Paper Citation Record · LEDGER

Towards a Characterization of Counting and Alternating Classes via Discrete Ordinary Differential Equations

As of 12 August 2026, this Paper Citation Record lists 39 of 39 outbound references and 0 inbound Pith citation observations for arXiv:2608.05431.

A citation records a reference. It does not transfer a finding from one paper to another.

pith.paper-citation-record.v1
2608.05431 v1

Coverage vector

measured 39 of 39 reference resolution

Typed states for the displayed outbound observations.

Source: paper_references, paper_reference_links, observed 2026-08-08T13:37:19.287270Z

measured 39 of 39 standing notices

One-hop event checks from named stored sources.

Source: scholarly_work_events, retraction_status_cache, observed 2026-08-12T06:34:41.77262+00:00

measured 0 of 0 inbound itemization

Pith citing papers itemized under the disclosed page cap.

Source: paper_references, paper_reference_links

measured 0 of 1 external citation measurements

A source-named dated measurement, never combined with another source.

Source: cited_works

Reference resolution

39 of 39 outbound references displayed

verified exact0
verified fuzzy30
unresolved9
parse uncertain0
malformed identifier0
metadata mismatch0

External citation measurements

No source-named external measurement is stored.

Outbound references

Observation 9dc26974-121e-44ee-9a0b-4ab35365f14d · outbound

This paper cites A new characterization of fac 0 via discrete ordinary differential equations.

Towards a Characterization of Counting and Alternating Classes via Discrete Ordinary Differential Equations A new characterization of fac 0 via discrete ordinary differential equations

Reference 1

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T13:37:19.888052Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.

Observation 03dbc132-af37-4d1a-9153-3cf44feffdbe · outbound

This paper cites Characterizing small circuit classes from F AC0 to F AC1 via discrete ordinary differential equations.

Towards a Characterization of Counting and Alternating Classes via Discrete Ordinary Differential Equations Characterizing small circuit classes from F AC0 to F AC1 via discrete ordinary differential equations

Reference 2

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T13:37:19.873297Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.

Observation 13a44148-59fc-4f5d-ac83-417df71e81aa · outbound

This paper cites Towards new characterizations of small circuit classes via discrete ordinary differential equations.Theoretical Computer Science, 1062:115655, 2026.

Towards a Characterization of Counting and Alternating Classes via Discrete Ordinary Differential Equations Towards new characterizations of small circuit classes via discrete ordinary differential equations.Theoretical Computer Science, 1062:115655, 2026

Reference 3

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T13:37:19.857774Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.

Observation 641282fe-ed26-4efc-8c66-3f7ea26e8e01 · outbound

This paper cites Recursion and proof theoretical characteri- zations of small circuit classes with modulo counting via discrete differential equations.

Towards a Characterization of Counting and Alternating Classes via Discrete Ordinary Differential Equations Recursion and proof theoretical characteri- zations of small circuit classes with modulo counting via discrete differential equations

Reference 4

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T13:37:19.841963Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.

Observation 579a04fb-d913-4247-96fb-fc153dcad09b · outbound

This paper cites Cambridge University Press, 2009.

Towards a Characterization of Counting and Alternating Classes via Discrete Ordinary Differential Equations Cambridge University Press, 2009

Reference 5

Resolution: unresolved; no resolver link, observed 2026-08-08T13:37:19.133179Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 175d73bd-c3b0-4f39-850c-730866c3c392 · outbound

This paper cites Springer, 1988.

Towards a Characterization of Counting and Alternating Classes via Discrete Ordinary Differential Equations Springer, 1988

Reference 6

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T13:37:19.815006Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.

Observation 70c40485-703b-4526-82d1-7ef3cbf796f3 · outbound

This paper cites A new recursion-theoretic characterization of poly-time functions.Computational Complexity, 2:97–110, 1992.

Towards a Characterization of Counting and Alternating Classes via Discrete Ordinary Differential Equations A new recursion-theoretic characterization of poly-time functions.Computational Complexity, 2:97–110, 1992

Reference 7

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T13:37:19.799869Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.

