Decidable ${\exists }^*{\forall }^*$ First-Order Fragments of Linear Rational Arithmetic with Uninterpreted Predicates

Abstract

First-order linear rational arithmetic enriched with uninterpreted predicates yields an interesting and very expressive modeling language. However, already the presence of a single uninterpreted predicate symbol of arity one or greater renders the associated satisfiability problem undecidable. We identify two decidable fragments, both based on the Bernays–Schönfinkel–Ramsey prefix class. Due to the inherent infiniteness of the underlying domain, a finite model property in the usual sense cannot be established. Nevertheless, we show that satisfiable sentences always have a model that can be described by finite means. To this end, we restrict the syntax of arithmetic atoms. In the first fragment that is presented, arithmetic operations are only allowed over terms without universally quantified variables. In the second fragment, arithmetic atoms are essentially confined to difference constraints over universally quantified variables with bounded range. We will call such atoms bounded difference constraints. As bounded intervals over the rationals still comprise infinitely many values, a trivial instantiation procedure is not sufficient to solve the satisfiability problem. After a slight shift of perspective, the positive decidability result for the first fragment can be restated in the framework of combinations of theories over non-disjoint vocabularies. More precisely, we combine first-order theories that share a dense total order without endpoints.

Technical Appendix

1.1 Some Results Based on Ramsey Theory

In the present section we prove Lemmas 7 and 8.

Lemma 7

Let $n, m > 0$ be positive integers. Let $\chi : {\mathbb {Q}}^m \rightarrow {\mathcal {C}}$ be any coloring. There is some positive integer ${\widehat{n}}$ such that for every set $R \subseteq {\mathbb {Q}}$ with $|R| \ge {\widehat{n}}$, i.e. R needs to be sufficiently large, there exists a subset $Q \subseteq R$ of cardinality n such that all ascending tuples $\langle r_1, \ldots , r_m\rangle \in Q^m$ are assigned the same color by $\chi$.

Proof

This proof is an adaptation of the proof of Ramsey’s Theorem on page 7 in  [62]. For $n < m$ the lemma is trivially satisfied, since in this case $Q^m$ cannot contain any ascending tuple. Hence, we assume $n \ge m$. In order to avoid technical difficulties when defining the sequence of elements $s_{m-1}, s_{m}, s_{m+1}, \ldots$ below, we assume for the rest of the proof that R is finite but sufficiently large. This assumption does not pose a restriction, as we could always consider a sufficiently large finite subset of R, if R were to be infinite.

We proceed by induction on $m \ge 1$. The base case $m=1$ is easy, since $\chi$ can assign only finitely many colors to elements in R and thus some color must be assigned at least $\big \lfloor \tfrac{|R|}{|{\mathcal {C}}|} \big \rfloor$ times. Hence, if R contains at least $n |{\mathcal {C}}|$ elements, we find a uniformly colored subset Q of size n. Suppose $m>1$. At first, we pick the $m-2$ smallest rational numbers $s_1< \cdots < s_{m-2}$ from R and set $S_{m-2} := R {\setminus } \{ s_1, \ldots , s_{m-2}\}$. Thereafter, we simultaneously construct two sufficiently long but finite sequences $s_{m-1}, s_m, s_{m+1}, \ldots$ and $S_{m-1}, S_m, S_{m+1}, \ldots$ as follows:

Given $S_i$, we define $s_{i+1}$ to be the smallest rational number in $S_i$.

Given $S_i$ and the element $s_{i+1}$, we define an equivalence relation $\sim _i$ on the set $S'_i := S_i {\setminus } \{ s_{i+1} \}$ so that $s \sim _i s'$ holds if and only if for every sequence of indices $j_1, \ldots , j_{m-1}$ with $1 \le j_1< \cdots < j_{m-1} \le i+1$, we have $\chi (s_{j_1}, \ldots , s_{j_{m-1}}, s) = \chi (s_{j_1}, \ldots , s_{j_{m-1}}, s')$. This equivalence relation partitions $S'_i$ into at most $|{\mathcal {C}}|^{{i+1}\atopwithdelims (){m-1}}$ equivalence classes. We choose one such class with largest cardinality to be $S_{i+1}$.

By construction of the sequence $s_1, s_2, s_3, \ldots$, we must have $\chi (s_{j_1}, \ldots , s_{j_{m-1}}, s_{K}) = \chi (s_{j_1}, \ldots ,$ $s_{j_{m-1}}, s_{K'})$ for every sequence of indices $j_1< \ldots < j_{m-1}$ and all indices $K, K' \ge j_{m-1}+1$. Notice that this covers all ascending m-tuples in $\{s_1, s_2, s_3, \ldots \}^m$ starting with $s_{j_1}, \ldots , s_{j_m-1}$, i.e. they all share the same color. We now define a new coloring $\chi ' : \{s_1, s_2, s_3, \ldots \}^{m-1} \rightarrow {\mathcal {C}}$ so that $\chi '(s_{j_1}, \ldots ,$ $s_{j_{m-1}}) := \chi (s_{j_1}, \ldots , s_{j_{m-1}}, s_{j_{m-1}+1})$ for every sequence of indices $j_1< \cdots < j_{m-1}$ (in case of $j_{m-1}$ being the index of the last element in the sequence $s_1, s_2, s_3, \ldots$, $\chi '(s_{j_1}, \ldots , s_{j_{m-1}})$ shall be an arbitrary color from ${\mathcal {C}}$). By induction, there exists a subset $Q \subseteq \{s_1, s_2, s_3, \ldots \}$ of cardinality n, such that every ascending $(m-1)$-tuple ${\bar{r}} \in Q^{m-1}$ is colored the same by $\chi '$. The definition of $\chi '$ entails that now all ascending m-tuples ${\bar{r}}' \in Q^m$ are colored the same by $\chi$. Hence, Q is the sought set. $\square$

In order to prove Lemma 8, we formulate and prove two auxiliary results.

