element 𝑔 ( 𝜇 ) . e number of 𝑠 -subsets containing 𝑔 ( 𝜇 ) is 𝑚 − 1 𝑠 − 1 . Hence the total number of good pairs is at most 2 𝑏 · 𝑚 − 1 𝑠 − 1 . Dividing by the total 𝑚 𝑠 subsets gives Pr [ 𝑔 ( 𝑀 ) ∈ 𝑅 ] ≤ 2 𝑏 · 𝑚 − 1 𝑠 − 1 / 𝑚 𝑠 = 2 𝑏 · 𝑠 / 𝑚 . T aking the minimum with 1 gives the claim. Per-link bound. Process the uninformed links in order ℓ 1 , . . . , ℓ 𝑡 . At step 𝑗 , condition on all constraint functions ( 𝑃 1 , . . . , 𝑃 𝑘 ) except the row 𝑃 ℓ 𝑗 ( 𝑣 ℓ 𝑗 − 1 , · ) , on the outcomes at links ℓ 1 , . . . , ℓ 𝑗 − 1 , and on all messages from players other than ℓ 𝑗 . Under this conditioning, 𝑣 ℓ 𝑗 − 1 is xed (from earlier layers or the same layer), 𝑣 ℓ 𝑗 is a deterministic function of player ℓ 𝑗 ’s message, and 𝑃 ℓ 𝑗 ( 𝑣 ℓ 𝑗 − 1 , · ) remains a uniformly random 𝑠 -subset (by Denition 17 ). Applying the message-decoding lemma with 𝑏 = 𝑏 ℓ 𝑗 (wher e 𝑣 ℓ 𝑗 is determined by player ℓ 𝑗 ’s message and the xed side information under the current conditioning), the conditional success probability at link ℓ 𝑗 is at most min ( 1 , 2 𝑏 ℓ 𝑗 · 𝑠 / 𝑚 ) . Combining. By the chain rule across all uninformed links, the success probability of a single candidate is at most Î ℓ ∈ 𝑈 min ( 1 , 2 𝑏 ℓ · 𝑠 / 𝑚 ) . A union bound ov er 𝑤 candidates gives total success probability at most 𝑤 · Î ℓ ∈ 𝑈 min ( 1 , 2 𝑏 ℓ · 𝑠 / 𝑚 ) . For the strategy to succeed with positive probability , this upper bound must be at least 1. T aking logarithms: log 𝑤 + ℓ ∈ 𝑈 min ( 0 , 𝑏 ℓ − log ( 𝑚 / 𝑠 ) ) ≥ 0 . Since min ( 0 , 𝑥 ) ≤ 𝑥 , 0 ≤ log 𝑤 + ℓ ∈ 𝑈 min ( 0 , 𝑏 ℓ − log ( 𝑚 / 𝑠 ) ) ≤ log 𝑤 + ℓ ∈ 𝑈 ( 𝑏 ℓ − log ( 𝑚 / 𝑠 ) ) , which rearranges to log 𝑤 + Í ℓ ∈ 𝑈 𝑏 ℓ ≥ | 𝑈 | · log ( 𝑚 / 𝑠 ) . eorem 6 (Tightness of the conservation law) . For the 𝑘 -position constrained generation task (Denition 13 ) with 𝑚 ≥ 2 𝑠 , every determination using 𝑑 layers satises Í 𝑑 𝑖 = 1 ( 1 + 𝑐 𝑖 ) ≥ 𝑘 , and this bound is achieved with equality for every 𝑑 ∈ { 1 , . . . , 𝑘 } . Proof. Lower b ound. e dependency DA G for the constrained generation task is the chain 1 → 2 → · · · → 𝑘 (each p osition’s feasibility dep ends on the previous value via 𝑃 ℓ ( 𝑣 ℓ − 1 , · ) ), which has longest path 𝑘 . By eorem 3 , Í 𝑑 𝑖 = 1 ( 1 + 𝑐 𝑖 ) ≥ 𝑘 for any 𝑑 -layer determination. Upper bound. For any 𝑑 ∈ { 1 , . . . , 𝑘 } , partition the 𝑘 levels into 𝑑 contiguous blocks 𝐵 1 , . . . , 𝐵 𝑑 of sizes ⌊ 𝑘 / 𝑑 ⌋ or ⌈ 𝑘 / 𝑑 ⌉ . In layer 𝑖 , commit to all positions in 𝐵 𝑖 by computing them sequentially: given the value 𝑣 ℓ − 1 from the pr evious position (either from an earlier layer or fr om earlier in the same lay er’s computation), look up any 𝑣 ℓ ∈ 𝑃 ℓ ( 𝑣 ℓ − 1 , · ) . Each lookup has circuit depth 𝑂 ( 1 ) (it is a table scan of 𝑃 ℓ ), and the | 𝐵 𝑖 | lookups within layer 𝑖 are chained, giving 𝑐 𝑖 = | 𝐵 𝑖 | − 1. e total is Í 𝑑 𝑖 = 1 ( 1 + ( | 𝐵 𝑖 | − 1 ) ) = Í 𝑑 𝑖 = 1 | 𝐵 𝑖 | = 𝑘 . 19 D Applications to Classical Problems is section demonstrates the breadth of the determination framew ork by applying it to problems acr oss several domains: stable matching, extensive-form games, chain-of-thought reasoning, and distributed round complexity (BSP and LOCAL). In each case, the framework either recovers a known result with a new explanation or reveals structur e that existing models do not capture. Determination depth arises for two reasons across these examples: enablement , where each commitment creates the sub-problem for the next layer (constrained generation, e xtensive-form games); and interference , wher e commitments that are individually valid conict when applied in the same layer (stable matching under the rotation basis). Each example connects to known results in its domain: stable matching recovers Garg’s parallel algorithm [ Gar20 ] and witnesses the orthogonality of determination and computational depth (nding a stable matching is in P , yet determination depth can be arbitrarily large); extensiv e-form games measure the game ’s strategic depth—the moves wher e a player must br eak a tie among equally-optimal options— connecting to Selten’s trembling-hand renement [ Sel75 ]; chain-of-thought reasoning decomposes Co T length into determination depth, computational depth, and architectural overhead, explaining why longer chains sometimes degrade [ SMA + 25 , SZW + 25 ]; and BSP/LOCAL expose hidden modeling assumptions ( Ameloot et al. ’s non-obliviousness [ ANdB13 ]) that silently collapse determination layers. e examples progress from a purely oine seing (stable matching) through online single-agent generation (chain-of-thought) to adversarial online interaction (extensive-form games) and distribute d multi-agent computation (BSP and LOCAL), illustrating how determination depth provides exact diagnostics across increasing envir onmental complexity . D .1 Stable Matching and Determination Universality Stable matching is a canonical relational specication: given pr eference lists for two disjoint sets of 𝑛 agents (encoded as environment e vents in an oine history ), the specication Spec SM maps each history to the set of all stable matchings of the encoded instance [ GS62 ]. e number of stable matchings ranges from 1 to exponentially many , and the specication is inherently relational whenever mor e than one exists. e analysis below is in the oine seing: all preference lists are given and the spe cication has no extensions. W e show that determination depth for stable matching is exactly characterize d by a classical combinatorial invariant—the height of the rotation poset—and that stable matching is universal for determination depth: every nite depth arises as the rotation-poset height of some stable matching instance. D .1.1 Rotations as commitments e structural theory of stable matchings is organize d around rotations [ IL86 , GI89 ]. A rotation is a cy clic reassignment of partners that transforms one stable