Paper Citation Record · LEDGER

EXPSPACE-Completeness of the Logics K4xS5 and S4xS5 and the Logic of Subset Spaces, Part 2: EXPSPACE-Hardness

As of 17 August 2026, this Paper Citation Record lists 33 of 33 outbound references and 0 inbound Pith citation observations for arXiv:1908.03509.

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

pith.paper-citation-record.v1
1908.03509 v1

Coverage vector

measured 33 of 33 reference resolution

Typed states for the displayed outbound observations.

Source: paper_references, paper_reference_links, observed 2026-08-14T14:27:03.737886Z

measured 33 of 33 standing notices

One-hop event checks from named stored sources.

Source: scholarly_work_events, retraction_status_cache, observed 2026-08-16T06:30:59.297886+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

33 of 33 outbound references displayed

verified exact0
verified fuzzy18
unresolved15
parse uncertain0
malformed identifier0
metadata mismatch0

External citation measurements

No source-named external measurement is stored.

Outbound references

Observation b6c8d1fa-b1f5-4b0a-889b-4c7a5420da4b · outbound

This paper cites an unresolved cited work.

EXPSPACE-Completeness of the Logics K4xS5 and S4xS5 and the Logic of Subset Spaces, Part 2: EXPSPACE-Hardness Unresolved cited work

Reference 1

Resolution: unresolved; raw_fallback, observed 2026-08-14T14:27:04.136514Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation eefc75ac-48d0-4c0f-b960-9011f2d30996 · outbound

This paper cites an unresolved cited work.

EXPSPACE-Completeness of the Logics K4xS5 and S4xS5 and the Logic of Subset Spaces, Part 2: EXPSPACE-Hardness Unresolved cited work

Reference 2

Resolution: unresolved; raw_fallback, observed 2026-08-14T14:27:04.125392Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation 9a17224f-3a58-4e8b-a08a-a1a7ed1922af · outbound

This paper cites an unresolved cited work.

EXPSPACE-Completeness of the Logics K4xS5 and S4xS5 and the Logic of Subset Spaces, Part 2: EXPSPACE-Hardness Unresolved cited work

Reference 3

Resolution: unresolved; raw_fallback, observed 2026-08-14T14:27:04.115105Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation 88beadb4-c361-4a1b-a9ba-92e2861ff92b · outbound

This paper cites an unresolved cited work.

EXPSPACE-Completeness of the Logics K4xS5 and S4xS5 and the Logic of Subset Spaces, Part 2: EXPSPACE-Hardness Unresolved cited work

Reference 4

Resolution: unresolved; raw_fallback, observed 2026-08-14T14:27:04.104871Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation 3a05b98a-6393-45c8-9513-56a400e27490 · outbound

This paper cites Dabrowski, L.

EXPSPACE-Completeness of the Logics K4xS5 and S4xS5 and the Logic of Subset Spaces, Part 2: EXPSPACE-Hardness Dabrowski, L

Reference 5

Resolution: verified fuzzy; raw_fallback, observed 2026-08-14T14:27:04.094796Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation 4736bb7e-feb3-4414-ae3b-c77921844c2d · outbound

This paper cites Heinemann.

EXPSPACE-Completeness of the Logics K4xS5 and S4xS5 and the Logic of Subset Spaces, Part 2: EXPSPACE-Hardness Heinemann

Reference 6

Resolution: verified fuzzy; raw_fallback, observed 2026-08-14T14:27:04.084952Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation d28bbc7c-e7a7-406e-aff1-f1368cd40a7e · outbound

This paper cites Heinemann.

EXPSPACE-Completeness of the Logics K4xS5 and S4xS5 and the Logic of Subset Spaces, Part 2: EXPSPACE-Hardness Heinemann

Reference 7

Resolution: verified fuzzy; raw_fallback, observed 2026-08-14T14:27:04.074083Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation 7d04760c-9301-4f2f-8561-835264492b89 · outbound

This paper cites Hertling and G.

