Arithmetic Reasoning in Z3

TU Wien Guest Lectures October 2025

Nikolaj Bjørner
Microsoft Research
Lev Nachmanson
Microsoft Research

Contents

Arithmetic - a fundamental SMT theory

Frog

F*

EverParse

linear and non-linear arithmetic

Certora

  • 256 bit non-linear arithmetic

OpenZeppelinImage

Verus

  • linear arithmetic

AutoVerus

Several other

  • Dafny, Boogie, Viper
  • Dafny - first customer of non-linear arithmetic (Gröbner)
  • Verdis - ZK for prime fields

Many Arithmetic Theories

ArithmeticTheories

Other Fragments

Cardinality, Pseudo-Boolean, finite range LIA theories


Modular arithmetic, finite fields


Bi-linear real arithmetic $\vec{x}A\vec{y} \leq \vec{b}$


Non-unit two-variable per inequality $ax + by \leq c$


At most one (unit) positive variable per inequality (Horn) $x - 3y - 2z \leq 2$. What is the non-unit complexity?


max-atom: $\max(x,y) + k \geq z$. In NP $\cap$ co-NP, local strategy/policy iteration (Costan et al., 2005; Gawlitza and Seidl, 2007; Bezem et al., 2008; Karpenkov et al., 2016; Karpenkov, 2017)


Knapsack theories: all variables with positive coefficients, except for one inequality where all variables have negative coefficients.

Theoretical vs Practical Complexity

  • Conjunctions:
    • RDL, IDL solved by
      • Floyd-Warshal ($V^3$),
      • Bellman-Ford ($V\cdot E$).
    • LRA - in P
    • LIA - NP complete
    • Finite Fields ~ Bit Vectors - NEXPTIME
    • NIA - undecidable
    • Quantified NIA - highly undecidable
  • Clauses:
    • $\geq$ NP hard

CDCL(Arith) by example

BasicTheory6

Solver Architecture (N. S. Bjørner and Nachmanson, 2024)

Arch

LRA - Linear Real Arithmetic

LPArch

LRA

What: Variables range over reals (rationals). Addition, multiplication by constant

How: Dual Simplex

Why: A basis for everything more

\[\begin{mdmathpre}%mdk \mdmathindent{2}\mathid{x},~\mathid{y},~\mathid{z}~\in \mathcal{R}\\ \mdmathindent{2}0~<~\mathid{x}~<~\mathid{y}~<~\mathid{z}~\land \mathid{x}~+~\mathid{y}~+~\mathid{z}~=~1\\ \\ \mdmathindent{2}\mathid{x}~=~1/6,~\mathid{y}~=~1/3,~\mathid{z}~=~1/2 \end{mdmathpre}%mdk \]

Farkas and Dual Simplex (Dutertre and Moura, 2006)

SMT solving based on Dual Simplex maintain a tableau of the form:

\[\begin{array}{ll} x_i = \sum_{x_j \in \nonbasicvars} a_{ij} x_j & x_i \in \basicvars \\ l_j \leq x_j \leq u_j & x_j \in \nonbasicvars \cup \basicvars \\ val : x_j \mapsto {\mathcal R} & \forall x_j \in \nonbasicvars \\ nval : x_i \mapsto \sum_{x_j \in \nonbasicvars} a_{ij} val(x_j) & \forall x_i \in \basicvars \end{array} \]

where $\basicvars$ are basic and $\nonbasicvars$ are non-basic variables.

Compiling into tableau

  x, y = Reals('x y')
  solve([x >= 0, Or(x + y <= 2, x + 2*y >= 6), 
                 Or(x + y >= 2, x + 2*y > 4)])

Introduce slacks to define terms

\[\begin{mdmathpre}%mdk \mdmathindent{2}\mathid{s}_1~=~\mathid{x}~+~\mathid{y},~\mathid{s}_2~=~\mathid{x}~+~2\mathid{y},~\mathid{s}_1,~\mathid{s}_2~\in \basicvars,~\mathid{x},~\mathid{y}~\in \nonbasicvars \end{mdmathpre}%mdk \]