Lemma 62

Let $n, m, p > 0$ be positive integers and let $\chi : {\mathbb {Q}}^{m p} \rightarrow {\mathcal {C}}$ be an arbitrary coloring. Let $R_1, \ldots , R_p$ be sufficiently large but finite subsets of ${\mathbb {Q}}$. There exist subsets $Q_1 \subseteq R_1, \ldots , Q_p \subseteq R_p$, each of cardinality n and there is some color $C \in {\mathcal {C}}$, such that for all ascending m-tuples ${\bar{r}}_1 \in Q_1^m, \ldots , {\bar{r}}_p \in Q_p^m$ we have $\chi ({\bar{r}}_1, \ldots , {\bar{r}}_p) = C$.

Proof

This proof is an adaptation of the proof of Theorem 5 on page 113 in  [62]. As in the proof of Lemma 7, we assume $n \ge m$. We proceed by induction on $p \ge 1$. The case $p=1$ is covered by Lemma 7. Suppose $p > 1$. We define an equivalence relation $\sim _p$ over the set $R_p^m$ so that ${\bar{s}}\sim _p {\bar{s}}'$ holds if and only if for all ascending tuples ${\bar{r}}_1 \in R_1^m, \ldots , {\bar{r}}_{p-1} \in R_{p-1}^m$ the colors $\chi \big ({\bar{r}}_1, \ldots , {\bar{r}}_{p-1}, {\bar{s}}\big )$ and $\chi \big ({\bar{r}}_1, \ldots , {\bar{r}}_{p-1}, {\bar{s}}'\big )$ are identical. This equivalence relation induces at most $|{\mathcal {C}}|^{{{|R_1|}\atopwithdelims (){m}}\cdot \ldots \cdot {{|R_{p-1}|}\atopwithdelims (){m}}}$ equivalence classes over $R_p^m$. It thus induces a coloring of $\chi ' : R_p^m \rightarrow {\mathcal {C}}'_p$ where ${\mathcal {C}}'_p$ contains one color for each of these equivalence classes. By virtue of Lemma 7, we can construct a subset $Q_p \subseteq R_p$ with n elements such that all ascending m-tuples ${\bar{r}}\in Q_p^m$ are colored identically by $\chi '$. Let the coloring $\chi ''$ be defined by $\chi ''({\bar{r}}_1, \ldots , {\bar{r}}_{p-1}) := \chi ({\bar{r}}_1, \ldots , {\bar{r}}_{p-1}, {\bar{s}})$ for some fixed ascending m-tuple ${\bar{s}}\in Q_{p}^m$. By induction, we find subsets $Q_1 \subseteq R_1, \ldots , Q_{p-1} \subseteq R_{p-1}$, each containing n elements, such that for all ascending m-tuples ${\bar{r}}_1 \in R_1^m, \ldots , {\bar{r}}_{p-1} \in R_{p-1}^m$ the colors $\chi ''({\bar{r}}_1, \ldots , {\bar{r}}_{p-1})$ are identical. But then, the definition of $\chi ''$ and $\chi '$ entail that the sets $Q_1, \ldots , Q_p$ satisfy the requirements posed by the lemma. $\square$

Recall that we write [K] to address the set $\{1, \ldots , K\}$ for any positive integer $K > 0$.

Lemma 63

Let $n, m, p > 0$ be positive integers, let $K \ge 0$ be a nonnegative integer and let $\chi : {\mathbb {Q}}^m \rightarrow {\mathcal {C}}$ be an arbitrary coloring. Let $R_1, \ldots , R_p$ be sufficiently large but finite subsets of ${\mathbb {Q}}$. Let $q_1, \ldots , q_K$ be fixed rational numbers. Let $\rho : [m] \rightarrow [p+K]\times [m]$ be some mapping such that $\rho (i) = \langle K,\ell \rangle$ with $K > p$ implies $\ell = 1$.

There exist subsets $Q_1 \subseteq R_1, \ldots , Q_p \subseteq R_p$, each of cardinality n, and there exists some color $C \in {\mathcal {C}}$ such that for all ascending tuples

$$\begin{aligned} {\bar{r}}_1 = \;&\langle r_{\langle 1,1\rangle }, \ldots , r_{\langle 1,m\rangle }\rangle \in Q_1^{m}\\&\vdots \\ {\bar{r}}_p = \;&\langle r_{\langle p,1\rangle }, \ldots , r_{\langle p,m\rangle }\rangle \in Q_p^{m} \\ {\bar{r}}_{p+1} = \;&\langle r_{\langle p+1, 1\rangle }\rangle := \langle q_1\rangle \\&\vdots \\ {\bar{r}}_{p+K} = \;&\langle r_{\langle p+K, 1\rangle }\rangle := \langle q_K\rangle \end{aligned}$$

we have $\chi ({\bar{r}}_{\rho (1)}, \ldots , {\bar{r}}_{\rho (m)}) = C$.