Observation f8e8d817-faf0-41b0-862a-dfcb734632f4 · outbound

This paper cites Monotonicity constraints in charac- terizations of pspace.Journal of Logic and Computation, 22(2):179–195, 2012.

Towards a Characterization of Counting and Alternating Classes via Discrete Ordinary Differential Equations Monotonicity constraints in charac- terizations of pspace.Journal of Logic and Computation, 22(2):179–195, 2012

Reference 8

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T13:37:19.785202Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.

Observation 923ceb72-a5fb-4c1c-ae4e-89fc417749e4 · outbound

This paper cites an unresolved cited work.

Towards a Characterization of Counting and Alternating Classes via Discrete Ordinary Differential Equations Unresolved cited work

Reference 9

Resolution: unresolved; raw_fallback, observed 2026-08-08T13:37:19.770567Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.

Observation 68d716c0-4181-4f80-a9e6-3a21a270b7cd · outbound

This paper cites Recursion schemes, discrete differential equations and characterization of polynomial time computations.

Towards a Characterization of Counting and Alternating Classes via Discrete Ordinary Differential Equations Recursion schemes, discrete differential equations and characterization of polynomial time computations

Reference 10

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T13:37:19.756540Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.

Observation 87b6cbff-8921-4743-98a5-42920679fe0c · outbound

This paper cites an unresolved cited work.

Towards a Characterization of Counting and Alternating Classes via Discrete Ordinary Differential Equations Unresolved cited work

Reference 11

Resolution: unresolved; raw_fallback, observed 2026-08-08T13:37:19.742621Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.

Observation b06d3555-db67-4713-ae60-9bbb0024e498 · outbound

This paper cites Polynomial time corresponds to solutions of polynomial ordinary differential equations of polynomial length.J.

Towards a Characterization of Counting and Alternating Classes via Discrete Ordinary Differential Equations Polynomial time corresponds to solutions of polynomial ordinary differential equations of polynomial length.J

Reference 12

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T13:37:19.728578Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.

Observation 082815c3-a7b7-442e-a643-4d9e80a9c8a0 · outbound

This paper cites Two queries.Journal of Computer and System Sciences, 59(2):182–194, 1999.

Towards a Characterization of Counting and Alternating Classes via Discrete Ordinary Differential Equations Two queries.Journal of Computer and System Sciences, 59(2):182–194, 1999

Reference 13

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T13:37:19.713621Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.

Observation 46f230c3-9676-4dca-a393-3c38a62b03b5 · outbound

This paper cites Buss.Bounded Arithmetic.

Towards a Characterization of Counting and Alternating Classes via Discrete Ordinary Differential Equations Buss.Bounded Arithmetic

Reference 14

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T13:37:19.698318Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.

Observation c7ead456-6886-4eba-bbed-73fd8ecb0d0b · outbound

This paper cites Texts in Theoretical Computer Science.

Towards a Characterization of Counting and Alternating Classes via Discrete Ordinary Differential Equations Texts in Theoretical Computer Science

Reference 15

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T13:37:19.684006Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.

Observation 66616a9e-6ddb-41f4-9172-2f7c160ab655 · outbound

This paper cites an unresolved cited work.

Towards a Characterization of Counting and Alternating Classes via Discrete Ordinary Differential Equations Unresolved cited work

Reference 16

Resolution: unresolved; raw_fallback, observed 2026-08-08T13:37:19.669408Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.

Observation 6349b41c-fa07-404c-a753-68810d21584a · outbound

This paper cites Clote and Gaisi Takeuti.

Towards a Characterization of Counting and Alternating Classes via Discrete Ordinary Differential Equations Clote and Gaisi Takeuti

Reference 17

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T13:37:19.655280Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.

Observation 2ad2d10d-5b66-4906-b57a-4cca0b0f1270 · outbound

This paper cites The intrinsic computational difficulty of functions.

Towards a Characterization of Counting and Alternating Classes via Discrete Ordinary Differential Equations The intrinsic computational difficulty of functions

Reference 18

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T13:37:19.640547Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.

Observation 5aafe422-bd69-487d-b976-76857ac40de2 · outbound

This paper cites Implicit recursion-theoretic characteri- zations of counting classes.Archive for Mathematical Logic, 61(7):1129–1144, 2022.

