## Small substructures and decidability issues for first-order logic with two variables.(English)Zbl 1284.03136

The authors study a partial case of the decidability problem for FO$$^2$$, the part of the first-order language over a finite relational signature consisting of formulas all of whose variables (not only the free ones) are contained in the two-element set $$\{x,y\}$$. It is a known fact that the satisfiability problem for this language is decidable, and that its set of identically true sentences is decidable.
The authors give a characterization of the satisfiability problems for FO$$^2$$ for classes of structures of the kind $${\mathcal {EQ}}[\tau_0;\tau_1]$$, where $$\tau_0$$ and $$\tau_1$$ are disjoint relational signatures and $$\tau_1$$ consists of at most three binary symbols; these classes consist of all structures of signatures $$\tau_0\cup\tau_1$$ in which all the symbols of $$\tau_1$$ are interpreted as equivalences. It follows from the obtained results that the study of the case $$|\tau_1|\leqslant 3$$ is enough to have a complete picture about decidability and satisfiability for any finite number of symbols in $$\tau_1$$.
Let $$\mathcal K$$ be a class of structures. Denote the satisfiability problem for FO$$^2$$ on $$\mathcal K$$ by SAT(FO$$^2$$, $${\mathcal K}$$) and the finite satisfiability problem for FO$$^2$$ by FINSAT(FO$$^2$$, $${\mathcal K}$$). We say that FO$$^2$$ has a finite (exponential) model property over a class of structures $$\mathcal K$$ if every of its satisfiable sentence has a finite model in $$\mathcal K$$ (of exponentially bounded cardinaility in the length of this sentence).
The main results of the paper are the following.
Theorem.
(i)
FO$$^2$$ has an exponential model property over $${\mathcal {EQ}}[\tau_0;E]$$. Hence SAT(FO$$^2$$, $${\mathcal {EQ}}[\tau_0;E]$$) and FINSAT(FO$$^2$$, $${\mathcal {EQ}}[\tau_0;E]$$) are Nexptime-complete.
(ii)
FO$$^2$$ does not possess the finite model property over $${\mathcal {EQ}}[\tau_0;E_1,E_2]$$.
However, SAT(FO$$^2$$, $${\mathcal {EQ}}[\tau_0;E_1,E_2]$$) is decidable in 3-Nexptime.
(iii)
SAT(FO$$^2$$, $${\mathcal {EQ}}[\tau_0;E_1,E_2,E_3]$$) and FINSAT(FO$$^2$$, $${\mathcal {EQ}}[\tau_0;E_1,E_2,E_3]$$) are undecidable; in fact FO$$^2$$ over $${\mathcal {EQ}}[\tau_0;E_1,E_2,E_3]$$ forms a conservative reduction class.

### MSC:

 03B25 Decidability of theories and sets of sentences 03C07 Basic properties of first-order languages and structures 68Q25 Analysis of algorithms and problem complexity
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### References:

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