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Non-Asymptotic Length Generalization

T0 review · 2 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper proves that the ability to length-generalize with computable guarantees is exactly the decidability of the language-equivalence problem, with bounds O(T^{O(K)}) for C-RASP2, 2n-2 for DFAs, and no computable bound for linear CFGs.

desk verdict A solid conceptual framework and a clean decidability equivalence, with the main C-RASP2 bound resting on a deferred lemma that needs referee verification. read the letter →

arxiv 2506.03085 v2 pith:OVNINDKW submitted 2025-06-03 cs.LG

Thomas Chen , Tengyu Ma , Zhiyuan Li This is my paper
classification cs.LG MSC 68Q3268Q45
keywords lengthgeneralizationnon-asymptoticboundscomplexityminimum-complexityinterpolatorlanguageequivalenceproblemcontext-freegrammarsC-RASPtransformers
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper asks a quantitative version of a classic question: given a hypothesis class, how long must the training data be before a learner is guaranteed to generalize to all longer inputs? It formalizes this as non-asymptotic length generalization, measured by a length complexity that depends on the complexity of the ground-truth function. The central characterization is that a function class admits non-asymptotic length generalization if and only if the language-equivalence problem for its encoding system is decidable, which makes the presence or absence of computable guarantees a computability phenomenon. The paper then supplies concrete bounds: 2n-2 for deterministic finite automata, O($T^{2}$) for one-layer C-RASP, O($T^{{O(K)}}$) for two-layer C-RASP (a transformer-related class), and a proof that linear context-free grammars have no computable bound at all.

What carries the argument

The central object is the length complexity N(F), the smallest n such that every two distinct functions in F are separated by some input string of length at most n; all learning-algorithm bounds in the paper are reduced to bounding this class-level quantity. The algorithm that realizes the optimum is the Minimum-Complexity Interpolator (Amci), which enumerates programs by increasing C and returns the first one consistent with the training set. For the C-RASP2 upper bound, the machinery is geometric: strings are represented as discrete test-functions (prefix-sum paths), continuous test-functions are used as a proxy, the achievable activation vectors are decomposed into a finite union of polytopes via basis schemas, and margin lower bounds on the average of a polytope's vertices produce a low-precision point that can be discretized into a short distinguishing string.

What would settle it

Find an infinite family of unequal pairs of C-RASP2 functions whose complexity T(f)^{K(f)} is at most $\alpha$ and whose shortest distinguishing string has super-polynomial length in $\alpha$; the simplest version would be a pair of 1-head functions with precision T that agree on all strings up to length 2^T, which would contradict the claimed O($T^{{O(1)}}$) bound for K=1.

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Extended reading notes

Core claim

The paper establishes that non-asymptotic length generalization is equivalent to decidability of the language-equivalence problem: for any encoding system R and complexity measure C satisfying Assumption 3.7, the length complexity N(F^R_c) is computably bounded in c if and only if it is decidable whether R(p)=R(q) for two descriptions p and q (Lemma 4.5). Because the Minimum-Complexity Interpolator achieves exactly N(F^R_c) (Theorem 4.2), this equivalence also identifies the optimal learning algorithm. The main positive result is that for C-RASP2 functions, the class of two-layer C-RASP programs with complexity C(f)=T(f)^{K(f)}, the length complexity is O($alpha^{{O(1)}}$) when the ground truth has complexity at most $\alpha$; in the more concrete parametrization, inputs of length O($T^{{O(K)}}$) suffice when the ground truth has precision T and K heads.

Load-bearing premise

The O($T^{{O(K)}}$) upper bound for two-layer C-RASP rests on a deferred lemma asserting that any two genuinely different programs are separated by some continuous test signal with strict margin, not merely by a discrete string; if that lemma fails, the bound no longer follows.