Proof

We again assume $n \ge m$. We define a new coloring $\chi ' : {\mathbb {Q}}^{m p} \rightarrow {\mathcal {C}}$ by

$$\begin{aligned} \chi '(r_{\langle 1,1\rangle }, \ldots , r_{\langle 1,m\rangle }, \ldots , r_{\langle p,1\rangle }, \ldots , r_{\langle p,m\rangle }) := \chi (r_{\rho (1)}, \ldots , r_{\rho (m)}) \end{aligned}$$

for every mp-tuple $\langle {\bar{r}}_1, \ldots , {\bar{r}}_p\rangle \in R_1^m \times \ldots \times R_p^m$ with ascending ${\bar{r}}_1, \ldots , {\bar{r}}_p$. By Lemma 62, there exist subsets $Q_1 \subseteq R_1, \ldots , Q_p \subseteq R_p$, each with n elements, such that for all ascending tuples ${\bar{r}}_1 \in Q_1^m, \ldots , {\bar{r}}_p \in Q_p^m$ the colors $\chi '({\bar{r}}_1, \ldots , {\bar{r}}_p)$ are the same. By definition of $\chi '$, the sets $Q_1, \ldots , Q_p$ satisfy the requirements of the lemma. $\square$

Now we have the right tools at hand to prove Lemma 8

Lemma 8

Let $n, m, p > 0$ be positive integers, let $K \ge 0$ be a nonnegative integer and let $\chi : {\mathbb {Q}}^m \rightarrow {\mathcal {C}}$ be an arbitrary coloring. Let $R_1, \ldots , R_p$ be sufficiently large but finite subsets of ${\mathbb {Q}}$. Let $q_1, \ldots , q_K$ be fixed rational numbers. Let $\rho _1, \ldots , \rho _L$ be some enumeration of all mappings $\rho _j : [m] \rightarrow [p+K]\times [m]$ for which $\rho _{j}(i) = \langle K,\ell \rangle$ with $K > p$ entails $\ell = 1$. Then, there exist subsets $Q_1 \subseteq R_1, \ldots , Q_p \subseteq R_p$, each of cardinality n, such that for all ascending tuples

$$\begin{aligned} {\bar{r}}_1, {\bar{r}}'_1 \;&\in Q_1^m \\&\vdots \\ {\bar{r}}_p, {\bar{r}}'_p \;&\in Q_p^m\\ {\bar{r}}_{p+1} \;&:= \langle r_{p+1, 1}\rangle := \langle q_1\rangle \\&\vdots \\ {\bar{r}}_{p+K} \;&:= \langle r_{p+K, 1}\rangle := \langle q_K\rangle \end{aligned}$$

and every index j, $1\le j\le L$, we have

$$\begin{aligned} \chi \big (r_{\rho _j(1)}, \ldots , r_{\rho _j(m)}\big ) = \chi \big (r'_{\rho _j(1)}, \ldots , r'_{\rho _j(m)}\big ). \end{aligned}$$

Proof

We again assume $n \ge m$. We construct sequences of subsets $S_{\ell ,0} \supseteq \ldots \supseteq S_{\ell ,L}$ for every $\ell$, $1\le \ell \le p$, such that

• $S_{\ell ,0} = R_\ell$, and

• $S_{\ell , j+1} \subseteq S_{\ell , j}$ is a subset of sufficient cardinality that is constructed by application of Lemma 63 for $\rho := \rho _{j+1}$, i.e. for all ascending tuples

  $$\begin{aligned} \langle s_{\langle 1,1\rangle }, \ldots , s_{\langle 1,m\rangle }\rangle&\in S_{1, j+1}^m\\ \vdots \quad&~\\ \langle s_{\langle p,1\rangle }, \ldots , s_{\langle p,m\rangle }\rangle&\in S_{p, j+1}^m \end{aligned}$$

  the colors $\chi ({\bar{s}}_{\rho _{j+1}(1)}, \ldots , {\bar{s}}_{\rho _{j+1}(m)})$ are the same.

Then the sets $S_{1, L}, \ldots , S_{p, L}$ are the sought $Q_1, \ldots , Q_p$. $\square$

1.2 Proof of Lemma 13

We start with some auxiliary results that will be useful to analyze the blowup that we incur during the normal-form transformation.

The following is standard methodology in the area of difference logic, see, e.g. Section 5.7 in  [82], Section 2.1 in  [37], or Section 11.4.5 in  [14]. Let m be any positive integer. Let $\bar{\mathrm {x}}$ be any m-tuple of pairwise-distinct first-order variables $x_1, \ldots , x_m$ of sort ${\mathbb {Q}}$ and let $x_0$ be any first-order variable of sort ${\mathbb {Q}}$ that does not occur in $\bar{\mathrm {x}}$. Let $\varLambda (x_0, \bar{\mathrm {x}}) :=\, x_0 \!=\! 0 \wedge \varLambda '(x_0, \bar{\mathrm {x}})$ be a conjunction where $\varLambda '(x_0, \bar{\mathrm {x}})$ is a conjunction of atoms of the form $x - y \le c$ or $x - y < c$ with $x,y \in \bar{\mathrm {x}}\cup \{x_0\}$ and $c \in {\mathbb {Z}}$.

Definition 64

(Difference constraint graph ${\mathcal {G}}_\varLambda$, cf. Definition 5.17 and Excercise 5.16 in  [82]) The difference constraint graph ${\mathcal {G}}(\varLambda )$ is a weighted directed graph $\langle V, E\rangle$ with $V = \{x_0, x_1, \ldots ,$ $x_m\}$ and $E \subseteq V \times V \times {\mathbb {Q}}$ such that

• $\langle x, x, 0\rangle \in E$ for all $x \in V$,

• $\langle x, y, c\rangle \in E$ if and only if $\varLambda$ contains the constraint $x - y \le c$, and

• $\langle x, y, c-\delta \rangle \in E$ if and only if $\varLambda$ contains the constraint $x - y < c$,

where we set $\delta := \tfrac{1}{2}(m+1)^{-1}$.