matching into an adjacent one in the laice of stable matchings. Formally , a r otation 𝜌 = ( 𝑎 0 , 𝑏 0 ) , ( 𝑎 1 , 𝑏 1 ) , . . . , ( 𝑎 𝑟 − 1 , 𝑏 𝑟 − 1 ) is a sequence of matched pairs in some stable matching such that 𝑏 𝑖 + 1 mo d 𝑟 is the ne xt partner on 𝑎 𝑖 ’s preference list (aer 𝑏 𝑖 ) with whom 𝑎 𝑖 appears in some stable matching (this can b e computed without enumerating all stable matchings [ IL86 ]); applying 𝜌 reassigns each 𝑎 𝑖 to 𝑏 𝑖 + 1 mo d 𝑟 , producing an adjacent stable matching in the laice. e set of rotations, partially ordered by precedence ( 𝜌 ≺ 𝜌 ′ if 𝜌 must be applied before 𝜌 ′ can be exposed), forms the rotation poset Π . e key structural facts (Ir ving and Leather [ IL86 ], Guseld and Irving [ GI89 ]) are: (i) e downsets (closed subsets) of Π are in bijection with the stable matchings of the instance. (ii) Every nite poset is realizable as the rotation poset of some stable matching instance. W e take rotations as the commitment basis Φ rot . e commitment for rotation 𝜌 is ambiguity-sensitive (non-pointwise): whether 𝜌 can be applied depends on the current admissible set, not just on individual 20 outcomes. W riting 𝑆 = Spe c ( 𝐻 ) for the admissible set at histor y 𝐻 : 𝜑 𝜌 ( 𝑆 ) ≜ { 𝜇 ∈ 𝑆 | 𝜌 ∈ ds ( 𝜇 ) } , if 𝜌 is exposed in 𝑆 (all predecessors of 𝜌 in Π are in ds ( 𝜇 ) for every 𝜇 ∈ 𝑆 ), 𝑆 , otherwise, where ds ( 𝜇 ) denotes the do wnset of Π corresponding to matching 𝜇 (by fact (i) abov e, each stable matching corresponds to a unique downset). e commitment is irrevocable and satises shrinkage (it lters the admissible set rather than transforming matchings; in the determination framework, “applying a rotation” means retaining only matchings consistent with that rotation having been applied). Any determination that applies rotations in a valid topological order preserves feasibility . D .1.2 Determination depth equals rotation p oset height Proposition 6 (Determination depth of stable matching) . For any stable matching instance with r otation poset Π , Depth Φ rot ( Spe c SM ) = height ( Π ) , where height ( Π ) is the length of the longest chain in Π . Proof. Upper bound. Partition the rotations of Π into layers by a longest-path layering: layer 𝑖 contains all rotations whose longest chain of predecessors has length 𝑖 . Aer layers 1 , . . . , 𝑖 − 1 have been applie d, every rotation in lay er 𝑖 is exposed: by the do wnset–matching bijection (i), applying a r otation 𝜌 ′ retains only matchings whose downsets contain 𝜌 ′ , so aer all predecessors of a layer- 𝑖 rotation have be en applied, every surviving matching’s do wnset contains them. Within layer 𝑖 , the r otations are pair wise incomparable in Π ; since all are simultaneously exposed, applying any subset does not aect the exposure status of the others. (Exposure of 𝜌 ′ requires all predecessors of 𝜌 ′ to be in ds ( 𝜇 ) for every surviving 𝜇 ∈ 𝑆 . Applying an incomparable 𝜌 only shrinks 𝑆 , which can only make this condition easier to satisfy , not harder .) Hence the commitments within each layer commute, and the number of layers equals height ( Π ) . Lower bound. Let 𝜌 1 ≺ 𝜌 2 ≺ · · · ≺ 𝜌 ℎ be a longest chain in Π . Each 𝜑 𝜌 𝑖 + 1 depends on 𝜑 𝜌 𝑖 (Denition 20 ): before 𝜌 𝑖 is applied, the admissible set contains matchings whose downsets do not include 𝜌 𝑖 , so 𝜌 𝑖 + 1 is not exposed and 𝜑 𝜌 𝑖 + 1 acts as the identity . is gives a dependency chain of length ℎ , so Depth Φ rot ( Spe c SM ) ≥ ℎ by Proposition 4 . Remark 5 ( Universality ) . By the Ir ving–Leather realization theorem [ GI89 ], every nite poset arises as a rotation p oset. Since determination depth e quals p oset height (Proposition 6 ), stable matching realizes every p ossible determination depth: for ev ery 𝑘 , there exists an instance with determination depth exactly 𝑘 . In particular , nding a stable matching is in P, yet the determination depth can be arbitrarily large— computational cost is polynomial regardless of determination depth. Depth-optimality of strategies. e Gale–Shapley algorithm [ GS62 ] nds a stable matching in 𝑂 ( 𝑛 2 ) sequential steps, but it is not depth-optimal: it uses 𝑂 ( 𝑛 2 ) sequential steps even when the rotation poset has small height. A depth-optimal strategy—applying all exposed rotations simultaneously at each layer— achieves depth equal to the poset height, potentially much less than 𝑂 ( 𝑛 2 ) . is is pr ecisely the parallel algorithm derived independently by Garg’s laice-linear predicate framework [ Gar20 ], which the determi- nation framework r ecovers as a consequence of the depth characterization. Whether the conservation law yields new low er bounds on parallel stable matching algorithms is an open question. 21 D .2 Strategic Depth in Extensive-Form Games How much arbitrary choice does a game for ce upon a player? Game-tree depth overcounts: at forced moves, where only one option is optimal, the player faces a purely computational burden, not a relational one. W e use determination depth to measure a game ’s strategic depth : the number of moves where the player must break a tie among multiple equally-optimal options—an irreducible relational cost that no amount of computation can eliminate. Specication. Consider a two-player extensive-form game with p erfect information. Player 1 (the determiner) and P layer 2 (the environment) alternate moves in a game tree 𝑇 ; this is an online spe cication, since player 2’s mo ves are environment ev ents that extend the histor y between player 1’s commitments. e outcome set 𝑂 is the set of all root-to-leaf paths (complete plays), and the initial admissible set is Spec ( 𝐻 0 ) = 𝑂 . W e restrict the admissible set to plays consistent with subgame-perfe ct equilibrium (SPE) [ Sel65 ]: at each node, the acting player’s move must be optimal given optimal play in all subsequent subgames. Formally , the admissible set