Formula using slacks:

\[\begin{mdmathpre}%mdk \mdmathindent{2}\mathid{x}~\geq 0,~(\mathid{s}_1~\leq 2~\vee \mathid{s}_2~\geq 6),~(\mathid{s}_1~\geq 2~\vee \mathid{s}_2~>~4) \end{mdmathpre}%mdk \]

Constraints := Tableau Equalities + bounds

Equalities are global definitions

\[\begin{mdmathpre}%mdk \mdmathindent{2}\mathid{s}_1~=~\mathid{x}~+~\mathid{y},~\mathid{s}_2~=~\mathid{x}~+~2\mathid{y},~\mathid{s}_1,~\mathid{s}_2~\in \basicvars,~\mathid{x},~\mathid{y}~\in \nonbasicvars \end{mdmathpre}%mdk \]

Bounds (e.g., $s_1 \leq 2$) are added/deleted during search.

From Definitions to a Tableau

${\color{red}s_1} := x + y, {\color{red}s_2} := x + 2y$



${\color{red}s_1} = x + y$
${\color{red}s_2} = x + 2y$



${\color{red}s_1} - x - y = 0$ - ${\color{red}s_1, s_2}$ are basic (dependent)
${\color{red}s_2} - x - 2y = 0$ - $x, y$ are non-basic

Pivoting

  • Value of non-basic variable $x_j$ can be chosen between $lo_j$ and $hi_j$.
  • Value of basic variable is a function of non-basic variable values.
  • Pivoting swaps basic and non-basic variables.
    • used to get values of basic variables within bounds.

Pivoting example

${\color{red}s_1} - x - y = 0$ - ${\color{red}s_1, s_2}$ are basic (dependent)
${\color{red}s_2} - x - 2y = 0$ - $x, y$ are non-basic $x \geq 0, (s_1 \leq 2 \vee s_2 \geq 6), (s_1 \geq 2 \vee s_2 > 4)$


Initial values: $x = y = s_1 = s_2 = 0$


Bounds $x \geq 0, s_1 \leq 2, s_1 \geq 2$:


$s_1 := 2$, make $s_1$ non-basic.
${\color{red}y} + x - s_1 = 0$ - ${\color{red}y, s_2}$ are basic (dependent)
${\color{red}s_2} + x - 2s_1 = 0$ - $s_1, y$ are non-basic

Infeasible Tableau

A tableau is infeasible if there is a row $x_i = \sum_{x_j \in \nonbasicvars} a_{ij} x_j$ such that

\[\begin{array}{lll} & a_{ij} < 0 \implies val(x_j) = l_j, & \forall x_j \\ & a_{ij} > 0 \implies val(x_j) = u_j, & \forall x_j \\ & nval(x_i) < l_i % \\ %or \\ % & a_{ij} > 0 \implies val(x_j) = l_j, & \forall x_j \\ % & a_{ij} < 0 \implies val(x_j) = u_j, & \forall x_j \\ % & nval(x_i) > u_i \end{array} \]

conflict explanation: literals that are used for tight bounds.

LRA is Convex

Convexity: $A\vec{x} \leq b \implies x_1 = y_1 \lor x_2 = y_2$, then $A\vec{x} \leq b \implies x_i = y_i$ for $i = 1$ or $i = 2$.

  • Suppose $val_1$ is a solution to $A\vec{x} \leq b$, then $val_1(x_i) = val_1(y_i)$ for $i = 1$.
  • Similar for solution $val_2$ but with $i = 2$.
  • Set $val_3 := (val_1 + val_2) / 2$.
  • Then $A \cdot val_3(\vec{x}) \leq b$ too and $val_3$ has to agree with either $val_1$ or $val_2$ on implied equality.

LRA - next

  • Z3 uses bignum (infinite precision arithmetic), always.

    • A show-stopper for LP centric applications (Marabou, traffic engineering)
    • Many scenarios can be solved at fixed precision or floats with guarantees (SCIP).

  • Z3 also contains difference logic solvers. They are used sparingly for SMTLIB categories IDL/RDL.

  • Integration of policy/strategy iteration?

