[Q1] I'm wondering if there are some "official" SHORT existential second order sentences with ONE binary relation, over strings (over a small alphabet) that define an NP-complete set.

(Something similar to 3-colorability that over graphs can be expressed with a short sentence)

All examples that come to my mind are rather LONG/complex even for simple NP-complete problems; for example we can define the instances of the STRING TO STRING CORRECTION PROBLEM $L$ using alphabet $\{ e, 0, 1, \# \}$

$L \subseteq \{ e^k (0|1)^n \# (0|1)^m \}$

$w \in L$ if the first string $S_1$ (binary sequence before $\#$) matches the second string $S_2$ (binary sequence after $\#$) up to at most $k$ edit/delete operations.

For example: $ee001100\#000000$ (with two edits we can transfor $001100 \to 000000$)

In order to define it we can use a SO sentence that begins with $\exists M$ but we need to use a lot of first order constraints and "details":

  • $\#$ occurs once
  • $M$ is a function (represents the copy or delete/insert operations)
  • $x$ belongs to $S_1$: $S_1(x) \equiv \exists y > x \;.\; U_{\#}(y)$
  • $x$ belongs to $S_2$: $S_2(x) \equiv \exists y < x \;.\; U_{\#}(y)$
  • $Diff(x,y)$ iff chars at $x,y$ are different
  • $M$ maps chars from $S_1$ to chars of $S_2$ or from a $e$ to a char of $S_1,S_2$
  • order is preserved: if $x_1, x_2 \in S_1 \land y_1,y_2 \in S_2 \land M(x_1,y_1) \land M(x_2, y_2)$ then $x_1 < x_2 \land y_1 < y_2$
  • all chars are copied/edited/deleted: $\forall x \in S_i \exists y . M(x,y) \lor M(y,x)$
  • edited chars handling: if $S_1(x) \land S_2(y) \land M(x,y) \land Diff(x,y)$ then $\exists z . U_{e}(z) \land ( M(z,x) \lor M(z,y)$
  • deleted chars handling: if $S_i(x) \land \forall y . \neg ( M(x,y) \lor M(y,x) )$ then $\exists z . U_{e}(z) \land M(z,x)$

[Q2.1] Can we get an even shorter sentences that define an NP-complete language if we allow ONE (existential) ternary relation or [Q2.2] more (existential) binary relations?.

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