A path $\pi$ in ${\mathcal {G}}_\varLambda$ is any finite, nonempty sequence $\langle x_{i_1}, x_{i_2}, c_1\rangle \langle x_{i_2}, x_{i_3}, c_2\rangle \ldots \langle x_{i_{\ell -1}}, x_{i_\ell }, c_\ell \rangle$ of edges from ${\mathcal {G}}_\varLambda$. We call $\pi$ simple, if the indices $i_1, \ldots , i_{\ell -1}$ are pairwise distinct, i.e. $\pi$ traverses every vertex in ${\mathcal {G}}_\varLambda$ at most once, except for the end point which may coincide with the starting point but does not have to. A simple cycle in ${\mathcal {G}}_\varLambda$ is any simple path whose start and end point coincide. The length of a path in ${\mathcal {G}}_\varLambda$ is the sum of the weights associated with the edges the path traverses. Notice that $\delta$ in Definition 64 is chosen such that the following property is satisfied. Let $\pi$ be any simple path in ${\mathcal {G}}_\varLambda$. Let $c_1, \ldots , c_\ell$ be the weights associated with the edges $\pi$ traverses. We have $\big ( \sum _{1 \le i \le \ell } \lceil c_i \rceil \big ) -1 < \sum _{1 \le i \le \ell } c_i \le \sum _{1 \le i \le \ell } \lceil c_i \rceil$.

Proposition 65

(cf. Theorem 1 in  [37]) Consider the difference constraint graph ${\mathcal {G}}_\varLambda$ and suppose that we have ${\mathbb {Q}}\models \exists x_0 \bar{\mathrm {x}}.\, \varLambda (x_0, \bar{\mathrm {x}})$. Then, for every pair $x,y \in \bar{\mathrm {x}}\cup \{x_0\}$ and every rational number r we have

• (a) ${\mathbb {Q}}\models \forall x_0 \bar{\mathrm {x}}.\, \varLambda (x_0, \bar{\mathrm {x}}) \rightarrow x - y \le r$ if and only if y is reachable from x in ${\mathcal {G}}_\varLambda$ and $\lceil d_{x,y} \rceil \le r$, and

• (b) ${\mathbb {Q}}\models \forall x_0 \bar{\mathrm {x}}.\, \varLambda (x_0, \bar{\mathrm {x}}) \rightarrow x - y < r$ if and only if y is reachable from x in ${\mathcal {G}}_\varLambda$ and we have either $\lceil d_{x,y} \rceil < r$ or $d_{x,y} < \lceil d_{x,y} \rceil = r$,

where $d_{x,y}$ is the length of a shortest simple path from x to y in ${\mathcal {G}}_\varLambda$.

In fact, a variant of Proposition 65 yields a deterministic decision procedure for the sentence $\psi := \exists x_0 \bar{\mathrm {x}}.\, \varLambda (x_0, \bar{\mathrm {x}})$ under ${\mathbb {Q}}$ that runs in polynomial time  [14, 82]: $\psi$ is satisfied by ${\mathbb {Q}}$ if and only if there is some simple cycle in ${\mathcal {G}}_\varLambda$ that has a negative length. In other words, we then have ${\mathbb {Q}}\models \forall x_0 \bar{\mathrm {x}}.\, \varLambda (x_0, \bar{\mathrm {x}}) \rightarrow x - x \le -1$ for some $x \in \bar{\mathrm {x}}\cup \{x_0\}$.

Now we have all pieces together to prove the existence of BSR(SLR) and BSR(BD) normal forms.

Lemma 13

For every BSR(SLR) (or BSR(BD)) clause set N there is an equisatisfiable BSR(SLR) (or BSR(BD)) clause set $N'$ in BSR(SLR) normal form (BSR(BD) normal form) such that

• (a) the length of $N'$ is at most exponential in the length of N,

• (b) for any clause C in $N'$ the number of variables occurring in C is not larger than the number of variables occurring in any clause in N,

• (c) if N is a BSR(SLR) clause set, the number of distinct rational numbers and Skolem constants occurring in $N'$ is linear in the length of N,

• (d) if N is a BSR(BD) clause set, then

  • (d.1) the number of clauses in $N'$ grows at most exponentially in the number of atoms $s \ne t$ occurring in any clause in N,

  • (d.2) the length of any clause in $N'$ is at most polynomial in the length of the longest clause in N,

  • (d.3) every free-sort Skolem constant occurring in $N'$ also occurs in N, and

  • (d.4) the absolute value of any integer in $N'$ is linear in $\kappa \cdot \lambda$, where $\kappa$ is the smallest positive integer that is larger than the absolute value of any integer occurring in N, and $\lambda$ is the smallest positive integer that is larger than the maximal number of universally quantified variables occurring in any clause in N.