at histor y 𝐻 is the set of all complete plays extending 𝐻 that arise under some subgame-perfe ct equilibrium of the full game. e spe cication is relational whenever multiple SPE plays pass through the current history . Commitment basis. At each player-1 node 𝑣 , the commitment “ cho ose child 𝑐 ” excludes all plays not passing through 𝑐 : a pointwise lter . Player-2 moves are environment ev ents—outside the determiner’s control, r egardless of player-2’s e quilibrium rationality . Player-1 mo ves at subgame-trivial nodes (wher e the SPE prescribes a unique move) ar e also eectively environment ev ents: the move is uniquely determined, so no choice is involved. Only moves at subgame-non-trivial nodes—where multiple children lead to plays with the same SPE value for player 1—are genuine commitments r equiring an arbitrar y choice. Denition 22 (Subgame-non-trivial no de) . A player-1 node 𝑣 is subgame-non-trivial if at least two children of 𝑣 are each consistent with some (possibly dierent) subgame-perfect equilibrium of the full game. It is subgame-trivial if exactly one child is SPE-consistent. Proposition 7 (Strategic depth of extensive-form games) . For a two-player extensiv e-form game with perfect information, the determination depth from play er 1’s perspe ctive (under the SPE-r estricted spe cication) equals the maximum number of subgame-non-trivial player-1 nodes on any root-to-leaf path. Proof. Upper bound. A strategy that auto-plays the unique SPE move at subgame-trivial no des and commits at subgame-non-trivial nodes uses one layer p er non-trivial node along any realized play . Forced moves (at trivial nodes) and player-2 responses occur between commitment layers as environment events, contributing no determination depth. Lower bound. Consider two consecutive subgame-non-trivial player-1 nodes 𝑢 and 𝑣 on a root-to-leaf path, with 𝑣 deeper than 𝑢 . Node 𝑣 has SPE-consistent children 𝑐 1 , 𝑐 2 , so there exist SPEs 𝜎 1 , 𝜎 2 of the full game inducing plays through 𝑐 1 and 𝑐 2 respectively . Between 𝑢 and 𝑣 , player-2 moves and trivial player-1 mov es occur as environment events. e SPEs 𝜎 1 and 𝜎 2 may prescribe dierent player-2 mo ves at nodes between 𝑢 and 𝑣 , so the admissible set at 𝑣 —which plays remain SPE-consistent—depends on which environment ev ents materialize aer the commitment at 𝑢 . Under one sequence of play er-2 moves, only plays through 𝑐 1 may remain admissible at 𝑣 ; under another , only plays through 𝑐 2 . Since these environment events have not o ccurred when the commitment at 𝑢 is made, no commitment at 𝑢 can validly determine the choice at 𝑣 : any xe d choice risks selecting a child whose plays ar e inadmissible under some SPE-consistent continuation. e choice at 𝑣 is therefor e a forced dependency on the histor y up to 𝑣 . Chaining these forced dependencies across all non-trivial nodes on the path gives a dependency chain of length equal to their count (Proposition 4 ). 22 e result decomposes game-tree depth into two components: strategic depth (the non-trivial nodes, where the player must br eak a tie among equally-optimal moves) and for ced depth (the trivial nodes, where the optimal move is uniquely determine d and no choice is involved). is decomp osition is distinct from classical game-tr ee depth and alternating T uring machine quantier depth, b oth of which count all player-1 moves—including forced ones that involve no relational choice . e distinction is analogous to the BSP diagnostic (Section D.4 ): just as BSP overcharges by counting communication rounds that resolve no ambiguity , game-tree depth overcharges by counting moves that present no relational choice. e analysis is restricted to nite perfect-information games (Zermelo ’s seing); extending to imperfect information requires handling information sets and is le open. Remark 6 (Error amplication under bounded rationality) . Strategic depth has consequences beyond the relational burden itself. If a bounded player can compute SPE values but trembles when breaking ties—erring at each non-trivial node with indep endent probability 𝑝 —the probability of perfect play is ( 1 − 𝑝 ) 𝑑 ( 𝑑 = strategic depth, not game-tree depth 𝑘 ). Forced moves pr esent no relational risk; computational failures at forced moves are an orthogonal axis of bounded rationality . Strategic depth is per-player and need not b e symmetric: one player can oen inuence the other’s strategic depth by cho osing which subtree to enter . In chess, this is “playing for complications”: steering toward positions where the opp onent faces more tie-breaking choices. Balancing payo maximization against error amplication is an open question that connects to Selten’s trembling-hand renement [ Sel75 ], with inuence over determination depth as a new degree of strategic freedom. Remark 7 (Mechanism design as basis enrichment) . Beyond extensive-form games, the determination framework applies to mechanism design. A mechanism (e.g., an auction format) can be viewed as enriching the commitment basis: a direct-rev elation me chanism allows a player to submit a full strategy as a single commitment, collapsing multiple sequential choices into one layer . A dominant-strategy mechanism (e .g., a second-price auction [ Vic61 ]) makes every node subgame-trivial—each player’s optimal action is independent of others’—collapsing strategic depth to 0. An indirect me chanism (e .g., an ascending auction [ Mil00 ]) trades strategic depth for communication: more rounds of bidding, but less information per round. is echoes the depth–width–communication tradeo (eorem 2 ): formalizing the precise relationship—whether indirect mechanisms obey an analogous conservation law trading determination depth against per-round communication and outcome-set width—is an open problem. D .3 De composing Chain-of- ought