Strategy/Policy iteration (Costan et al., 2005)

Goal is to find minimal $x_i$ such that:

\[\begin{mdmathpre}%mdk \mdmathindent{3}\mathid{x}_\mathid{i}~\geq \mathid{e}_\mathid{i}~~\mathid{for}~\mathid{each}~\mathid{x}_\mathid{i}\\ \mdmathindent{3}\mathid{where}~\mathid{e}~::=~\mathid{x}_\mathid{j}~|~\mathid{a}\cdot \mathid{e}~|~\mathid{e}~+~\mathid{e}~|~\min(\mathid{e},\mathid{e})~|~\max(\mathid{e},\mathid{e})\\ \mdmathindent{3}\mathid{where}~\mathid{a}~>~0~\mathid{is}~\mathid{a}~\mathid{constant} \end{mdmathpre}%mdk \]


Policy Initially $\pi$ replaces $\min(e_1,e_2)$ by $e_1$ (arbitrary choice).


Solve the resulting unit-Horn system with solution $M$.


Evaluate each $\min(e_1, e_2)$ subterm wrt. $M$ to check if it preserves $\min$.


Repeat For each $\min(e_1, e_2)$, if $M(e_1) > M(e_2)$, but $\pi(\min(e_1,e_2)) = e_1$, then update $\pi(\min(e_1,e_2)) := e_2$.

Bounds and Equality Propagation

BoundsArch

Bounds

What: Extract Boolean propagation axioms from arithmetic.

How: Index bounds atoms that are created and infer bounds from tableau.

Why: It is much more efficient to have the SAT solver propagate (binary) clauses than theory reasoning.

\[\begin{mdmathpre}%mdk \mdmathindent{3}(\mathid{x}~\geq 1~\lor \mathid{P})~\land (\mathid{x}~\leq 0~\lor \mathid{Q})~\land \ldots \end{mdmathpre}%mdk \]

If $x \geq 1$ is asserted, then directly propagate $x \leq 0$ to false.

LP Bounds Propagation

  • For atoms $x \geq 3, x \geq 2, x \geq 1$ add axioms $x \geq 3 \implies x \geq 2,\ \ \ \ x \geq 2 \implies x \geq 1$.

  • Infer new bounds of variables: $x - 2y = 0, y \geq 1 \implies x \geq 2$.

    • For every tableau row $r$,
      • For every variable $x$ in row $r$,
        • Check if bounds of other variables strengthen current variable bound,
        • Use implied bound for T-propagation of bounds atoms.

Equalities

What: Extract equalities between variables and propagate equalities to other theories.

How: By cheap tests on the tableau.

Why: Useful to propagate congruences for arithmetic variables used under non arithmetical function symbols.

\[\begin{mdmathpre}%mdk \mdmathindent{4}\mathid{x}~\leq \mathid{y}~+~1~\land \mathid{z}~\geq \mathid{y}~+~1~\land \mathid{z}~\leq \mathid{x}~\land \mathid{p}(\mathid{x})~\land \neg \mathid{p}(\mathid{z})\\ \\ \mdmathindent{4}\leadsto \mathid{x}~=~\mathid{z}~~\\ \mdmathindent{4}\leadsto \mathid{p}(\mathid{x})~\land \neg \mathid{p}(\mathid{x}) \end{mdmathpre}%mdk \]

Finding implied equalities

\[\begin{mdmathpre}%mdk \mdmathindent{3}\mathid{x}~+~\mathid{u}~\leq \mathid{z}~\land \mathid{z}~-~1~\leq \mathid{y}~\land \mathid{y}~\leq \mathid{x}~\land 1~\leq \mathid{u}~\leq 1 \end{mdmathpre}%mdk \]

Solved tableau:

\[\begin{mdmathpre}%mdk \mdmathindent{4}\mathid{x}~=~\mathid{z}~-~\mathid{u}~-~\mathid{s}_1,~\mathid{y}~=~\mathid{z}~-~\mathid{u}~-~\mathid{s}_2,~\\ \mdmathindent{4}1~\leq \mathid{u}~\leq 1,~0~\leq \mathid{s}_1~\leq 0,~0~\leq \mathid{s}_2~\leq 0 \end{mdmathpre}%mdk \]

assignment $z = 0, x = y = -1$. Necessary condition for $x = y$. Sufficient (incomplete) condition:

  • $x$ is basic, and the tableau has row $x - y + \alpha = 0$,
  • $x, y$ are connected through a non-basic variable $z$ in a pair of the tableau rows in one of the following forms
    1. $x - z + \alpha = 0, y - z + \alpha' = 0$,
    2. $x + z + \alpha = 0, y + z + \alpha' = 0$,

$\alpha, \alpha'$ are linear combinations of fixed variables. We never have to calculate $\alpha$!

