Cited By
View all- Kasymov N(2025)Separable numberings of universal algebrasMathematics and Theoretical Computer Science10.26907/2949-3919.2024.4.66-1022:4(66-102)Online publication date: 24-Jan-2025
Defining algorithmically presented structures in first order logic
Article No.: 47, Pages 1 - 13
Abstract
We aim to describe the isomorphism types of infinite structures in the language of first-order logic. This pursuit holds importance in logic in computer science, encompassing model theory, descriptional complexity, and the foundations of computability. We introduce the notion of quasi-axiomatizability aimed at describing the isomorphism types of structures. Our focus centers on two classes of algorithmically presented structures. The first is the class of structures for which the positive atomic diagrams are computably enumerable. We call these structures positive structures. The second is the class of structures for which the negative atomic diagrams are computably enumerable. We call these structures negative structures. We study quasi-axiomatizability of structures from these classes by ∃, ∀, ∃∀, and ∀∃-sentences in expansions of languages. Our work is a contribution to the interplay between expressive power of first-order logic, computability, and model theory.
References
[1]
U. Andrews, S. Lempp, J. Miller, K. Ng, L. S. Mauro and A. Sorbi. Universal computably enumerable equivalence relations. Journal of Symbolic Logic, V 79, Issue 1, 60-88, 2014.
[2]
R. Alvir, J. Knight, and C. McCoy. Complexity of Scott sentences. Fund. Math., 251(2):109?129, 2020.
[3]
R. Alvir, Noam Greenberg, M. Harrison-Trainor, and D. Turetsky. Scott complexity of countable structures. The Journal for Symbolic Logic, To appear.
[4]
J. A. Bergstra, J. V. Tucker: On the adequacy of finite equational methods for data type specification. ACM SIGPLAN Notices 14(11): 13-18 (1979).
[5]
J. A. Bergstra, J. V. Tucker: A Characterisation of Computable Data Types by Means of a Finite Equational Specification Method. Proceedings of ICALP 1980, p.76-90.
[6]
J. A. Bergstra, M. Broy, J. V. Tucker, M. Wirsing: On the Power of Algebraic Specifications. MFCS 1981: 193-204.
[7]
J. A. Bergstra, J. V. Tucker: The Completeness of the Algebraic Specification Methods for Computable Data Types. Inf. Control. 54(3): 186-200 (1982).
[8]
J. A. Bergstra, J. V. Tucker: Initial and Final Algebra Semantics for Data Type Specifications: Two Characterization Theorems. SIAM J. Comput. 12(2): 366-387 (1983)
[9]
J. A. Bergstra, J. V. Tucker: Algebraic Specifications of Computable and Semicomputable Data Types. Theor. Comput. Sci. 50: 137-181 (1987)
[10]
J. A. Bergstra and J. V. Tucker. A characterization of computable data types by means of a finite, equational specification method, Lecture Notes in Comput. Sci., Vol. 85, 76--90 (1980).
[11]
J. A. Bergstra, J. V. Tucker: Equational Specifications, Complete Term Rewriting Systems, and Computable and Semicomputable Algebras. J. ACM 42(6): 1194-1230 (1995).
[12]
C. Bernardi and A. Sorbi. Classifying positive equivalence relations. The Journal of Symbolic Logic, 48(3):529--538, 1983.
[13]
A. Blumensath, E. Grädel: Automatic Structures. Proceedings of Logic in Computer Science Conference, p. 51--62, 2000.
[14]
D. Cenzer, J. Remmel. Complexity theoretic model theory and algebra. in Handbook of recursive mathematics, Volume 1. Studies in Logic and the Foundations of Mathematics Vol. 138, p.381-513, 1998.
[15]
R. Downey (editor). Turing's legacy (Developments from Turing's ideas in Logic). Lecture Notes in Logic, Cambridge University Press, 2014.
[16]
R. Downey, A. Kach, S. Lempp, A. Lewis-Pye, A. Montalbán, and D. Turetsky. The complexity of computable categoricity. Adv. Math., p: 268:423, 2015.