EXPSPACE-Completeness of the Logics K4xS5 and S4xS5 and the Logic of Subset Spaces, Part 2: EXPSPACE-Hardness Hertling and G

Reference 8

Resolution: verified fuzzy; raw_fallback, observed 2026-08-14T14:27:04.062917Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation 60763da2-70c1-472f-a2bd-ce8d91ba6fb2 · outbound

This paper cites an unresolved cited work.

EXPSPACE-Completeness of the Logics K4xS5 and S4xS5 and the Logic of Subset Spaces, Part 2: EXPSPACE-Hardness Unresolved cited work

Reference 9

Resolution: unresolved; raw_fallback, observed 2026-08-14T14:27:04.052769Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation e456e1e7-fbba-4326-a4e0-30a81a4f7541 · outbound

This paper cites an unresolved cited work.

EXPSPACE-Completeness of the Logics K4xS5 and S4xS5 and the Logic of Subset Spaces, Part 2: EXPSPACE-Hardness Unresolved cited work

Reference 10

Resolution: unresolved; raw_fallback, observed 2026-08-14T14:27:04.042402Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation 15f6692f-817c-48bc-b5e6-f25eb27b19f5 · outbound

This paper cites an unresolved cited work.

EXPSPACE-Completeness of the Logics K4xS5 and S4xS5 and the Logic of Subset Spaces, Part 2: EXPSPACE-Hardness Unresolved cited work

Reference 11

Resolution: unresolved; raw_fallback, observed 2026-08-14T14:27:04.032241Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation e33f879e-e3bf-4eec-a24c-9abcc42966d7 · outbound

This paper cites an unresolved cited work.

EXPSPACE-Completeness of the Logics K4xS5 and S4xS5 and the Logic of Subset Spaces, Part 2: EXPSPACE-Hardness Unresolved cited work

Reference 12

Resolution: unresolved; raw_fallback, observed 2026-08-14T14:27:04.021314Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation 4fa05b4e-d6fa-4d4f-a952-c20c15022dd1 · outbound

This paper cites Kurucz, F.

EXPSPACE-Completeness of the Logics K4xS5 and S4xS5 and the Logic of Subset Spaces, Part 2: EXPSPACE-Hardness Kurucz, F

Reference 13

Resolution: verified fuzzy; raw_fallback, observed 2026-08-14T14:27:04.011222Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation bdf74bb5-1363-4aac-8b42-03ae59debefe · outbound

This paper cites Lange and C.

EXPSPACE-Completeness of the Logics K4xS5 and S4xS5 and the Logic of Subset Spaces, Part 2: EXPSPACE-Hardness Lange and C

Reference 14

Resolution: verified fuzzy; raw_fallback, observed 2026-08-14T14:27:04.000767Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation c52e3a21-1489-4262-b720-2a5d02701c13 · outbound

This paper cites an unresolved cited work.

EXPSPACE-Completeness of the Logics K4xS5 and S4xS5 and the Logic of Subset Spaces, Part 2: EXPSPACE-Hardness Unresolved cited work

Reference 15

Resolution: unresolved; raw_fallback, observed 2026-08-14T14:27:03.989689Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation 69954da4-0488-47fc-ac42-9151482639a9 · outbound

This paper cites an unresolved cited work.

EXPSPACE-Completeness of the Logics K4xS5 and S4xS5 and the Logic of Subset Spaces, Part 2: EXPSPACE-Hardness Unresolved cited work

Reference 16

Resolution: unresolved; raw_fallback, observed 2026-08-14T14:27:03.979050Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation 3e7ebd9d-70dd-49c5-b3b9-4ab85c8c18bf · outbound

This paper cites Parikh, L.