Editorial extensions

If this is right

  • If the equivalence is correct, then a computable training-length guarantee exists for a class exactly when two descriptions of that class can be tested for semantic equality by a program.
  • Linear context-free grammars are thereby proved to resist any computable length-generalization guarantee, sharpening the known asymptotic learnability of CFGs.
  • Deterministic finite automata length-generalize from all strings of length at most 2n-2, tying the training length to state-minimization.
  • One-layer C-RASP programs need O(T^2) training length and two-layer programs need O(T^{O(K)}), giving the first provable non-asymptotic guarantees for a transformer-related class.
  • Because C-RASP covers functions computable by finite-precision transformers, these bounds offer a candidate explanation for when transformers can be expected to extend to longer inputs.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The decidability equivalence suggests that if a trained transformer's hypothesis space has undecidable equivalence, then no algorithm, however powerful, can certify a finite training length that guarantees length generalization; observed failures would then be intrinsic rather than optimization artifacts.
  • The O(T^{O(K)}) bound is almost certainly loose: the authors state they expect a lower bound of order T^K, which would imply the true difficulty lies in the exponent being linear in K rather than polynomial.
  • The framework could be instantiated on other transformer-inspired programming languages; for any such language, deciding the equivalence problem would immediately become the key quantity to known whether length generalization is even possible.
  • The quantitative guarantees depend on the chosen complexity measure (description length, precision, heads), so the framework makes the modeling choice of a measure an explicit part of the learning-theoretic statement.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper introduces a non-asymptotic notion of length generalization, defining the length complexity of a function class as the minimum input length needed to distinguish any two functions of bounded complexity. It proves that the Minimum-Complexity Interpolator achieves optimal length complexity, that non-asymptotic length generalization is equivalent to decidability of the language equivalence problem for the encoding system, and that linear CFGs therefore have no computable length-complexity bound. It gives a constructive DFA bound of 2n-2 states, an O(T^2) bound for C-RASP1, and an O(T^{O(K)}) bound for C-RASP2. The C-RASP2 bound is the headline contribution; the paper sketches a geometric proof that any two unequal C-RASP2 functions have a short distinguishing string.

Significance. The conceptual framework is valuable: replacing asymptotic identification-in-the-limit with a computable, non-asymptotic length-complexity bound is a real step forward, and the equivalence between non-asymptotic length generalization and decidability of the equivalence problem is elegant and cleanly proved. The DFA bound 2n-2 is supported by a constructive state-merging argument and is convincing. If the C-RASP2 upper bound is correct, it would be the first non-asymptotic length-generalization guarantee for a nontrivial transformer-related class, which is a substantial result. However, the proof of that bound depends on a deferred technical lemma, and the available manuscript does not contain the lemma's proof; the central claim is therefore not yet fully verifiable.

major comments (2)
  1. [Section 6, Lemma 6.11, Appendix D.6] The C-RASP2 upper bound rests on an unverified lemma. In the proof of Theorem 6.1, the text invokes 'the non-trivial Lemma D.37' to pass from a discrete distinguisher x0 to a continuous test-function Y0 whose activations lie strictly inside H1+∩H2+ (or the opposite pair). This strict-margin property is used in Step 2 to lower-bound the margin γ of the average of vertices, in Step 3 to perturb to a low-precision point while preserving margin, and in Step 4 to ensure that a discrete approximation preserves the signs of both second-layer inequalities. If the discrete activation lies exactly on the boundary of one halfspace, the argument collapses unless Lemma D.37 is proved. The manuscript itself calls the lemma non-trivial and defers its proof to Appendix D.6, which is not present in the submitted version, so the main theorem cannot be checked as written. The same lemma is also used to assert M≤2k for the basis schema; without it, Corollary D.15 gives only M≤k^2, which would degrade the final length bound from O(T^{O(K)}) to O(T^{O(K^2)}). Please provide the full proof of Lemma D.37 (and Lemma D.30) or state precisely which variant is assumed, and reconcile the bound with the abstract.
  2. [Appendix D.1, Proof of Theorem 5.5, type (ii) case] There is a concrete error in the C-RASP1 proof. In the case b/a<0 and b'/a'<0, the text claims that 'after x coordinate at most T+1, then lines ... go below y=0' and hence any distinguisher has length at most T. This is false under the parameter bound |a|,|b|,|d|≤T: taking a=T, b=-1, d=T gives the line y = -(1/T)x + T, which stays nonnegative for x up to T^2. The final O(T^2) bound may still be salvageable by replacing T+1 with O(T^2), but the proof as written contains a false assertion and should be corrected.
minor comments (4)
  1. [Section 4.3] The sentence 'Interestingly, Definition 4.9 is equivalent to non-asymptotic length generalization (Definition 4.7)' appears to cite the wrong definition: the equivalence is to Definition 3.9, not to Definition 4.7, which is the stronger notion without complexity information.
  2. [Definition 4.9] The definition is titled 'Finite Length Generalization w.r.t. encoding system R and complexity measure C' but the body omits the phrase 'with Complexity Information'; for consistency with the surrounding text, the title should include that phrase.
  3. [Section 6, Equations (3)-(4)] Equations (3) and (4) use non-strict inequalities (≤ and >) for the distinguishing conditions, while the later halfspaces H1+, H2+ etc. use strict inequalities; the relationship between these two formulations should be stated explicitly to avoid confusion.
  4. [Appendix D.5, Lemma D.23] The expression 'min0≤a,b,c,d≤T' should be read as a minimum over the relevant parameters with the additional condition that the two fractions are unequal; as written, the notation is ambiguous. A subscript or clarifying phrase would help.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper's central equivalence is a genuine logical characterization and the C-RASP bounds are derived from definitions and lemmas, not from fitted or self-referential inputs.