[17]
R. Downey, D. Hirschfeldt, B. Khoussainov. Uniformity in Computable Structure Theory, Algebra and Logic, volume 42, p. 318--332, 2003.
[18]
Y. L. Ershov. Theory of numberings. Nauka, Moscow, 1977 (in Russian).
[19]
Yu. L. Ershov, S.S. Goncharov. Constructive models. Transl. from the Russian. (English) (Siberian School of Algebra and Logic) Siberian School of Algebra and Logic. New York, NY: Consultants Bureau. xii, 293 p. (2000).
[20]
Yu. Ershov, A. Nerode, S. Goncharov, J. Remmel (eds). Handbook of Recursive Mathematics, Volume 1. Studies in Logic and Foundations of Mathematics, Elsevier, 1998.
[21]
Yu. Ershov, A. Nerode, S. Goncharov, J. Remmel (eds). Handbook of Recursive Mathematics, Volume 2. Studies in Logic and Foundations of Mathematics, Elsevier, 2001.
[22]
J. Esparza, O. Kupferman, M. Y. Vardi: Verification. Handbook of Automata Theory (II.) 2021: 1415-1456.
[23]
E Fokina, B Khoussainov, P Semukhin, D Turetsky. Linear orders realized by c.e. equivalence relations. The Journal of Symbolic Logic 81 (2), 463-482, 2016.
[24]
E. Fokina, V. Harizanov, and A. Melnikov. Computable model theory, In book Turing's Legacy, Cambridge University Press, p. 124--194, 2014.
[25]
S. Gao and P. Gerdes. Computably enumerable equivalence relations. Studia Logica, 67(1):27--59, 2001.
[26]
A Gavryushkin, B Khoussainov, F Stephan. Reducibilities among equivalence relations induced by recursively enumerable structures. Theoretical Computer Science 612, 137-152, 2016.
[27]
A Gavruskin, S Jain, B Khoussainov, F Stephan. Graphs realised by re equivalence relations. Annals of Pure and Applied Logic 165 (7-8), 1263-1290, 2014.
[28]
Z. Gao, S. Jain, B. Khoussainov, W. Li, A. Melnikov, K. Seidel, F. Stephan. Random subgroups of rationals. Proceedings of the MFCS, p: 25:1--25:14, 2019.
[29]
S. S. Goncharov. Data models and languages for their description. Vych. Sist. (Tr. IM SO AN SSSR), 107, 52?70, 1985.
[30]
S. Goncharov, B. Khoussainov. Open problems in The Theory of Constructive Algebraic Systems, Contemporary Mathematics 257, Computability Theory and Its Applications (Current Trends and open Problems), p.145-170, 2000.
[31]
M.T. Godziszewski and J. D. Hamkins. Computable Quotient Presentations of Models of Arithmetic and Set Theory. In: Logic, Language, Information, and Computation: 24th International Workshop, Proceedings of WoLLIC 2017, pp. 140--152, Ed. by Juliette Kennedy and Ruy J.G.B. de Queiroz. Springer, 2017.
[32]
M. T. Godziszewski. Formal Theories of Truth and Nonstandard Models of Arithmetic. PhD Thesis. University of Warsaw.
[33]
G. Gratzer, Universal Algebra, Van Nostrand, Princeton, NJ, 1968.
[34]
M. Harrison-Trainor. An introduction to the Scott complexity of countable structures and a survey of recent results. Bulletin of Symbolic Logic, Volume 28, Issue 1, p.71-103, 2022.
[35]
M. Harrison-Trainor and M. Ho. On optimal Scott sentences of finitely generated algebraic structures. Proc. Amer. Math. Soc., 146(10):4473- 4485, 2018.
[36]
D. Hirschfeldt and B. Khoussainov. Finitely presented expansions of computably enumerable semigroups, Algebra and Logic, Vol. 51, No. 5, 435--444 (2012).
[37]
W. Hodges, Model Theory. Encyclopaedia of Mathematics and Its Applications, No 42. Cambridge University Press, 1993.
[38]
N.K. Kassymov. On Finitely Approximable and c.e. Representable Algebras, Algebra and Logic, 26, No 6,1986.