Proof

(Sketch) We start with the BSR(SLR) case. First, we show how make sure that every base-sort variable that occurs in $\varLambda$ in a clause $\varLambda \wedge \varGamma \rightarrow \varDelta$ also occurs in $\varGamma$ or in $\varDelta$. Consider any BSR(SLR) clause $\varLambda \wedge \varGamma \rightarrow \varDelta$ and let $\bar{\mathrm {x}}$ be some nonempty tuple of base-sort variables that occur in $\varLambda$ but neither in $\varGamma$ nor in $\varDelta$. Recall that all variables in clauses are implicitly universally quantified. We observe that $\forall \bar{\mathrm {x}}.\, (\varLambda \wedge \varGamma \rightarrow \varDelta )$ is equivalent to $(\exists \bar{\mathrm {x}}.\, \varLambda ) \wedge \varGamma \rightarrow \varDelta$. Since $\varLambda$ is a conjunction of LRA atoms, we may apply virtual substitution  [90, 131] to eliminate the quantifier block $\exists \bar{\mathrm {x}}$ and compute some disjunction of conjunctions of LRA atoms $\bigvee _i \varLambda '_i$ that is ${\mathbb {Q}}$-equivalent to $\exists \bar{\mathrm {x}}.\, \varLambda$. Then, the clause $(\exists \bar{\mathrm {x}}.\, \varLambda ) \wedge \varGamma \rightarrow \varDelta$ is equivalent to the conjunction of clauses $\bigwedge _i (\varLambda '_i \wedge \varGamma \rightarrow \varDelta )$. The length of $\bigwedge _i (\varLambda '_i \wedge \varGamma \rightarrow \varDelta )$ is at most exponential in the length of $\varLambda \wedge \varGamma \rightarrow \varDelta$ (cf. Theorem 3.7 in  [90]), the length of each $\varLambda _i$ is at most linear in the length of $\varLambda$, and the set of variables occurring in any $\varLambda '_i$ is a subset of the variables occurring freely in $\exists \bar{\mathrm {x}}.\, \varLambda$.

In BSR(SLR) clauses the used elimination sets contain only testpoints of the form t, $t + \varepsilon$, or $- \infty$, where t is some LRA term occurring in N in some atom $x \mathrel {\triangleleft }t$. Virtually substituting such a testpoint in any arithmetic atom that is admitted in BSR(SLR) yields again an atom admitted in BSR(SLR).

Next, we describe how to modify N in such a way that it can be partitioned into $N_{{\mathbb {Q}}}$ and $N_\mathrm {BSR}$ as required in Definition 11. Clauses of the form $s \mathrel {\triangleleft }t \wedge \varLambda ' \wedge \varGamma \rightarrow \varDelta$, where t is neither a variable nor a Skolem constant, are replaced—under preservation of (un)satisfiability—with two clauses $t \ne c \rightarrow {\texttt {false}}$ (which is equivalent to $t = c$) and $s \mathrel {\triangleleft }c \wedge \varLambda ' \wedge \varGamma \rightarrow \varDelta$ for some fresh uninterpreted constant symbol c of sort ${\mathbb {Q}}$. Doing this exhaustively for all clauses with nonempty part $\varGamma$ or $\varDelta$ leads to the desired partition of N in $N_{{\mathbb {Q}}}$ and $N_\mathrm {BSR}$.

Now we treat the BSR(BD) case. Again, we first show how to make sure that every base-sort variable that occurs in $\varLambda$ in a clause $\varLambda \wedge \varGamma \rightarrow \varDelta$ also occurs in $\varGamma$ or in $\varDelta$. Clauses of the form $s \not =t \wedge \varLambda ' \wedge \varGamma \rightarrow \varDelta$ are equivalently replaced with two clauses $s < t \wedge \varLambda ' \wedge \varGamma \rightarrow \varDelta$ and $s > t \wedge \varLambda ' \wedge \varGamma \rightarrow \varDelta$. We do this exhaustively for all atoms $s \ne t$ that contain at least one variable not occurring in the $\varGamma$ or $\varDelta$ part of the respective clause. In the worst case, treating a clause in N in this way produces $2^k$ clauses if the original clause contains k atoms $s \not =t$ that need to be replaced.

Consider any BSR(BD) clause $C := \varLambda ' \wedge \varLambda \wedge \varGamma \rightarrow \varDelta$ where every atom in $\varLambda '$ contains a variable x that does not occur in $\varLambda$, $\varGamma$, and $\varDelta$. Let $\bar{\mathrm {x}}$ be some tuple listing all these variables exactly once and let $\bar{\mathrm {v}}$ be some tuple listing all the other variables occurring in C. We assume that $\varLambda '$ does not contain any atoms of the from $s \not =t$. Moreover, we assume that all atoms $s = t$ in $\varLambda '$ have been replaced with conjunctions $s \le t \wedge t \le s$. We observe that $\forall \bar{\mathrm {x}}.\, (\varLambda '(\bar{\mathrm {x}}, \bar{\mathrm {v}}) \wedge \varLambda (\bar{\mathrm {v}}) \wedge \varGamma (\bar{\mathrm {v}}) \rightarrow \varDelta (\bar{\mathrm {v}}))$ is equivalent to $\big (\exists \bar{\mathrm {x}}.\, \varLambda '(\bar{\mathrm {x}}, \bar{\mathrm {v}})\big ) \wedge \varLambda (\bar{\mathrm {v}}) \wedge \varGamma (\bar{\mathrm {v}}) \rightarrow \varDelta (\bar{\mathrm {v}})$. Since $\varLambda '(\bar{\mathrm {x}}, \bar{\mathrm {v}})$ is a conjunction of LRA atoms, we may apply the Fourier-Motzkin elimination procedure to eliminate the variables $\bar{\mathrm {x}}$ in $\big (\exists \bar{\mathrm {x}}.\, \varLambda '(\bar{\mathrm {x}}, \bar{\mathrm {v}})\big )$ one by one.