Complete Equality Propagation

Will this method find all implied equalities?

Propagating Equality Assumptions

If $x, y$ are shared variables and $val(x) = val(y)$, but the equality between $x, y$ has not been propagated. Then we can assume the equality $x \simeq y$ atom. It is satisfied in the theory of arithmetic, and the equality atom will propagate to other theories that have to reconcile on $x = y$.

\[\begin{mdmathpre}%mdk \mdmathindent{2}0~\leq \mathid{x},~\mathid{y},~\mathid{z}~\leq 1~\land \mathid{x},~\mathid{y},~\mathid{z}~\in \mathcal{Z}~\land \\ \mdmathindent{2}\mathid{P}(\mathid{x})~\land \neg \mathid{P}(\mathid{y})~\land \mathid{Q}(\mathid{y})~\land \neg \mathid{Q}(\mathid{z})~\land \neg \mathid{S}(\mathid{x})~\land \mathid{S}(\mathid{z}) \end{mdmathpre}%mdk \]

If $val(x) = val(y) = 0$ then assume $x \simeq y$: conflicts with $P(x) \land \neg P(y)$.

Propagating Equality Assumptions (II)

  • Arithmetic solver: $x = y = 0, z = 1$

  • Add atom $x \simeq y$ to search space and assign it to true.

  • EUF solver $x \simeq y \land P(x) \land \neg P(y) \land Q(y) \land \neg Q(z) \land \neg S(x) \land S(z)$

  • Conflict: $x \simeq y \land P(x) \land \neg P(y)$

  • Lemma $x \not\simeq y$

  • Arithmetic solver: $x = 0, y = z = 1$.

  • New lemma $y \not\simeq z\ldots$

Propagation - next

  • Z3 exposes solve_for API function to extract all equalities through Gaussian elimination.

    • If variables are equal modulo current $val$ it performs a local solver call to check if disequality is feasible.

  • Integrate stronger inequality propagation eagerly?

    • Such as bounds tightening.

Special Arithmetic

Difference Arithmetic

(set-logic QF_IDL) ; optional in Z3
(declare-fun t11 () Int)
(declare-fun t12 () Int)
(declare-fun t21 () Int)
(declare-fun t22 () Int)
(declare-fun t31 () Int)
(declare-fun t32 () Int)

(assert (and (>= t11 0) ...))
(assert (and (>= t21 0) ...))
(assert (and (>= t31 0) (>= t32 (+ t31 2)) (<= (+ t32 3) 8)))
(assert (or ... (>= t21 (+ t11 2))))
(assert (or (>= t21 (+ t31 2)) ...))
(check-sat)
(get-model) ; display the model

Bellman-Ford

Solve difference logic using graph $O(E\cdot V)$ Bellman-Ford network flow algorithm. Negative cycle $\Rightarrow$ unsat.

BellmanFord

Convexity and Complexity

  • IDL poly time but is not convex. $0 \leq x, y, z \leq 1$: two out of three have to be equal.

  • UF is poly time.

  • Quiz [Vaughan Pratt]: The combination IDL+UF is NP hard.