[39]
B. Khoussainov. Quantifier Free Definability on Infinite Algebras. Proceedings of Logic in Computer Science Conference, LICS 2016, p.730-738, 2016.
[40]
N. Kasymov and B. Khoussainov. Finitely generated enumerable and absolutely locally finite algebras. Vychislitelnye Sistemy, 116:3--15, 1986 (in Russian).
[41]
N. Kh. Kasymov. Algebras with residually finite positively presented enrichments, Algebra and Logic, Vol. 26, No. 6, 715--730 (1987).
[42]
N. Kh. Kasymov. Homomorphisms onto negative algebras, Algebra and Logic, Vol. 31, No. 2, 132--144 (1992).
[43]
N. Kh. Kasymov. Algebras over negative equivalences, Algebra and Logic, Vol. 33, No. 1, 46--48 (1994).
[44]
N. Kh. Kasymov N. Kh. Positive algebras with congruences of finite index, Algebra and Logic, Vol. 30, No. 3, 293--305 (1991).
[45]
N. Kh. Kasymov. Positive algebras with countable congruence lattices, Algebra and Logic, Vol. 31, No. 1, 21--37 (1992).
[46]
N. K. Kasymov and A. S. Morozov. Definability of linear orders over negative equivalences, Algebra and Logic, Vol. 55, No. 1, 24--37 (2016).
[47]
N. Kh. Kasymov. Recursively separable enumerated algebras, Russian Math. Surveys, Vol. 51, No. 3, 509--538 (1996).
[48]
B. Khoussainov, T. Slaman, and P. Semukhin. [EQUATION]-Presentations of algebras, Arch. Math. Logic, Vol. 45, No. 6, 769--781 (2006).
[49]
B. Khoussainov. Randomness, computability, and algebraic specifications. Annals of Pure and Applied Logic, V 91, Issue 1, p.1-15, 1998.
[50]
B. Khoussainov. A quest for algorithmically random infinite structures. Proceeding of Logic in Computer Science (LICS 2014) conferences, p. 561--569, 2014.
[51]
B. Khoussainov and A. Miasnikov. Finitely presented expansions of groups, semigroups, and algebras. Trans. Amer. Math. Soc., 366(3):1455--1474, 2014.
[52]
B. Khoussainov, A. Nerode. Automatic presentations of structures. D. Lievant (editor) Proceedings, conference on Logic and Computational Complexity, p 367-393 (1994).
[53]
B. Khoussainov, J. Liu. M. Minnes. Unary automatic graphs: an algorithmic perspective. Mathematical Structures in Computer Science, Volume 19, Issue 1, p. 133--152, 2009.
[54]
B. Khoussainov, S. Lempp, and T. A. Slaman, Computably enumerable algebras, their expansions, and isomorphisms, Internat. J. Algebra Comput. 15, 437--454, 2005.
[55]
B. Khoussainov, S. Rubin. Automatic structures: overview and future directions. Journal of Automata, Languages and Combinatorics, Volume 8, Issue 2, p. 287--301, 2003.
[56]
B. Khoussainov, M.Minnes. Model-theoretic complexity of automatic structures. Annals of Pure and Applied Logic Volume 161, Issue 3, p. 416--426, 2009.
[57]
J. F. Knight and J. Millar. Computable structures of rank [EQUATION]. J. Math. Log., 10(1-2):31-43, 2010.
[58]
J. F. Knight and C. McCoy. Index sets and Scott sentences. Arch. Math. Logic, 53(5-6):519-524, 2014.
[59]
A. H. Lachlan. A note on positive equivalence relations. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 33:43--46, 1987.
[60]
A. I. Mal'cev. Constructive algebras. I. Uspehi Mat. Nauk, 16(3 (99)):3-60, 1961.
[61]
L. S. Moss, J. Meseguer, and J. A. Goguen, Final Algebras, Cosemicomputable Algebras, and Degrees of Unsolvability. Theoretical Computer Science 100, 267-302, 1992.
[62]
A. Nies. Separating classes of groups by first-order formulas. Intern. J. Algebra Computation. 13, no 3 (2003) 287-302.
[63]
M. O. Rabin. Computable algebra, general theory and theory of computable fields. Trans. Amer. Math. Soc., 95:341-360, 1960.