Towards a Characterization of Counting and Alternating Classes via Discrete Ordinary Differential Equations Implicit recursion-theoretic characteri- zations of counting classes.Archive for Mathematical Logic, 61(7):1129–1144, 2022

Reference 19

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T13:37:19.625505Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.

Observation cb447d03-6284-4073-93e9-e8d18792c914 · outbound

This paper cites Generalized first-order spectra and polynomial-time recognizable sets.

Towards a Characterization of Counting and Alternating Classes via Discrete Ordinary Differential Equations Generalized first-order spectra and polynomial-time recognizable sets

Reference 20

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T13:37:19.610462Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.

Observation 420640b1-c22c-4ebf-939e-87443db0a8ec · outbound

This paper cites Computational complexity of probabilistic Turing machines.SIAM Journal of Computing, 6(4):675–695, 1977.

Towards a Characterization of Counting and Alternating Classes via Discrete Ordinary Differential Equations Computational complexity of probabilistic Turing machines.SIAM Journal of Computing, 6(4):675–695, 1977

Reference 21

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T13:37:19.594673Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.

Observation 98674877-53b3-41ee-89a1-51c5f4906539 · outbound

This paper cites an unresolved cited work.

Towards a Characterization of Counting and Alternating Classes via Discrete Ordinary Differential Equations Unresolved cited work

Reference 22

Resolution: unresolved; raw_fallback, observed 2026-08-08T13:37:19.579801Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.

Observation 2386de2e-de7c-473e-9479-9731120e764f · outbound

This paper cites Monotone complexity.

Towards a Characterization of Counting and Alternating Classes via Discrete Ordinary Differential Equations Monotone complexity

Reference 23

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T13:37:19.564770Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.

Observation c17875d1-1c9b-429a-a675-9a0d318fdb7e · outbound

This paper cites Expressibility and parallel complexity.SIAM Journal on Computing, 18(3):625–638, 1989.

Towards a Characterization of Counting and Alternating Classes via Discrete Ordinary Differential Equations Expressibility and parallel complexity.SIAM Journal on Computing, 18(3):625–638, 1989

Reference 24

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T13:37:19.550268Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.

Observation a83e3905-913e-4a8f-9574-0d67f8eab94b · outbound

This paper cites Springer Science & Business Media, 2012.

Towards a Characterization of Counting and Alternating Classes via Discrete Ordinary Differential Equations Springer Science & Business Media, 2012

Reference 25

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T13:37:19.535935Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.

Observation 0202eee1-c7b8-459e-b04b-885c6c726e58 · outbound

This paper cites Function algebraic characterizations of the polytime functions.Comput.

Towards a Characterization of Counting and Alternating Classes via Discrete Ordinary Differential Equations Function algebraic characterizations of the polytime functions.Comput

Reference 26

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T13:37:19.521462Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.

Observation 40547830-3425-4cb8-ba56-361bf883cf64 · outbound

This paper cites A foundational delineation of poly-time.Information and Computation, 110(2):391–420, 1994.

Towards a Characterization of Counting and Alternating Classes via Discrete Ordinary Differential Equations A foundational delineation of poly-time.Information and Computation, 110(2):391–420, 1994

Reference 27

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T13:37:19.506941Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.

Observation 41c5de8a-84cb-41fb-a9d4-478a74152654 · outbound

This paper cites Ramified recurrence and computational complexity II: substitution and poly-space.

Towards a Characterization of Counting and Alternating Classes via Discrete Ordinary Differential Equations Ramified recurrence and computational complexity II: substitution and poly-space

Reference 28

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T13:37:19.492416Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.

Observation 39959339-ff73-4756-b092-27a9c2b5e537 · outbound

This paper cites A characterization of alternating log time by ramified recurrence.Theoretical Computer Science, 236(1-2):193–208, 2000.

Towards a Characterization of Counting and Alternating Classes via Discrete Ordinary Differential Equations A characterization of alternating log time by ramified recurrence.Theoretical Computer Science, 236(1-2):193–208, 2000

Reference 29

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T13:37:19.477927Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.

