TY - GEN

T1 - Extensions and Limits of the Specker-Blatter Theorem

AU - Fischer, Eldar

AU - Makowsky, Johann A.

N1 - Publisher Copyright:
© Eldar Fischer and Johann A. Makowsky.

PY - 2024/2

Y1 - 2024/2

N2 - The original Specker-Blatter Theorem (1983) was formulated for classes of structures C of one or several binary relations definable in Monadic Second Order Logic MSOL. It states that the number of such structures on the set [n] is modularly C-finite (MC-finite). In previous work we extended this to structures definable in CMSOL, MSOL extended with modular counting quantifiers. The first author also showed that the Specker-Blatter Theorem does not hold for one quaternary relation (2003). If the vocabulary allows a constant symbol c, there are n possible interpretations on [n] for c. We say that a constant c is hard-wired if c is always interpreted by the same element j ∈ [n]. In this paper we show: (i) The Specker-Blatter Theorem also holds for CMSOL when hard-wired constants are allowed. The proof method of Specker and Blatter does not work in this case. (ii) The Specker-Blatter Theorem does not hold already for C with one ternary relation definable in First Order Logic FOL. This was left open since 1983. Using hard-wired constants allows us to show MC-finiteness of counting functions of various restricted partition functions which were not known to be MC-finite till now. Among them we have the restricted Bell numbers Br,A, restricted Stirling numbers of the second kind Sr,A or restricted Lah-numbers Lr,A. Here r is an non-negative integer and A is an ultimately periodic set of non-negative integers.

AB - The original Specker-Blatter Theorem (1983) was formulated for classes of structures C of one or several binary relations definable in Monadic Second Order Logic MSOL. It states that the number of such structures on the set [n] is modularly C-finite (MC-finite). In previous work we extended this to structures definable in CMSOL, MSOL extended with modular counting quantifiers. The first author also showed that the Specker-Blatter Theorem does not hold for one quaternary relation (2003). If the vocabulary allows a constant symbol c, there are n possible interpretations on [n] for c. We say that a constant c is hard-wired if c is always interpreted by the same element j ∈ [n]. In this paper we show: (i) The Specker-Blatter Theorem also holds for CMSOL when hard-wired constants are allowed. The proof method of Specker and Blatter does not work in this case. (ii) The Specker-Blatter Theorem does not hold already for C with one ternary relation definable in First Order Logic FOL. This was left open since 1983. Using hard-wired constants allows us to show MC-finiteness of counting functions of various restricted partition functions which were not known to be MC-finite till now. Among them we have the restricted Bell numbers Br,A, restricted Stirling numbers of the second kind Sr,A or restricted Lah-numbers Lr,A. Here r is an non-negative integer and A is an ultimately periodic set of non-negative integers.

KW - MC-finiteness

KW - Monadic Second Order Logic

KW - Specker-Blatter Theorem

UR - http://www.scopus.com/inward/record.url?scp=85185226813&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.CSL.2024.26

DO - 10.4230/LIPIcs.CSL.2024.26

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AN - SCOPUS:85185226813

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 32nd EACSL Annual Conference on Computer Science Logic, CSL 2024

A2 - Murano, Aniello

A2 - Silva, Alexandra

T2 - 32nd EACSL Annual Conference on Computer Science Logic, CSL 2024

Y2 - 19 February 2024 through 23 February 2024

ER -