EXPSPACE-Completeness of the Logics K4xS5 and S4xS5 and the Logic of Subset Spaces, Part 2: EXPSPACE-Hardness Parikh, L

Reference 17

Resolution: verified fuzzy; raw_fallback, observed 2026-08-14T14:27:03.968956Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation b0e4a1ec-6554-4d8d-90e3-c0218e0b0638 · outbound

This paper cites shared variables.

EXPSPACE-Completeness of the Logics K4xS5 and S4xS5 and the Logic of Subset Spaces, Part 2: EXPSPACE-Hardness shared variables

Reference 18

Resolution: verified fuzzy; raw_fallback, observed 2026-08-14T14:27:03.956913Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation ebf7d1ab-c267-4da9-9aee-508db5dadcdb · outbound

This paper cites an unresolved cited work.

EXPSPACE-Completeness of the Logics K4xS5 and S4xS5 and the Logic of Subset Spaces, Part 2: EXPSPACE-Hardness Unresolved cited work

Reference 19

Resolution: unresolved; raw_fallback, observed 2026-08-14T14:27:03.945400Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation fb1bd509-6460-4d9f-8a5c-ed2a58bb9811 · outbound

This paper cites Note that the value ofX is copied fromp′ i to its ♦ →-successorpi+1 because by the formula persistent(X) we force the vectorX of propositional variables to be persistent.

EXPSPACE-Completeness of the Logics K4xS5 and S4xS5 and the Logic of Subset Spaces, Part 2: EXPSPACE-Hardness Note that the value ofX is copied fromp′ i to its ♦ →-successorpi+1 because by the formula persistent(X) we force the vectorX of propositional variables to be persistent

Reference 20

Resolution: verified fuzzy; raw_fallback, observed 2026-08-14T14:27:03.932811Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation 8ed8394e-a37e-4c91-ba54-ee4a29610679 · outbound

This paper cites an unresolved cited work.

EXPSPACE-Completeness of the Logics K4xS5 and S4xS5 and the Logic of Subset Spaces, Part 2: EXPSPACE-Hardness Unresolved cited work

Reference 21

Resolution: unresolved; raw_fallback, observed 2026-08-14T14:27:03.919465Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation 5d191d34-f38d-4f1a-b7f1-f607dd7ae147 · outbound

This paper cites We deﬁne an S4-frame(W1,R ♦) by W1 :={0,..., 2n− 1} and, fori,i′∈W1, iR♦i′ :⇐⇒ i≤i′.

EXPSPACE-Completeness of the Logics K4xS5 and S4xS5 and the Logic of Subset Spaces, Part 2: EXPSPACE-Hardness We deﬁne an S4-frame(W1,R ♦) by W1 :={0,..., 2n− 1} and, fori,i′∈W1, iR♦i′ :⇐⇒ i≤i′

Reference 22

Resolution: verified fuzzy; raw_fallback, observed 2026-08-14T14:27:03.907785Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation efc4f3b7-0dd2-4990-81ea-00e92ee66227 · outbound

This paper cites staircase.

EXPSPACE-Completeness of the Logics K4xS5 and S4xS5 and the Logic of Subset Spaces, Part 2: EXPSPACE-Hardness staircase

Reference 23

Resolution: verified fuzzy; raw_fallback, observed 2026-08-14T14:27:03.896023Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation 40c87fb8-22ff-490d-b545-57ad7db796d1 · outbound

This paper cites an unresolved cited work.

EXPSPACE-Completeness of the Logics K4xS5 and S4xS5 and the Logic of Subset Spaces, Part 2: EXPSPACE-Hardness Unresolved cited work

Reference 24

Resolution: unresolved; raw_fallback, observed 2026-08-14T14:27:03.884458Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation ee393f9b-c58e-4036-9ac0-f9e3f0b747fa · outbound

This paper cites can be mapped to.