Consider any $x \in \bar{\mathrm {x}}$. In order to eliminate x from $\exists x.\, \varLambda '(\bar{\mathrm {x}}, \bar{\mathrm {v}})$, we proceed as follows. Let $\varLambda '_0, \varLambda '_1, \varLambda '_2$ be the shortest conjunctions satisfying the following properties:

• (i) every atom from $\varLambda '$ that does not contain x occurs in $\varLambda '_0$,

• (ii) for every atom in $\varLambda '$ that contains x there is a ${\mathbb {Q}}$-equivalent atom in $\varLambda '_1 \wedge \varLambda '_2$,

• (iii) every atom in $\varLambda '_1$ has the form $s \le x$ or $s < x$ where s is either an integer, a variable, or an LRA term $y + c$ for some variable $y \in \bar{\mathrm {x}}\cup \bar{\mathrm {v}}$ of sort ${\mathbb {Q}}$ and some integer c, and

• (iv) every atom in $\varLambda '_2$ has the form $x \le t$ or $x < t$ where t is either an integer, a variable, or an LRA term $y + c$ for some variable $y \in \bar{\mathrm {x}}\cup \bar{\mathrm {v}}$ of sort ${\mathbb {Q}}$ and some integer c.

Let $\varLambda ''$ be the conjunction of the following set of atoms

$$\begin{aligned}&\Big \{ s< t \, \Big | \, (s \mathrel {\triangleleft }_1 x) \in \varLambda '_1 \text { and } (x \mathrel {\triangleleft }_2 t) \in \varLambda '_2 \text { where at least one of } \mathrel {\triangleleft }_1, \mathrel {\triangleleft }_2\text { is the strict }< \Big \} \\&\cup \, \Big \{ s \le t \, \Big | \, (s \le x) \in \varLambda '_1 \text { and } (x \le t) \in \varLambda '_2 \Big \}. \end{aligned}$$

Let $\bar{\mathrm {x}}' := \bar{\mathrm {x}}{\setminus } \{x\}$. It is well known that the two formulas $\exists x.\, \varLambda '_1(x, \bar{\mathrm {x}}', \bar{\mathrm {v}}) \wedge \varLambda '_2(x, \bar{\mathrm {x}}', \bar{\mathrm {v}})$ and $\varLambda ''(\bar{\mathrm {x}}', \bar{\mathrm {v}})$ are ${\mathbb {Q}}$-equivalent (see, e.g.  [122], Section 12.2). Hence, $\exists x.\, \varLambda '(x, \bar{\mathrm {x}}', \bar{\mathrm {v}})$ can be replaced with the ${\mathbb {Q}}$-equivalent formula $\varLambda '_0(\bar{\mathrm {x}}', \bar{\mathrm {v}}) \wedge \varLambda ''(\bar{\mathrm {x}}', \bar{\mathrm {v}})$.

Concerning the atoms in $\varLambda ''(\bar{\mathrm {x}}, \bar{\mathrm {v}})$ we find that every atom therein can be transformed into an equivalent atom of the form $y \mathrel {\triangleleft }c$, $y \mathrel {\triangleleft }z$, or $y - z \mathrel {\triangleleft }c$ where $y, z \in \bar{\mathrm {x}}' \cup \bar{\mathrm {v}}$, c is some integer, and ${\mathrel {\triangleleft }} \in \{<, \le , \ge , >\}$. As we need to keep at most $4 \cdot |\bar{\mathrm {x}}' \cup \bar{\mathrm {v}}| + 8 \cdot |\bar{\mathrm {x}}' \cup \bar{\mathrm {v}}|^2$ of these atoms (at most one atom $y \mathrel {\triangleleft }c$ for each pair $y,\mathrel {\triangleleft }$ and at most two atoms $y - z \mathrel {\triangleleft }d$ and $z - y \mathrel {\triangleleft }e$ for every triple $y, z, \mathrel {\triangleleft }$), we may assume that the length of $\varLambda ''(\bar{\mathrm {x}}', \bar{\mathrm {v}})$ is at most polynomial in the number of variables in $\bar{\mathrm {x}}', \bar{\mathrm {v}}$.

We apply the described elimination procedure to eliminate the other variables in $\bar{\mathrm {x}}$ as well, in a variable-by-variable fashion. Hence, the final conjunction $\varLambda '''(\bar{\mathrm {v}})$ contains at most $4 \cdot |\bar{\mathrm {v}}| + 8 \cdot |\bar{\mathrm {v}}|^2$ atoms, and we replace the clause $C(\bar{\mathrm {x}}, \bar{\mathrm {v}})$ in N with the equivalent clause $\varLambda '''(\bar{\mathrm {v}}) \wedge \varLambda (\bar{\mathrm {v}}) \wedge \varGamma (\bar{\mathrm {v}}) \rightarrow \varDelta (\bar{\mathrm {v}})$. In addition, we can bound the absolute value of the integers occurring in $\varLambda '''$ as follows. It is easy to verify that we can transform $\varLambda (\bar{\mathrm {x}}, \bar{\mathrm {v}})$ into a ${\mathbb {Q}}$-equivalent conjunction $\varLambda _\mathrm {diff}(\bar{\mathrm {x}}, \bar{\mathrm {v}})$ of difference constraints in the sense of Definition 64 and Proposition 65 (see the paragraph preceding Definition 64). We have mentioned right after Proposition 65 that we can check in polynomial time whether $\exists \bar{\mathrm {x}}\bar{\mathrm {v}}.\, \varLambda _\mathrm {diff}(\bar{\mathrm {x}}, \bar{\mathrm {v}})$ is satisfied under ${\mathbb {Q}}$. In the opposite case, $\varLambda '''$ can be replaced by ${\texttt {false}}$. Henceforth, we assume that ${\mathbb {Q}}\models \exists \bar{\mathrm {x}}\bar{\mathrm {v}}.\, \varLambda _\mathrm {diff}(\bar{\mathrm {x}}, \bar{\mathrm {v}})$.