full rationale

The derivation chain is self-contained. Theorem 4.2 proves the optimality of the Minimum-Complexity Interpolator by a two-sided distinguishability argument: any algorithm with access only to inputs of length below N(F_R_c) must fail on some pair in F_R_c, while at length N(F_R_c) the interpolator is forced to output the ground truth because every lower-or-equal-complexity function has already been separated from it. This is a real proof from Definition 4.1, not a restatement of the conclusion. Lemma 4.5 gives explicit constructions in both directions: an equivalence oracle is used to compute an upper bound on N(F_R_c), and a computable bound on N(F_R_c) is used to decide equivalence by checking agreement up to that bound. The equivalence to decidability is therefore constructive rather than definitional. The CFG impossibility uses the external undecidability of linear-CFG equivalence from Baker and Book (1974), and the DFA result uses a state-minimization argument; neither relies on the paper's own claims. The C-RASP upper bounds are obtained by bounding the minimum distinguishing length through parameter precision, polytope margin lower bounds, and discrete approximation lemmas; no parameter is fitted to data and no quantity called a prediction is a renamed input. The one potentially load-bearing unverified step, Lemma D.37 (Lemma D.30 in the appendix), is a geometric strict-margin conversion whose proof is deferred; if it failed the bound would collapse, but that is a proof-completeness or correctness risk, not circularity. The only self-citation, Mahankali et al. (2023) on the difficulty of analyzing SGD, is a motivating aside and does not support any theorem, so it does not raise the circularity score.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

No new entities are postulated; continuous test-functions and schemas are proof devices, not entities with independent falsifiable handles. The free-parameter list is empty because the arguments contain no fitted constants: all bounds are derived from the stated complexity measures and definitions.

assumptions (7)
  • domain assumption Encoding system R is computable and total: every binary string maps to a computable function, invalid encodings map to the empty language.
    Definitions 3.1, 3.3, 3.4; required so learning algorithms can evaluate and compare hypotheses.
  • domain assumption Complexity measure C satisfies Assumption 3.7: computable, a TM enumerates programs in non-decreasing C, finite programs per complexity level.
    Assumption 3.7; needed for Amci to be computable (Lemma 4.3) and for the equivalence results (Lemma 4.5, 4.10).
  • standard math Language equivalence for linear Context-Free Grammars is undecidable (Baker & Book, 1974).
    Used in Proposition 4.6 to rule out non-asymptotic length generalization for linear CFGs.
  • standard math In a minimal DFA, any two distinct states are distinguished by some finite string.
    Used in Lemma C.1 to count equivalence classes; from Hopcroft & Ullman (1979).
  • ad hoc to paper The technical Lemma D.37 holds: for unequal C-RASP2 functions there exists a continuous test-function whose activations strictly separate the two second-layer halfspaces.
    Invoked in the proof of Theorem 6.1 (Section 6, 'Lemma D.37 implies...'); proof deferred to Appendix D.6 and not verifiable in the provided text.
  • ad hoc to paper C-RASP1 and C-RASP2 are defined with the restrictive forms given in Definitions 5.3 and 5.4 (integer parameters, distinct slopes, sum lambda_i > z).
    The upper bounds apply only to these subclasses, not to the full C-RASP class; the conclusion section overstates this.
  • standard math Standard convex geometry results (e.g., Rockafellar) on polytopes, relative interiors, and margins are used in the margin lemmas.
    Used in Appendix D.4 (Lemma D.18, D.25-D.29) to lower-bound margins of averaged vertices.