Observation 619c3bef-b581-4ef5-8866-cbee0c199810 · outbound

This paper cites Characterizing PSPACE with pointers.Mathematical Logic Quarterly, 54(3):323–329, 2008.

Towards a Characterization of Counting and Alternating Classes via Discrete Ordinary Differential Equations Characterizing PSPACE with pointers.Mathematical Logic Quarterly, 54(3):323–329, 2008

Reference 30

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T13:37:19.463300Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.

Observation b54d9517-a213-415b-a8f9-ad7b8bd24779 · outbound

This paper cites A recursion-theoretic approach to np.Annals of Pure and Applied Logic, 162(8):661–666, 2011.

Towards a Characterization of Counting and Alternating Classes via Discrete Ordinary Differential Equations A recursion-theoretic approach to np.Annals of Pure and Applied Logic, 162(8):661–666, 2011

Reference 31

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T13:37:19.447988Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.

Observation 539ea7e2-6f40-4045-8472-b5ae36d171d6 · outbound

This paper cites The polynomial hierarchy of functions and its levels.Theoretical Computer Science, 900:25–34, 2022.

Towards a Characterization of Counting and Alternating Classes via Discrete Ordinary Differential Equations The polynomial hierarchy of functions and its levels.Theoretical Computer Science, 900:25–34, 2022

Reference 32

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T13:37:19.431601Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.

Observation 6424acfe-3d12-4c68-8bf4-f1b5bd35f14f · outbound

This paper cites Subrecursiveness: Machine-independent notions of computability in restricted time and storage.Mathematical Systems Theory, 6(1):3–15, 1972.

Towards a Characterization of Counting and Alternating Classes via Discrete Ordinary Differential Equations Subrecursiveness: Machine-independent notions of computability in restricted time and storage.Mathematical Systems Theory, 6(1):3–15, 1972

Reference 33

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T13:37:19.417006Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.

Observation 542b417a-2c07-48d6-ba9f-44b661ebb1f1 · outbound

This paper cites The complexity of computing the permanent.Theoretical Computer Science, 8(2):189–201, 1979.

Towards a Characterization of Counting and Alternating Classes via Discrete Ordinary Differential Equations The complexity of computing the permanent.Theoretical Computer Science, 8(2):189–201, 1979

Reference 34

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T13:37:19.401524Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.

Observation cc2f8c19-3f94-4c8d-a4ff-69d97d51837f · outbound

This paper cites an unresolved cited work.

Towards a Characterization of Counting and Alternating Classes via Discrete Ordinary Differential Equations Unresolved cited work

Reference 35

Resolution: unresolved; raw_fallback, observed 2026-08-08T13:37:19.385136Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.

Observation d3f951ec-18c6-47d4-99b6-45d40ba4095d · outbound

This paper cites beginning of the tape marker.

Towards a Characterization of Counting and Alternating Classes via Discrete Ordinary Differential Equations beginning of the tape marker

Reference 36

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T13:37:19.370575Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.

Observation 9403c9c4-676f-43fe-9175-681e7c5e12f7 · outbound

This paper cites an unresolved cited work.

Towards a Characterization of Counting and Alternating Classes via Discrete Ordinary Differential Equations Unresolved cited work

Reference 37

Resolution: unresolved; raw_fallback, observed 2026-08-08T13:37:19.355994Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.

Observation 38dcde8f-455b-4577-a14a-b5fa94b02b45 · outbound

This paper cites an unresolved cited work.

Towards a Characterization of Counting and Alternating Classes via Discrete Ordinary Differential Equations Unresolved cited work

Reference 38

Resolution: unresolved; raw_fallback, observed 2026-08-08T13:37:19.340808Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.

Observation dcbd3f29-98e9-45ea-bad0-e9c22b4b8a64 · outbound

This paper cites an unresolved cited work.

Towards a Characterization of Counting and Alternating Classes via Discrete Ordinary Differential Equations Unresolved cited work

Reference 39

Resolution: unresolved; raw_fallback, observed 2026-08-08T13:37:19.325080Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.

Pith citing papers

No inbound Pith citation observations are available.