Since $\varLambda '''(\bar{\mathrm {v}})$ is the result of applying Fourier-Motzkin elimination to $\exists \bar{\mathrm {x}}.\, \varLambda '(\bar{\mathrm {x}}, \bar{\mathrm {v}})$, we observe that for every atom of the form $u - v \le c$ occurring in $\varLambda '''(\bar{\mathrm {v}})$ we have ${\mathbb {Q}}\models \forall \bar{\mathrm {x}}\bar{\mathrm {v}}.\, \varLambda _\mathrm {diff}(\bar{\mathrm {x}}, \bar{\mathrm {v}}) \rightarrow u - v \le c$. Let $\kappa$ be the smallest positive integer that is larger than the absolute value of any integer occurring in $\varLambda '$. Then, by Proposition 65, we observe $c \ge - \kappa \cdot (|\bar{\mathrm {x}}\cup \bar{\mathrm {v}}|+1)$ and, in addition, that there exists some integer k satisfying the following properties:

• (1) $- \kappa \cdot (|\bar{\mathrm {x}}\cup \bar{\mathrm {v}}|+1) \;\le \; k \;\le \; \kappa \cdot (|\bar{\mathrm {x}}\cup \bar{\mathrm {v}}|+1)$, and

• (2) ${\mathbb {Q}}\models \forall \bar{\mathrm {x}}\bar{\mathrm {v}}.\, \varLambda '(\bar{\mathrm {x}}, \bar{\mathrm {v}}) \rightarrow u - v \mathrel {\triangleleft }k$.

This means, if $c > \kappa \cdot (|\bar{\mathrm {x}}\cup \bar{\mathrm {v}}|+1)$, then we can replace $u - v \le c$ in $\varLambda '''$ with the atom $u - v \le k$, which subsumes the former. Using similar arguments we can show the same for other atoms occurring in $\varLambda '''$. Consequently, we may assume that $\varLambda '''$ contains only integers whose absolute value is linear in $\kappa \cdot (|\bar{\mathrm {x}}\cup \bar{\mathrm {v}}|+1)$. $\square$

1.3 Proof of Lemma 59

Lemma 59

Let N be a finite clause set as described above and let $\lambda$ be the maximal number of rational-valued variables in any clause in N; if $\lambda < m$, we set $\lambda := m$. Let $(\sim _k)_{k \ge 1}$ be any family of downwards scalable equivalence relations that satisfies the following properties.

• (a) Each $\sim _k$ in the family has a finite index, i.e. it induces only finitely many equivalence classes over ${\mathbb {Q}}^k$.

• (b) Let $\varLambda (\bar{\mathrm {x}})$ be any conjunction of atoms $A(\bar{\mathrm {x}}) \in \varTheta (X)$ with $|\bar{\mathrm {x}}| \le \lambda$. For any two ${\bar{r}}, {\bar{r}}' \in {\mathbb {Q}}^{|\bar{\mathrm {x}}|}$ with ${\bar{r}}\sim _{|\bar{\mathrm {x}}|} {\bar{r}}'$ we have ${\mathcal {A}}\models \varLambda ({\bar{r}})$ if and only if ${\mathcal {A}}\models \varLambda ({\bar{r}}')$.

Moreover, let Q be any finite subset of ${\mathbb {Q}}$ such that for every k, $1 \le k \le \lambda$, every $S \in {\mathbb {Q}}^k/_{\sim _k}$, and every ${\bar{r}}\in S$ there is some ${\bar{q}}\in Q^k$ such that ${\bar{q}}\sim _k {\bar{r}}$.

Suppose we have ${\mathcal {A}}\models C({\bar{{\mathsf {e}}}}, {\bar{q}})$ for every clause $C(\bar{\mathrm {u}}, \bar{\mathrm {x}}) \in N$ and all ${\bar{{\mathsf {e}}}}\in \big ({\mathcal {S}}^{\mathcal {A}}\big )^{|\bar{\mathrm {u}}|}$, ${\bar{q}}\in Q^{|\bar{\mathrm {x}}|}$. If ${\mathcal {A}}$ is $\sim$-uniform over Q, then we can turn ${\mathcal {A}}$ into a model ${\mathcal {B}}$ of N that is $\sim$-uniform over ${\mathbb {Q}}$.

Proof

We construct the structure ${\mathcal {B}}$ as follows. We set ${\mathcal {S}}^{\mathcal {B}}:= {\mathcal {S}}^{\mathcal {A}}$, and for every constant symbol c occurring in N we set $c^{\mathcal {B}}:= c^{\mathcal {A}}$. Moreover, for every uninterpreted predicate symbol P occurring in N and for all tuples ${\bar{{\mathsf {e}}}}\in ({\mathcal {S}}^{\mathcal {A}})^{m'}$ and ${\bar{r}}\in {\mathbb {Q}}^m$ we pick some tuple ${\bar{q}}\in Q^m$ which is $\sim$-equivalent to ${\bar{r}}$, and we define $P^{\mathcal {B}}$ so that

$$\begin{aligned} \langle {\bar{{\mathsf {e}}}}, {\bar{r}}\rangle \in P^{\mathcal {B}}\quad \text {if and only if}\quad \langle {\bar{{\mathsf {e}}}}, {\bar{q}}\rangle \in P^{\mathcal {A}}. \end{aligned}$$

• Claim: The structure ${\mathcal {B}}$ is $\sim$-uniform.