[64]
H. Rogers. Theory of Recursive Functions and Effective Computability. The MIT press, 1967.
[65]
M. Roggenbach, A. Cerone, B. H. Schlingloff, and G. Schneider, S.A. Shaikh. Formal Methods for Software Engineering Languages, Methods, Application Domains. Texts in Theoretical computer science, EATCS series, Springer, 2022.
[66]
S. Scott. Logic with denumerably long formulas and finite strings of quantifiers. In Theory of Models (Proc. 1963 Internat. Sympos. Berkeley), pages 329?341. North-Holland, Amsterdam, 1965.
[67]
R. Soare. Recursively Enumerable Sets and Degrees: A Study of Computable Functions and Computably Generated Sets, Springer-Verlag, Berlin, Heidelberg (1987).
[68]
S. Tennenbaum. Non-archimedian models for arithmetic, Notices of the American Mathematical Society, 6, 1959, p. 270.
[69]
J. V. Tucker. Origins and Development of Formal Methods. In Formal Methods for Software Engineering Languages, Methods, Application Domains. Texts in Theoretical computer science, EATCS series, Springer, 2022.
[70]
J. V. Tucker, J. I. Zucker: Abstract computability and algebraic specification. ACM Trans. Comput. Log. 3(2): 279-333 (2002).
[71]
J. V. Tucker, J. I. Zucker. Computable functions and semicomputable sets on many-sorted algebras. Handbook of logic in computer science: Volume 5: Logic and algebraic methods. Pages 397-525, 2001.
[72]
J Loeckx, H -D Ehrich. Algebraic specification of abstract data types. Handbook of logic in computer science: Volume 5: Logic and algebraic methods. Pages 217-316, 2001.
Recommendations
First-Order Modal Logic: Frame Definability and a Lindström Theorem
We generalize two well-known model-theoretic characterization theorems from propositional modal logic to first-order modal logic (FML, for short). We first study FML-definable frames and give a version of the Goldblatt---Thomason theorem for this logic. ...
Tractable Transformations from Modal Provability Logics into First-Order Logic
CADE-16: Proceedings of the 16th International Conference on Automated Deduction: Automated DeductionWe define a class of modal logics LF by uniformly extending a class of modal logics L. Each logic L is characterised by a class of first-order definable frames, but the corresponding logic LF is sometimes characterised by classes of modal frames that ...
Comments
Information & Contributors
Information
Published In
July 2024
988 pages
ISBN:9798400706608
DOI:10.1145/3661814
- Conference Chair:
- Pawel Sobocinski,
- Program Chairs:
- Ugo Dal Lago,
- Javier Esparza
Copyright © 2024 Copyright is held by the owner/author(s). Publication rights licensed to ACM.
Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than the author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected].
Sponsors
- SIGLOG: ACM Special Interest Group on Logic and Computation
- IEEE Computer Society
In-Cooperation
- EACSL
Publisher
Association for Computing Machinery
New York, NY, United States
Publication History
Published: 08 July 2024
Check for updates
Author Tags
Qualifiers
- Research-article
Funding Sources
- NSFC
Conference
LICS '24
Sponsor:
LICS '24: 39th Annual ACM/IEEE Symposium on Logic in Computer Science
July 8 - 11, 2024
Tallinn, Estonia
Acceptance Rates
LICS '24 Paper Acceptance Rate 72 of 236 submissions, 31%;
Overall Acceptance Rate 215 of 622 submissions, 35%
Contributors
Other Metrics
Bibliometrics & Citations
Bibliometrics
Article Metrics
- View Citations1Total Citations
- 66Total Downloads
- Downloads (Last 12 months)66
- Downloads (Last 6 weeks)3
Reflects downloads up to 23 Feb 2025
Other Metrics
Citations
Cited By
View all- Kasymov N(2025)Separable numberings of universal algebrasMathematics and Theoretical Computer Science10.26907/2949-3919.2024.4.66-1022:4(66-102)Online publication date: 24-Jan-2025
View Options
Login options
Check if you have access through your login credentials or your institution to get full access on this article.
Sign in