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Pith. "Pith review of Non-Asymptotic Length Generalization." pith.science (2026). https://pith.science/paper/OVNINDKW

@misc{pith2026250603085,
  author       = {Pith},
  title        = {Pith review of: Non-Asymptotic Length Generalization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OVNINDKW}},
  note         = {Machine review of arXiv:2506.03085}
}
abstract

Length generalization is the ability of a learning algorithm to learn a hypothesis which generalizes to longer inputs than the inputs in the training set. In this paper, we provide provable guarantees of length generalization for various classes of functions in an idealized setting. First, we formalize the framework of non-asymptotic length generalization, which requires a computable upper bound for the minimum input length that guarantees length generalization, as a function of the complexity of ground-truth function under some given complexity measure. We refer to this minimum input length to length generalize as length complexity. We show the Minimum-Complexity Interpolator learning algorithm achieves optimal length complexity. We further show that whether a function class admits non-asymptotic length generalization is equivalent to the decidability of its language equivalence problem, which implies that there is no computable upper bound for the length complexity of Context-Free Grammars. On the positive side, we show that the length complexity of Deterministic Finite Automata is $2n - 2$ where $n$ is the number of states of the ground-truth automaton. Our main results are upper bounds of length complexity for a subset of a transformer-related function class called C-RASP (Yang & Chiang, 2024). We show that the length complexity of 1-layer C-RASP functions is $O(T^2)$ when the ground-truth function has precision $T$, and that the length complexity of 2-layer C-RASP functions is $O(T^{O(K)})$ when the ground-truth function has precision $T$ and $K$ heads.

Figures

Figures reproduced from arXiv: 2506.03085 by the authors.

Figure 1
Figure 1. Summary of equivalence results between different characterizations of length generalization in [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Depiction of a Test-Function consisting of 4 segments. [PITH_FULL_IMAGE:figures/full_fig_p039_2.png] view at source ↗
Figure 3
Figure 3. Four types of Segments, based on which lines their start-point and end-point lie on, and the sector [PITH_FULL_IMAGE:figures/full_fig_p040_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Depiction of a Basis Schema consisting of 4 monotone curves. [PITH_FULL_IMAGE:figures/full_fig_p045_4.png]
Figure 5
Figure 5. Figure 5: Depiction of A({si}i∈[k]) for k = 2 with two slopes s1 > s2. B1 is on the horizontal axis while B2 is on the vertical axis. When k = 2, there are only two basis schema: a single monotone (up) curve and a single monotone (down) curve. The dark blue triangle with vertice…

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. From Expressivity to Sample Complexity: Narrow Teachers for Transformers via C-RASP

    cs.LG 2026-07 conditional novelty 6.0 of 10

    Embedding a narrow C-RASP teacher into a wider quantized Transformer yields sample complexity O((L d log Q)/ε) under posterior sampling of zero-training-error models.

Reference graph

Works this paper leans on

29 extracted references · 28 canonical work pages · cited by 1 Pith paper

  1. [1]

    A outputs ˆp at least for one N ∈ N

    A can only output either “pass” or some program ˆp ∈ {0, 1}∗. A outputs ˆp at least for one N ∈ N

    work page

  2. [2]

    equivalent

    Whenever A outputs some program ˆp ∈ {0, 1}∗, it must be correct in the sense that R(ˆp) = f∗. We now prove the following equivalences. Lemma 4.5 (Equivalence to Decidability of Language Equivalence Problem) . For any encoding system R and complexity measure C satisfying Assumption 3.7, the Language Equivalence problem for R is decidable if and only if le...

    work page 1979

  3. [3]

    , m− 2}, Y restricted to the interval [ Ti, Ti+1] can be rearranged into an equivalent monotone curve by Lemma D.13