• Proof: By construction of ${\mathcal {B}}$ and by our assumption that ${\mathcal {A}}$ is $\sim$-uniform over Q. $\Diamond$

We next show ${\mathcal {B}}\models N$. Consider any clause $C = \varLambda \wedge \varGamma \rightarrow \varDelta$ in N and let $\beta$ be any variable assignment ranging over ${\mathcal {S}}^{\mathcal {B}}\cup {\mathbb {Q}}$. Starting from $\beta$, we derive a special variable assignment $\gamma _C$ as follows. Let $x_1, \ldots , x_{\lambda _C}$ be an enumeration of all base-sort variables occurring in C. Since $\lambda _C \le \lambda$, there is some tuple $\langle q_1, \ldots , q_{\lambda _C}\rangle \in Q^{\lambda _C}$ such that $\langle q_1, \ldots , q_{\lambda _C}\rangle \sim _{\lambda _C} \big \langle \beta (x_1), \ldots , \beta (x_{\lambda _C}) \big \rangle$. We define $\gamma _C(x_i) := q_i$ for every i, $1 \le i \le \lambda _C$. For all other base-sort variables, $\gamma _C$ can be defined arbitrarily. For every free-sort variable u we set $\gamma _C(u) := \beta (u)$. Then, we observe

$$\begin{aligned} \big \langle \beta (x_1), \ldots , \beta (x_{\lambda _C}) \big \rangle \sim _{\lambda _C} \big \langle \gamma _C(x_1), \ldots , \gamma _C(x_{\lambda _C}) \big \rangle . \end{aligned}$$

(7)

As we have assumed ${\mathcal {A}}\models C({\bar{{\mathsf {e}}}}, {\bar{q}})$ for all ${\bar{{\mathsf {e}}}}\in \big ({\mathcal {S}}^{\mathcal {A}}\big )^{|\bar{\mathrm {u}}|}$, ${\bar{q}}\in Q^{|\bar{\mathrm {x}}|}$, we in particular get ${\mathcal {A}},\gamma _C \models C$. By case distinction on why ${\mathcal {A}},\gamma _C \models C$ holds, we infer ${\mathcal {B}},\beta \models C$ as follows:

• Case ${\mathcal {A}}, \gamma _C \not \models t\mathrel {\triangleleft }t'$ for some LRA atom $t\mathrel {\triangleleft }t'$ in $\varLambda$. Recall that we assume that for all atoms in $\varLambda$ there are variable-renamed variants in $\varTheta$. Hence, by (7) in combination with Condition (b), we have that ${\mathcal {A}}, \gamma _C \not \models t \mathrel {\triangleleft }t'$ entails ${\mathcal {A}}, \beta \not \models t \mathrel {\triangleleft }t'$. Since ${\mathcal {B}}$ and ${\mathcal {A}}$ interpret arithmetic terms in the same way, we conclude ${\mathcal {B}}, \beta \not \models t \mathrel {\triangleleft }t'$.

• Case ${\mathcal {A}}, \gamma _C \not \models P(s_1, \ldots , s_{m'}, t_1, \ldots , t_m)$ for some atom $P(s_1, \ldots , s_{m'}, t_1, \ldots , t_m)$ in $\varGamma$, where the $s_i$ are terms (variables or constants) of sort ${\mathcal {S}}$ and the $t_j$ are variables over the rationals. By definition of $\gamma _C$, we have $\gamma _C(t_j) \in Q$ for every j, $1 \le j \le m$. Moreover, $\gamma _C$ and ${\mathcal {B}}$ are defined such that ${\mathcal {A}}(\gamma _C)(s_i) = {\mathcal {B}}(\beta )(s_i)$ for every i, $1 \le i \le m'$. This together with Observation (7) and $\sim$-uniformity of ${\mathcal {B}}$ entails ${\mathcal {B}}, \beta \not \models P(s_1, \ldots , s_{m'}, t_1, \ldots , t_m)$.

• Case ${\mathcal {A}}, \gamma _C \models P(s_1, \ldots , s_{m'}, t_1, \ldots , t_m)$ for some atom $P(s_1, \ldots , s_{m'}, t_1, \ldots , t_m)$ in $\varDelta$. In analogy to the previous case we infer ${\mathcal {B}}, \beta \models P(s_1, \ldots , s_{m'}, t_1, \ldots , t_m)$.

• Case ${\mathcal {A}}, \gamma _C \not \models t\approx t'$ for some atom $t\approx t' \in \varGamma$. Then, t and $t'$ are either variables or constant symbols of the free sorts. Since ${\mathcal {B}}$ and ${\mathcal {A}}$ behave identically on free-sort constant symbols and since $\beta (u) = \gamma _C(u)$ for every variable $u\in V_{\mathcal {S}}$, we get ${\mathcal {B}}, \beta \not \models t\approx t'$.

• Case ${\mathcal {A}}, \gamma _C \models t\approx t'$ for some $t\approx t' \in \varDelta$. In analogy to the above case, we obtain ${\mathcal {B}}, \beta \models t\approx t'$.

Altogether, we have shown ${\mathcal {B}}\models N$. $\square$