Andreas Krebs
ABSTRACT
We study an extension of FO2[<], first-order logic interpreted in finite words, in which formulas are restricted to use only two variables. We adjoin to this language two-variable atomic formulas that say, 'the letter a appears between positions x and y'. This is, in a sense, the simplest property that is not expressible using only two variables.
We present several logics, both first-order and temporal, that have the same expressive power, and find matching lower and upper bounds for the complexity of satisfiability for each of these formulations. We also give an effective necessary condition, in terms of the syntactic monoid of a regular language, for a property to be expressible in this logic. We show that this condition is also sufficient for words over a two-letter alphabet. This algebraic analysis allows us us to prove, among other things, that our new logic has strictly less expressive power than full first-order logic FO[<].
- Jorge Almeida. A syntactical prof of the locality of DA, Int. J. Alg. Comput. 6 (1996), 165--177.Google Scholar
Cross Ref
- Rajeev Alur and Thomas Henzinger. A really temporal logic, J. ACM 41.1 (Jan 1994), 181--203. Google ScholarDigital Library
- Volker Diekert, Paul Gastin and Manfred Kufleitner. First-order logic over finite words, Int. J. Found. Comp. Sci. 19 (2008), 513--548.Google ScholarCross Ref
- Kousha Etessami, Moshe Vardi and Thomas Wilke. First-order logic with two variables and unary temporal logic. Inform. Comput. 179.2 (2002), 279--295. Google ScholarDigital Library
- Martin Fürer. The computational complexity of the unconstrained domino problem (with implications for logical decision problems), Proc. Logic and machines: decision problems and complexity, Münster (Egon Börger, Gisbert Hasenjaeger and Dieter Rödding, eds.), LNCS 171 (1984), 312--319. Google ScholarDigital Library
- Erich Grädel. Decision procedures for guarded logics, Proc. 16th CADE, Trento (Harald Ganzinger, ed.), LNCS 1632 (1999), 31--51. Google ScholarDigital Library
- Neil Immerman and Dexter Kozen. Definability with bounded number of bound variables, Inform. Comput. 83.2 (1989), 236--244. Google ScholarDigital Library
- Andreas Krebs, Kamal Lodaya, Paritosh Pandya and Howard Straubing. Two-variable logic with a between predicate (Preprint, Arxiv, 2016) 1603.05625.Google Scholar
- François Laroussinie, Antoine Meyer and Eudes Petonnet. Counting LTL, Proc. 17th TIME, Paris (Nicolas Markey and Jef Wijsen, eds.) (IEEE, 2010), 51--58. Google ScholarDigital Library
- Kamal Lodaya, Paritosh Pandya and Simoni Shah. Around dot depth two, Proc. 14th DLT, London (Canada) (Yuan Gao, Hanlin Lu, Shinnosuke Seki and Sheng Yu, eds.), LNCS 6224 (2010), 303--314. Google ScholarDigital Library
- Robert McNaughton and Seymour Papert. Counter-free automata (MIT Press, 1971).Google Scholar
- Martin Otto. Two variable first-order logic over ordered domains, J. Symb. Log. 66.2 (2001), 685--702.Google ScholarCross Ref
- Jean-Éric Pin. Varieties of Formal Languages (Plenum, 1986). Google ScholarDigital Library
- Thomas Place and Luc Segoufin. Decidable characterization of FO2(<, +1) and locality of DA (Preprint, ENS Cachan, 2014).Google Scholar
- Marcel-Paul Schützenberger. On finite monoids having only trivial subgroups, Inform. Contr. 8 (1965), 190--194.Google ScholarCross Ref
- Marcel-Paul Schützenberger. Sur le produit de concaténation non ambigu, Semigroup Forum 13 (1976), 47--75.Google ScholarCross Ref
- Michael Sipser. Borel sets and circuit complexity, Proc. 15th STOC, Boston (David Johnson, Ronald Fagin, Michael Fredman, David Harel, Richard Karp, Nancy Lynch, Christos Papadimitriou, Ronald Rivest, Walter Ruzzo and Joel Seiferas, eds.), (ACM, 1983), 61--69. Google ScholarDigital Library
- Aravinda Prasad Sistla and Edmund Clarke. The complexity of propositional linear temporal logics, J. ACM 32.3 (1985), 733--749. Google ScholarDigital Library
- Howard Straubing. Semigroups and languages of dot-depth two, Theoret. Comp. Sci. 58 (1988), 361--378. Google ScholarDigital Library
- Howard Straubing. Finite Automata, Formal Languages, and Circuit Complexity (Birkhäuser, 1994). Google ScholarDigital Library
- Howard Straubing. On logical descriptions of regular languages, Proc. 5th Latin, Cancun (Sergio Rajsbaum, ed.), LNCS 2286 (2002), 528--538. Google ScholarDigital Library
- Bret Tilson. Categories as algebra, J. Pure Appl. Alg. 48 (1987), 83--198.Google ScholarCross Ref
- Denis Thérien and Thomas Wilke. Over words, two variables are as powerful as one quantifier alternation, Proc. 30th STOC, Dallas (Jeffrey Vitter, ed.) (ACM, 1998), 234--240. Google ScholarDigital Library
- Philipp Weis and Neil Immerman. Structure theorem and strict alternation hierarchy for FO2 on words, Log. Meth. Comp. Sci. 5.3:3 (2009), 1--23.Google Scholar
- Thomas Wilke. Classifying discrete temporal properties, Proc. 16th STACS, Trier (Christoph Meinel and Sophie Tison, eds.), LNCS 1563 (1999), 31--46. Google ScholarDigital Library
Index Terms
- Two-variable Logic with a Between Relation
Recommendations
Two-Variable Separation Logic and Its Inner Circle
Separation logic is a well-known assertion language for Hoare-style proof systems. We show that first-order separation logic with a unique record field restricted to two quantified variables and no program variables is undecidable. This is among the ...
Complexity of Two-Variable Dependence Logic and IF-Logic
LICS '11: Proceedings of the 2011 IEEE 26th Annual Symposium on Logic in Computer ScienceWe study the two-variable fragments D^2 and IF^2 of dependence logic and independence-friendly logic. We consider the satisfiability and finite satisfiability problems of these logics and show that for D^2, both problems are NEXPTIME-complete, whereas ...
Two-Variable First-Order Logic with Equivalence Closure
LICS '12: Proceedings of the 2012 27th Annual IEEE/ACM Symposium on Logic in Computer ScienceWe consider the satisfiability and finite satisfiability problems for extensions of the two-variable fragment of first-order logic in which an equivalence closure operator can be applied to a fixed number of binary predicates. We show that the ...
Comments