    MaxTB := {Ti}0≤i≤m−1 ⊂ [0, 1] is a set of real numbers where for each i ∈ {0, 1, , . . . , m− 2}, Y restricted to the interval [ Ti, Ti+1] can be rearranged into an equivalent monotone curve by Lemma D.13. Proving Termination. At each iteration α, Algorithm 5 maintains variables Spanα, tα−1, and bα−1, which we claim satisfy the property that Spanα holds t...

    work page

  4. [4]

    ˜y ≤ max( b a ˜x + d a , b′ a′ ˜x + d′ a′ )

    work page

  5. [5]

    ˜y > min( b a ˜x + d a , b′ a′ ˜x + d′ a′ )

    work page

  6. [6]

    functions

    ˜y ≥ 0 29 Since b a ∈ (0, 1), then it suffices to find such a lattice point either between y = b a x + d a and y = x, between y = b a x + d a and y = 0, or between y = b a x + d a and y = b′ a′ x + d′ a′ . For each of the 3 cases, any lattice point between the two lines specified in that case will satisfy all four constraints. By the argument in the previ...

    work page

  7. [7]

    S is 1-Lipschitz and monotone non-decreasing

    A segment is a restricted test-function S : [a, b] → [0, 1] where [ a, b] ⊂ [0, 1] which maps a continuous subset [a, b] to [0 , 1]. S is 1-Lipschitz and monotone non-decreasing. The segment’s start-point ( a, S(a)) and the end-point ( b, S(b)) each lie on one of the k lines, in the sense that there exists some i, j∈ [k] where S(a) = si · a and S(b) = sj ...

    work page

  8. [8]

    minimal segment

    A schema Y is a blueprint for a continuous test-function, specifying a sequence of lines {li}i∈[k] that any test-function of the schema must cross. It consists of an integer 0 < M <∞ and two tuples {idx(i)}i∈[M ] ⊂ [k]M , {seci}i∈[M ] ⊂ [k + 1]M , where |idx(i) − idx(i + 1)| ≤1 for all i ∈ [M − 1]. If |idx(i) − idx(i + 1)| = 1, then seci+1 is unique and m...

    work page

Show all 29 references
  1. [9]

    ∈ [k]2 indicates that the ith monotone curve in the schema will have start-point on line yi 1 and end-point on line yi

  2. [10]

    Corollary D.14

    The concatenation of all m − 1 monotone curves yield the basis schema. Corollary D.14. (Completeness of Basis Schema) Given a (k, T)-configuration {si}i∈[k], for any 1 ≤ m ≤ k, say that the list of tuples {(yi 1, yi 2)}i∈[m] ⊂ {1, . . . , k}m is valid if they satisfy the follo...

  3. [12]

    The 1st monotone curve has start-point at the origin, (0, 0)

    = (k, 1) For any m ∈ [k] and valid {(yi 1, yi 2)}i∈[m], define the basis schema, Y{(yi 1,yi 2)}i∈[m] as the concatenation of m − 1 monotone curves, where for i ∈ [m − 1], the ith monotone curve has start-point on line yi 1 ∈ [k] and end-point on line yi 2 ∈ [k]. The 1st monoto...

  4. [13]

    Finally, note that each monotone curve consists of multiple segments

    = (4, 3). Finally, note that each monotone curve consists of multiple segments. The first one has 5 segments, the second one has 4 segments, the third one has 3 segments, and the fourth one has 3 segments. A({si}i∈[k]) ⊂ [ m∈[k],valid {(yi 1,yi 2)}i∈[m] A(Y{(yi 1,yi 2)}i∈[m] )...

  5. [14]

    Algorithm 5 terminates

  6. [15]

    Algorithm 5 returns valid YPairs := {(yi 1, yi 2)i∈[m−1]} where m := |YPairs| + 1 with m ≤ k, and where the valid predicate is defined in the statement of Corollary D.14

  7. [17]

    = (1, k) or (y1 1, y1

  8. [18]

    The last issue about ensuring YPairs satisfies the property that ( y1 1, y1

    = (k, 1). The last issue about ensuring YPairs satisfies the property that ( y1 1, y1

  9. [20]

    The issue arises when Span(Y) ⊊ [k], so that Y does not span all k lines

    = (k, 1) is simple to deal with. The issue arises when Span(Y) ⊊ [k], so that Y does not span all k lines. However, we can note that any test-function which does not span all k lines can be thought of as part of a schema which does span all k lines, except that the segments of...