EXPSPACE-Completeness of the Logics K4xS5 and S4xS5 and the Logic of Subset Spaces, Part 2: EXPSPACE-Hardness can be mapped to

Reference 25

Resolution: verified fuzzy; raw_fallback, observed 2026-08-14T14:27:03.872589Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation c9aee480-0832-4917-a9bb-57db1d5ab08d · outbound

This paper cites So, we shall assume thatr1 ♦ →x, for allv∈W1, andRL =W2×W2.

EXPSPACE-Completeness of the Logics K4xS5 and S4xS5 and the Logic of Subset Spaces, Part 2: EXPSPACE-Hardness So, we shall assume thatr1 ♦ →x, for allv∈W1, andRL =W2×W2

Reference 26

Resolution: verified fuzzy; raw_fallback, observed 2026-08-14T14:27:03.861023Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation 7b775f9c-d060-4a93-9eb6-8233590a6ab2 · outbound

This paper cites 2.{(π(v),π (v′)) : v,v′∈V and vEv′}⊆ R♦.

EXPSPACE-Completeness of the Logics K4xS5 and S4xS5 and the Logic of Subset Spaces, Part 2: EXPSPACE-Hardness 2.{(π(v),π (v′)) : v,v′∈V and vEv′}⊆ R♦

Reference 27

Resolution: verified fuzzy; raw_fallback, observed 2026-08-14T14:27:03.849604Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation 1c1c1056-25a3-4dfa-bff2-1dd1cf9a9850 · outbound

This paper cites A Reduction of ATMs Working in Exponential Time to S4× S5 72.

EXPSPACE-Completeness of the Logics K4xS5 and S4xS5 and the Logic of Subset Spaces, Part 2: EXPSPACE-Hardness A Reduction of ATMs Working in Exponential Time to S4× S5 72

Reference 28

Resolution: verified fuzzy; raw_fallback, observed 2026-08-14T14:27:03.838143Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation e7a43fb8-dad2-47a9-93bb-1178d138922e · outbound

This paper cites partial tree ofM on inputw.

EXPSPACE-Completeness of the Logics K4xS5 and S4xS5 and the Logic of Subset Spaces, Part 2: EXPSPACE-Hardness partial tree ofM on inputw

Reference 29

Resolution: verified fuzzy; raw_fallback, observed 2026-08-14T14:27:03.825400Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation 9c969b4f-fae0-40dd-abaa-91fec108f2b4 · outbound

This paper cites an unresolved cited work.

EXPSPACE-Completeness of the Logics K4xS5 and S4xS5 and the Logic of Subset Spaces, Part 2: EXPSPACE-Hardness Unresolved cited work

Reference 30

Resolution: unresolved; raw_fallback, observed 2026-08-14T14:27:03.814390Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation 6bbc4635-f677-433f-9608-efd70de49d92 · outbound

This paper cites So, the second condition is satisﬁed.

EXPSPACE-Completeness of the Logics K4xS5 and S4xS5 and the Logic of Subset Spaces, Part 2: EXPSPACE-Hardness So, the second condition is satisﬁed

Reference 31

Resolution: verified fuzzy; raw_fallback, observed 2026-08-14T14:27:03.801326Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation a0daa93f-ad7b-4633-8053-588e4fc7acc0 · outbound

This paper cites an unresolved cited work.

EXPSPACE-Completeness of the Logics K4xS5 and S4xS5 and the Logic of Subset Spaces, Part 2: EXPSPACE-Hardness Unresolved cited work

Reference 32

Resolution: unresolved; raw_fallback, observed 2026-08-14T14:27:03.789212Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation e11118c3-f613-430d-912f-62b8d0fa04fd · outbound

This paper cites a partial tree ofM on inputw.

EXPSPACE-Completeness of the Logics K4xS5 and S4xS5 and the Logic of Subset Spaces, Part 2: EXPSPACE-Hardness a partial tree ofM on inputw

Reference 33

Resolution: verified fuzzy; raw_fallback, observed 2026-08-14T14:27:03.776836Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Pith citing papers

No inbound Pith citation observations are available.