Length An autoregressive transformer generates tokens sequentially , each an irre vocable commitment. Chain- of-thought (Co T) prompting increases the sequential lay ers available. Re cent work shows Co T provides exponential advantages [ FZG + 23 , LLZM24 , MAAN25 ], yet longer chains sometimes degrade [ SMA + 25 , SZW + 25 ]. W e use the determination framework to decompose Co T length into formally independent components. e decomp osition is not a theorem ab out any specic transformer architecture; it is a semantic lower bound that applies to any autoregressive procedure in which each token is an irrevocable commitment and the task is relational (multiple valid continuations exist). e constraine d generation task (Section 3 ) serves as the witness b ecause it isolates determination depth from computational depth: each commitment is a constant-time table lookup, so the se quential cost is entirely semantic. e total Co T length 𝑇 is lower-bounded by three formally indep endent quantities: (i) determination depth 𝑑 —non-commuting commitment layers, witnessed by the constrained generation task (Section 3 ); (ii) computational depth 𝑐 — inherent per-step sequentiality from b ounded transformer depth (TC 0 -like [ FZG + 23 ]), witnessed by graph connectivity [ MAAN25 ]; and (iii) architectural overhead —tokens sp ent on information management (context summarization, backtracking, re-derivation), reducible by changing the architecture without changing the task. 23 Proposition 8 (Co T lower bound from determination depth) . Co T length 𝑇 ≥ max ( 𝑑 , 𝑐 ) : the bound 𝑇 ≥ 𝑑 follows from eorem 1 and 𝑇 ≥ 𝑐 from [ FZG + 23 , MAAN25 ]. e two bounds are independent: there exist tasks with 𝑑 = 𝑘 , 𝑐 = 𝑂 ( 1 ) (eorem 1 ), and tasks with 𝑑 = 0, 𝑐 = 𝜔 ( log 𝑛 ) [ MAAN25 ]. In both cases, fewer than max ( 𝑑 , 𝑐 ) layers requires exponential parallel width. e decomposition explains the apparent conict. T asks where Co T provides exponential advantage are determination-bound or computation-b ound: their sequential cost is intrinsic, and more layers directly reduce it. T asks where longer chains degrade [ SMA + 25 , SZW + 25 ] are architecture-bound: 𝑑 and 𝑐 are small, so additional tokens are spent on overhead (conte xt management, backtracking, re-derivation) that introduces errors without reducing the boleneck. For determination-b ound tasks, the conditional-spread parameter 𝛾 governs ho w much distributional knowledge can substitute for chain length: a model facing a 𝛾 -spread distribution cannot b enet from additional training, while a model facing a more structured distribution can trade prediction quality for fewer lay ers (Remark 1 ). D .4 Diagnosing Distributed Round Complexity BSP [ V al90 ] organizes parallel computation into synchronous rounds separated by barriers. W e show that BSP round complexity and determination depth disagr ee in both directions, and that the disagreement is precisely explained by the gap between BSP ’s implicit commitment basis and the atomic basis that serves as our intrinsic reference (Section 2.1 ). e atomic basis as intrinsic refer ence. Under the atomic basis in the online seing, determination depth measures the irreducible commitment structure of a spe cication—the minimum number of non- commuting layers forced by forward validity constraints, with no assumptions about process identity , membership, or communication primitives. is is the “bare ” complexity of the spe cication. BSP departs from this reference in two ways: it charges for communication (adding rounds where no commitment occurs) and it implicitly enriches the basis (collapsing layers that the atomic basis e xposes). Overcharging: rounds without commitments. BSP charges one round for any operation requiring communication, whether or not the communication resolves semantic ambiguity . A single-valued specica- tion has determination depth 0 under any basis—the output is uniquely determined, so no commitment is needed. Relational join is the canonical example: parallel implementations that require no synchronization barriers have been known since the early 1990s [ W A91 ], yet BSP assigns one round to the data shue because BSP rounds account for communication uniformly . e unnecessary barrier has practical conse- quences (e .g., straggler sensitivity and inability to pipeline), which the determination framework diagnoses as over charging for a depth-0 task. Undercharging: basis enrichment collapses depth. BSP implicitly enriches the atomic basis by assuming xed, static membership : the process set is nite , globally known, and do es not change during execution. In a real system, establishing membership requires a prior commitment—a discovery protocol, a conguration step, or a leader’s decision ab out who participates—that BSP treats as given. is assumption hides exactly one unit of determination depth. Under the atomic basis, resolving membership costs at least one layer: a memb ership commitment (which processes participate?) must precede a value commitment (what do es each process output?), and the two do not commute, giving depth 2. BSP absorbs the memb ership commitment into its model assumptions, leaving only the value commitment visible—one BSP r ound for a problem with intrinsic depth 2. By the conser vation law (e orem 3 ), the total sequential depth is unchanged; BSP hides the membership layer rather than eliminating it (Corollary 2 in Section B , with 𝑗 = 1). e consequences are sharp. With static membership, universally quantied conditions (e.g., verifying that all processes have completed a round) r educe to nite conjunctions over a kno wn process set [ Imm86 ], 24 and all remaining non-monotonic reasoning can pr ocee d without distributed coordination by the CALM theorem [ ANdB13 ]. With dynamic memb ership, Ameloot’s preconditions fail: universal quantication over participants requires a prior determination of who participates, and each change in membership costs an additional layer . LOCAL model. e same diagnostic applies to the LOCAL mo del of distributed graph computation. Consider ( Δ + 1 ) -coloring of a graph 𝐺 = ( 𝑉 , 𝐸 ) on 𝑛 nodes. e sp ecication is relational: many valid colorings exist. e synchronous LOCAL model [ Lin92 ] provides synchronized rounds, simultaneous neighbor-state revelation, and unique node identiers. Under this basis, ( Δ + 1 ) -coloring has determination depth 1: deterministic symmetr y-breaking algorithms (e.g., Cole– Vishkin [ CV86 ]) compute a proper coloring as a function of the ID-labeled neighborho od in Θ ( log ∗ 𝑛 ) rounds. e round complexity is entirely computational—it measures the cost of symmetr y breaking, not of semantic commitment. Common knowledge of unique identiers collapses the determination structure entirely ( Corollary 2 ). IDs alone account for the collapse—synchronization and within-layer communication ar e not needed. Each node waits (asynchronously ) for its lower-ID neighbors to commit, then takes the smallest available color . e ID ordering prev ents circular dependencies, so no barrier or broadcast is needed. In the language of Ameloot et al. [ ANdB13 ], both static membership (Section D.4 ) and unique identiers are instances of non-obliviousness : shared knowledge that eliminates the need for distributed coordination. Establishing common knowledge of identiers lets all remaining non-monotonic reasoning procee d locally , but the conservation law (e orem 3 ) ensures that the sequential cost persists as local computation within layers. For coloring, IDs happen to collapse both costs (gr eedy coloring along the ID-induced orientation is a single-layer local computation), but this is a special property of the coloring specication, not a general consequence of having IDs. Remark 8 (Randomized strategies) . e analysis above applies to deterministic strategies. With random- ization, the Lov ´ asz Local Lemma yields 𝑂 ( 1 ) -round distributed algorithms for ( Δ + 1 ) -coloring with high probability [ MT10 ], collapsing determination depth to 𝑂 ( 1 ) even without IDs. Randomization thus pro vides a mechanism for collapsing determination depth by breaking symmetry probabilistically . Characterizing which specications admit randomized depth collapse is an op en question. E Metacomplexity Proofs How hard is it to compute the determination depth of a given specication? is section shows that the answer ranges from NP-har d to PSP A CE-complete. In the oine seing, computing determination depth under a pointwise basis is alr eady NP-hard (via de cision tree synthesis). In the online seing, the alternation between the determiner’s commitments and the environment’s histor y extensions produces quantier alternation: “is depth ≤ 𝑘 ?” is Σ 𝑃 2 𝑘 -complete for each xed 𝑘 and PSP ACE-complete for unbounde d 𝑘 , so the polynomial hierarchy is precisely the hierarchy of determination depths. Proposition 9 (Metacomplexity of determination depth) . Computing determination depth is NP-hard in the oine seing: for sp ecications arising from decision tree synthesis (“given a truth table, output an optimal decision tree”), determination depth equals the minimum de cision tree depth, which is NP-hard to compute [ HR76 ]. Proof. Decision tree synthesis. e environment events encode a truth table 𝑓 : { 0 , 1 } 𝑛 → { 0 , 1 } (an oine seing with no further extensions), and the outcome set 𝑂 is the family of all decision trees on 𝑛 variables. e sp ecication maps each complete histor y—encoding a truth table—to the set of de cision trees that compute it: Spec ( 𝐻 ) = { 𝑇 : 𝑇 computes 𝑓 𝐻 } , where 𝑓 𝐻 is the truth table encoded in 𝐻 . e commitment basis consists of pointwise lters “test variable 𝑥 𝑖 at the current node”: each such commitment restricts 25 the admissible set to decision trees that test 𝑥 𝑖 at that node. Because each no de of a de cision tree tests exactly one variable, two test commitments for dierent variables at the same node hav e no common tree in their intersection—the admissible set be comes empty—so any valid determination must select exactly one variable per node. A determination of depth 𝑑 corresponds to a de cision tree of depth 𝑑 : each lay er selects a variable to test at each node of that layer , and the two test outcomes branch into sub-problems resolved by subsequent layers. Any depth- 𝑑 determination yields a depth- 𝑑 tree, and vice v ersa, so determination depth equals minimum decision tree depth, which is NP-hard to compute [ HR76 ]. Depth here r eects the tree structure: a depth- 𝑑 decision tree has 𝑑 levels of nodes, and each level is one round of variable-test commitments. e NP-hardness result ab ove is an oine result: the sp ecication is fully given and there is no adversarial environment. In the online seing, an adversarial environment adds universal quantication— the environment xes some variables, the determiner xes others—and the metacomplexity rises through the full polynomial hierarchy . T o state the complexity result precisely , consider the following class of online specications. Given a Boolean formula 𝜃 ( 𝑥 1 , . . . , 𝑥 𝑛 ) (encoded by a polynomial-size circuit), dene a specication with outcome set 𝑂 = { 0 , 1 } 𝑛 (all variable assignments) and initial admissible set Spec ( 𝐻 0 ) = 𝑂 (e very assignment is initially admissible). e commitment basis consists of pointwise lters “set variable 𝑥 𝑖 = 𝑏 ”: each excludes all assignments with 𝑥 𝑖 ≠ 𝑏 , shrinking the admissible set without resolving it completely . e game has 𝑛 rounds. A level assignment partitions the 𝑛 rounds between two play ers: at each determiner round the determiner xes one variable (an existential choice); at each environment round the