  10. [21]

    positive

    ̸= ((yi 1)′, (yi 2)′) will be such that yi 2 ̸= (yi 2)′, and one of these schema will have visited Sectoryi 2 or Sector(yi 2)′ for the last time while the other will return to it later. The surjectivity of the mapping can be checked easily. There are i − 1 elements in {1, 2, ....

  11. [22]

    , O(T d)}M −1

    ∀1 ≤ i ≤ min(cK, M− 1), Ai ∈ {−O(T d), . . . , O(T d)}M −1

  12. [23]

    crossing points

    ∀ min(cK, M− 1) ≤ i ≤ M − 1, Ai ∈ {−O((T ′)d), . . . , O((T ′)d)}M −1 51 Then |A| ≤O(T O(K) · (T ′)O(K′)). Proof. If T, T′ ≥ 2, by the homogeneity of the determinant, we can factor out a factor of O(T d) from each of the first min( cK, M− 1) rows of A and a factor of O((T ′)d)...

    1970

  13. [24]

    = (1, k) or ( y1 1, y1

  14. [25]

    adjacent

    = (k, 1) We can the test-function schema, Y{(yi 1,yi 2)}i∈[m] as the concatenation of m − 1 monotone curves, where the ith monotone curve goes from lines yi 1 to yi 2 for i ∈ [m − 1]. The set of these test-function schemas over all valid {(yi 1, yi 2)}i∈[m] satisfying the abov...

  15. [26]

    (Type I) Aj is a convex subset of a ( k − 1)-dimensional plane that has nonzero intersection with the (k − 2)-dimensional intersection of O = L1 ∩ L2 and that is a subset of cl(H1) ∩ cl(H2) ∪ H c 1 ∩ H c 2

  16. [27]

    (Type II) Aj is a convex subset of the quadrant cl( H1) ∩ cl(H2), where: KX i=1 λiBord(1,i) ≥ z K′ X i=1 λ′ iBord(2,i) ≥ z′

  17. [28]

    Lemma D.34 implies that either all Aj are type II or all Aj are type III, which can be seen as follows

    (Type III) Aj is a convex subset of the quadrant, H c 1 ∩ H c 2, where: KX i=1 λiBord(1,i) ≤ z K′ X i=1 λ′ iBord(2,i) ≤ z′ Lemma D.33 implies that it is impossible for any Aj to be Type I since the dimension of Type I sets is k − 1 but Aj is dimension k. Lemma D.34 implies tha...

  18. [29]

    Thus, O is disjoint from S j∈[Nk] int(Aj)

    and H2 ∩ int(H c 1). Thus, O is disjoint from S j∈[Nk] int(Aj). Because there is an Aj of type II and a separate Aj′ of type III, part of SNk j=1 int(Aj) is contained in cl(H1) ∩ cl(H2) and part of it is contained in H c 1 ∩ H c

  19. [30]

    x + y >2, x+ y >1

    However, since no point in O can be in SNk j=1 int(Aj), then SNk j=1 int(Aj) is not connected (i.e. it is the union of two non-empty, separated sets). This contradicts Lemma D.34. Thus, all Aj must be type II or they must all be type III. 66 Finally, observe that (0 , . . . ,0...

  20. [31]

    Once again, {Aj}j∈[Nk] are either all type II or all type III, contradicting that z, z′ > 0, and PK i=1 λi > z andPK′ i=1 λ′ i > z′ respectively

    ̸= ∅. Once again, {Aj}j∈[Nk] are either all type II or all type III, contradicting that z, z′ > 0, and PK i=1 λi > z andPK′ i=1 λ′ i > z′ respectively. Thus, Condition I cannot hold. ■ Here is a Corollary of Lemma D.34 which will be useful in strengthening Lemma D.30, which wi...

  21. [2021]

    Nye, M., Andreassen, A

    URL https://arxiv.org/abs/2102.13019. Nye, M., Andreassen, A. J., Gur-Ari, G., Michalewski, H., Austin, J., Bieber, D., Dohan, D., Lewkowycz, A., Bosma, M., Luan, D., Sutton, C., and Odena, A. Show your work: Scratchpads for intermediate computation with language models, 2021....

    1967 arXiv

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