environment xes one variable adversarially (a universal choice). Aer all 𝑛 rounds every variable is xed and the admissible set is a singleton. e determination depth is the numb er 𝑘 of determiner rounds; the metacomplexity question is: given 𝜃 and 𝑘 , does there exist a level assignment with 𝑘 determiner rounds and a strategy for the determiner that guarantees the nal assignment satises 𝜃 , regardless of the environment’s choices? e input size is polynomial in 𝑛 (the circuit encoding of 𝜃 ) ev en though | 𝑂 | = 2 𝑛 . eorem 7 (Determination depth captures the polynomial hierarchy) . For the class of specications above: (i) For each xed 𝑘 , the problem “is determination depth ≤ 𝑘 ?” is Σ 𝑃 2 𝑘 -complete. (ii) For unbounded 𝑘 (given as input), the problem is PSP A CE-complete. is is the standard setup for QBF and PSP ACE-complete game e valuation. Proof. Upper bound. e determiner guesses which 𝑘 of the 𝑛 rounds to contr ol. Each determiner choice is an existential quantier; each environment choice is a universal quantier . Aer all 𝑛 rounds, every variable is xed and satisfaction of 𝜃 is checkable in polynomial time by evaluating the cir cuit. Consecutive rounds controlled by the same player collapse into a single quantier blo ck, so the quantier paern has at most 2 𝑘 alternations, placing the problem in Σ 𝑃 2 𝑘 . When 𝑘 is part of the input, the number of alternations 2 𝑘 is at most 2 𝑛 ; since the circuit encoding of 𝜃 has at least 𝑛 input wires, 2 𝑛 is linear in the input size. A single alternating polynomial-time machine can therefore handle every 𝑘 , placing the problem in APTIME = PSP A CE [ Sip83 ]. Hardness. Re duce fr om Σ 2 𝑘 -QBF for xed 𝑘 , or from TQBF for unbounded 𝑘 . Given a quantied Boolean formula ∃ 𝑦 1 ∀ 𝑥 1 · · · ∃ 𝑦 𝑘 ∀ 𝑥 𝑘 . 𝜃 ( 𝑥 , 𝑦 ) , construct a specication with 2 𝑘 variables and outcome set 𝑂 = { 0 , 1 } 2 𝑘 . Odd- numbered rounds (1 , 3 , . . . , 2 𝑘 − 1) ar e controlled by the determiner , xing the existential variables 𝑦 1 , . . . , 𝑦 𝑘 ; even-numbered rounds (2 , 4 , . . . , 2 𝑘 ) are controlled by the environment, xing the universal variables 𝑥 1 , . . . , 𝑥 𝑘 . e quantier alternation of the QBF maps directly onto the round structure: each ∃ becomes a determiner round, each ∀ an environment round. e initial admissible set is Spec ( 𝐻 0 ) = 𝑂 (all assignments); 26 each round’s commitment narrows it by xing one variable. Aer all 2 𝑘 rounds the assignment is fully determined; the determiner’s strategy succeeds i the resulting assignment satises 𝜃 . Correctness (the QBF is true i the determiner has a depth- 𝑘 strategy). If the QBF is true, the existential player has a winning strategy: a choice of each 𝑦 𝑖 (possibly dep ending on 𝑥 1 , . . . , 𝑥 𝑖 − 1 ) such that 𝜃 is satised for every univ ersal assignment. e determiner plays this strategy at the odd rounds, using 𝑘 layers (one per existential variable), so depth ≤ 𝑘 . Conversely , any depth- 𝑘 determiner strategy is an adaptive assignment to the existential variables that satises 𝜃 against every universal response—exactly a witness that the QBF is true. e polynomial hierarchy is thus precisely the hierarchy of metacomplexity for determination depth: deciding whether a specication has depth ≤ 𝑘 is Σ 𝑃 2 𝑘 -complete, and PSP ACE-complete for unbounde d 𝑘 . In the oine seing (Proposition 9 ), the metacomplexity is NP-hard but does not capture the full hierar chy . F Related W ork is section positions determination depth among existing notions of staged computation and semantic dependency . Closest antecedents treat staging either as an operational resource (rounds, adaptivity , oracle calls) or as a syntactic discipline (stratication, xpoint nesting). Our contribution is to identify staging as a semantic invariant of resolution—dened prior to any particular model or language. Parallelism, depth, and inherent sequentiality . Depth as a complexity measure has a long history in models of parallel computation. Circuit comple xity distinguishes size from depth, isolating irreducible sequential structure even when unbounded parallelism is available [ Pip80 , Bar90 ]. Circuit depth measures the inherent sequentiality of evaluating a function: data dependencies force some gates to wait for oth- ers. Determination depth measures the inherent sequentiality of choosing an outcome from a relation: commitment dep endencies force some de cisions to wait for others. e two are orthogonal: a function has determination depth 0 regardless of its circuit depth, while a relation can have high determination depth even when ev ery individual commitment is a constant-depth circuit (Section 3 ). is orthogonality is witnessed in the oine seing by the constrained generation task (Section 3 ); in the online seing, forward validity constraints add a further source of determination depth that circuit models do not capture at all. PRAM and BSP models similarly separate local computation from global synchronization, charging depth to barriers or rounds [ BC82 , V al90 ]. In the online (distributed) seing, these models can both under- charge (when a r ound boundar y hides an irrev ocable commitment that the model treats as primitive) and over charge (when communication structur e is conated with semantic resolution); Section D .4 develops this diagnostic in detail. More broadly , classical hierarchy theorems establish strict separations based on time, space, or alternation depth [ Sip83 ]; our determination depth hierarchies are orthogonal, arising from semantic dependency rather than resource bounds. Trace monoids and partial commutation. e layered normal form (Section A ) resembles the Foata normal form of Mazurkiewicz trace theor y [ Maz77 ]: a canonical factorization of a word in a partially commutative monoid into maximal independent steps. e resemblance is structural but the theories diverge in three ways that produce qualitatively dierent phenomena. First, in classical trace the ory the indep endence relation is xe d : two leers either commute or they do not, regardless of context. In determination theor y , commutation is dynamic : whether two commitments commute dep ends on the current rened specication, and applying one commitment can cr eate or destroy independence among the remaining ones. is dynamic commutation arises for dierent reasons in the two seings: in the online seing, forward validity constraints (a commitment that is safe alone may be come unsafe aer another 27 commitment narrows the admissible set at some extension) create and destroy independence; in the oine seing, enablement (commiing to one position determines which choices remain feasible at the next) produces the same eect. Second, this state-dependence makes the analogue of the Foata normal form non-unique—dierent layering choices lead to dierent rened sp ecications and potentially dierent minimum determination depths—and computing the minimum height becomes NP-hard (Proposition 9 ), in contrast to the polynomial-time computability of static Foata height. ird, the main results of this paper—the exponential depth–width separation (oine), the oracle characterization (online), and the conservation law ( both seings)—have no analogues in classical trace theor y . ey arise from the semantic content of commitments (feasibility , resolution, forward validity) rather than from the algebraic structur e of partial commutation alone. In short, trace theory provides the algebraic skeleton; the semantic content of determination lls it with phenomena that the skeleton alone cannot express. Oracle mo dels and adaptivity . e use of a disclosure oracle to characterize depth parallels classical distinctions between adaptive and non-adaptive computation. Decision tree complexity , oracle T uring machines, and communication complexity all exhibit hierarchies based on the number of adaptive rounds [ KL80 , PS84 , Nis91 ]. Our oracle characterization (Proposition 1 ) is specic to the online seing under a commutative basis: oracle invocations witness irreducible semantic dep endency—points at which the strategy must obser ve the ev olving histor y before commiing further—rather than quer y adaptivity or information ow . Local computation between oracle calls is unrestricted, emphasizing that determination depth cannot be collapsed by parallelism or control ow alone. In the oine seing, a single oracle call reveals the complete input and the oracle count collapses to one; the sequential cost reappears as computational depth within layers (the conservation law , eorem 3 ). In this sense, determination depth and adaptivity depth both measure staged dependency , but the dependency has a dierent character . An adaptive query reveals information ab out a xe d, predetermine d answer; a determination commitment creates the answer by irrevocably excluding alternatives. Adaptivity depth measures how many times a strategy must look; determination depth measur es how many times it must choose. e two ar e orthogonal (Observation 3 , Section 3 ): pointer chasing at sparsity 𝑠 = 1 has high adaptivity depth but determination depth zero (the answer is unique); constrained generation in the oine seing has determination depth 𝑘 but adaptivity depth zero (the entire input is giv en). Round hierarchies in communication complexity . e closest technical ante cedent to our exponential separation is the Nisan– Wigderson round hierarchy for p ointer chasing [ N W93 ]; our constrained generation task extends the same chain structure fr om functions to relations (Observation 3 , eorem 2 ). e round- elimination te chnique—reducing a 𝑘 -round protocol to a ( 𝑘 − 1 ) -round protocol with a communication blowup—is the standard tool for proving round low er bounds in communication complexity [ MNSW98 , Sen18 ]. Mao, Y ehudayo, and Zhang [ MYZ25 ] recently obtained tight pointer-chasing b ounds via gadgetless liing, bypassing round elimination entir ely . Both techniques—round elimination and gadgetless liing— operate in the functional regime ( 𝑠 = 1), where hardness is informational: the answer is unique but distributed across players. Our lower bound targets the relational regime ( 𝑠 ≥ 2), where the answer is not unique and hardness arises from the cost of commiing to one answer among many , not from distributing information ab out a xed answ er . Our exponential separation is an oine result (the entir e constraint chain is given to a centralized machine), while p ointer chasing is an online communication problem (the chain is distributed across players who communicate in rounds). e three-way tradeo (eorem 2 ) unies both: it interpolates between the oine width bound (seing communication to zero) and the online communication bound (seing width to one). Our low er-bound te chnique (conditional spread and union bounds over uninformed links) exploits the r elational structure of the specication rather than information-theoretic arguments about message content. e key qualitative dierence is that round-elimination lower bounds 28 are polynomial in the communication parameter , while our depth–width tradeo is exponential —and the transition occurs precisely at the boundar y between functional and relational specications ( 𝑠 = 1 vs. 𝑠 ≥ 2). Logic, stratication, and xpoints in databases. Layered semantics appear in database theory thr ough stratied negation, well-founded semantics, and alternating xpoints [ GRS91 ]; in descriptive complexity , xpoint nesting and alternation depth classify expressiv e power [ Imm99 ]. ese layerings ar e properties of a program under a chosen semantics : a stratied Datalog program with 𝑘 strata requires 𝑘 non-commuting sealing commitments (one per stratum), but applying w ell-founded or stable semantics to the same program yields a dierent commitment structure and potentially a dierent determination depth. Determination depth, by contrast, is a property of the specication (the mapping from histories to admissible outcome sets), independent of any particular logical formalism or semantics used to dene it. T wo logic programs that dene the same specication have the same determination depth, e ven if they dier in strata count, xpoint nesting, or use of negation. Conversely , applying dierent negation semantics to the same program can yield dierent specications and hence dierent determination depths—a distinction that strata counts alone cannot express. Leaf languages. e leaf language framework of Bovet, Cr escenzi, and Silvestri [ BCS92 ] characterizes complexity classes by the language-theoretic complexity of the string of outcomes at the leaves of a nondeterministic computation tree: NP corresponds to testing membership in { 0 , 1 } ∗ 1 { 0 , 1 } ∗ , PSP ACE to a regular language recognizable with 𝑂 ( log 𝑛 ) memory , and so on. Both framew orks use the structure of a tree of possibilities to classify pr oblems, but the objects and questions dier . Leaf languages classify the acceptance condition applied to a xed nondeterministic computation; determination depth classies the commitment structure required to resolve a relational specication. In particular , leaf languages op erate on a single computation tree whose branching is xed by the machine, while determination depth measures layered commitment across histories whose extensions are chosen by an adversarial environment—a distinction that is specic to the online seing. e PH characterization (eorem 4 ) recovers the same hierarchy that leaf languages capture, but thr ough alternation of commitment and environment moves in the online game rather than thr ough the complexity of a leaf-string acceptance condition. In the oine seing, the adversarial environment disappears and the exponential separation (e orem 1 ) provides a dierent route to determination depth hierarchies, via commitment dependency rather than alternation. More broadly , the leaf language program provided an elegant classication of existing complexity classes but did not yield new separations or lower b ounds. Our framework is aimed at a dierent target: not reclassifying known classes, but identifying an axis of complexity orthogonal to computation (Section 3 ), producing new tradeos (eorems 1 – 2 ), and diagnosing existing models (Sections D .4 – D .4 ). Coordination and monotonicity . e Coordination Criterion and the CALM theorem relate mono- tonicity to the absence of coordination requirements in distributed systems [ Hel26 , ANdB13 ]. ese ar e inherently online r esults: coordination cost arises fr om the need to commit before the full history is kno wn. In our framework, monotone (future-monotone) spe cications correspond to the depth-zero fragment, in which no irrev ocable commitment is nee ded at any prex. Deeper spe cications require staged com- mitments in the online seing, ev en in the presence of powerful coordination primitives; the distributed extension (Section B ) makes this precise. Inference-time compute and chain-of-thought reasoning. A growing b ody of work studies the power and limitations of chain-of-thought reasoning in transformers [ FZG + 23 , LLZM24 , MAAN25 ]. Feng, Zhang, Gu, Y e, He, and W ang [ FZG + 23 ] and Li and Liu [ LLZM24 ] show that Co T enables b ounded-depth transformers (TC 0 ) to solve problems requiring greater computational depth. Mirtaheri, Anil, Agarwal, 29 and Neyshabur [ MAAN25 ] prove an exponential separation between sequential and parallel scaling for graph connectivity . ese results concern computational depth—the inherent sequentiality of evaluating a function. Our exponential separation (Section 3 ) establishes an orthogonal source of sequential advantage: determination depth, arising fr om non-commuting commitments in relational specications. e separation is an oine result (the full constraint chain is given); the decomp osition of Appendix D.3 extends to the online autoregressiv e seing, unifying computational depth, determination depth, and architectural overhead into a single framework. Complementary empirical work identies failur e modes of e xtended r easoning [ SMA + 25 , SZW + 25 ] and studies optimal compute allocation [ SLXK24 ]; our decomposition provides a formal account of why these phenomena arise. Summary . Across these domains, notions of depth have appeared as proxies for sequentiality , staging, or adaptivity . Our contribution is to identify determination depth and determination cost as semantic complexity measur es—arising from forward validity constraints in the online seing and from commitment dependency in the oine seing—that are orthogonal to computational complexity . Determination depth measures the minimum sequential layers of irre vocable commitment; determination cost measures the total number of commitments; the exponential separation (eorem 1 ) sho ws that reducing one without increasing the other is impossible. Acknowledgments anks to Peter Alvaro, T yler Hou, Sarah Morin and Christos Papadimitriou for helpful fee dback on earlier dras. AI Disclosure. W e use d Claude ( Anthropic), ChatGPT (OpenAI), and Gemini (Google) to assist with exposition, structural organization, proof review , and adversarial fee dback throughout the manuscript. e tools materially aected the prose and presentation in all sections, including section ordering, framing of arguments, and selection of open problems. All mathematical content, denitions, theorems, and pr oofs are the authors’ own. e authors veried the correctness and originality of all content including references. 30
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