Bit-vectors as arithmetic (Zohar et al., 2022)

Add bit-vector operators together with on-demand expansion.

\[\begin{mdmathpre}%mdk \mdmathindent{3}\mathsf{bvand}(\mathid{x},~\mathid{y},~\mathid{N})~~\mbox{bit-wise conjunction of $x, y$ defined for $[0, 2^N-1]$}\\ \mdmathindent{3}\displaystyle \mathsf{bvand}(\sum_\mathid{i}~2^\mathid{i}~\mathid{x}_\mathid{i},~\sum_\mathid{i}~2^\mathid{i}~\mathid{y}_\mathid{i},~\mathid{N})~=~\sum_{\mathid{i}~=~0}^{\mathid{N}-1}~2^\mathid{i}~(\mathid{x}_\mathid{i}\cdot \mathid{y}_\mathid{i})~ \end{mdmathpre}%mdk \]

  • Native arithmetic operators and decision procedures for bit-wise and, shift left, shift right logical and shift right arithmetical.

  • Quiz: express bitwise negation.

Bit-vectors as arithmetic in Z3.

  • Eager axioms:

    • $0 \leq \mathsf{bvand}(x, y, N) < 2^N$
    • $0 \leq x \implies \mathsf{bvand}(x, y, N) \leq x$.

  • On-demand axioms - the rest

    • based on current LIA model.

Pseudo-Booleans

  • Pseudo-Boolean constraints are treated as a theory.
  • There are two implementations: one in the main SMT core, and one as part of a tuned SAT solver.
  • The PB theory solvers perform Boolean propagation on cardinality and PB constraints.
  • Conflict analysis enabled by hijacking the SAT solver conflict analysis process.

EUF or Arithmetic?

An instance from a plant capacity problem:

  • Number of Stations = O(1K)
  • Number of Tasks = O(10K)
  • Up to O(10) different operators per station

Direct MIP-style encoding: $t_{i,s,op}$ - Task $i$ is at station $s$ attended by operator $op$.

\[\begin{mdmathpre}%mdk \mdmathindent{3}10\mathid{K}~\times 1\mathid{K}~\times 10~=~100\mathid{M}~\mbox{variables} \end{mdmathpre}%mdk \]

Instead use EUF $\mathit{station}(i) = s, \mathit{operator}(i) = op$, $\mathit{taskof}(s, op) = i$.

Scales encoding, but loses propagation efficiency.

Claim: Ackerman reduction simulate indicator variables.

LIA - Linear Integer Arithmetic

LIAArch

LIA

What: Variables range over Integers. Addition, multiplication by constant.

How: Conservative: If unsat in LRA then unsat in LIA. Cuts, Branch and Bound.

Why: A practical basis for program verification (few verification systems use bit-vectors) and many applications.

\[\begin{mdmathpre}%mdk \mdmathindent{1}\mathid{x},~\mathid{y},~\mathid{z}~\mathid{in}~\mathcal{Z}\\ \mdmathindent{1}0~<~\mathid{x}~<~\mathid{y}~<~\mathid{z}~\land \mathid{x}~+~\mathid{y}~+~\mathid{z}~=~1~\leadsto \mathid{unsat}\\ \mdmathindent{1}2\mathid{x}~+~6\mathid{y}~+~3~=~0~\leadsto \mathid{unsat} \end{mdmathpre}%mdk \]

From LRA to LIA

When can a solution to LRA be used for LIA?

  • Patching and Cubes: If assignment from LRA is already integral or can be fixed.

  • GCD Test and bounds tightening: Detect when there are no integer solutions.

  • Cuts and branches: Block current LRA solution.

Patching (Moura and Bjørner, 2008)

\[\begin{mdmathpre}%mdk \mdmathindent{1}\mathid{y}~-~\frac{1}{2}~\mathid{x}~=~0~~~~&~~~~~~~~~~~~\\ \mdmathindent{1}\mathid{z}~-~\frac{1}{3}~\mathid{x}~=~0~~~~&~~~~~~~~~~~\\ 3~\leq \mathid{x}~\leq 10,~-3~\leq \mathid{y}~\leq 4,~-4~\leq \mathid{z}~\leq 12~\\~ \end{mdmathpre}%mdk \]

\[\begin{mdmathpre}%mdk \mdmathindent{2}\mbox{Initial assignment:}~~~~~~~~&~~~\mathid{x}~:=~3,~\mathid{y}~:=~\frac{3}{2},~\mathid{z}~:=~1~\\ \mdmathindent{2}\mbox{Move $x$ from $3$ to $8$:}~~&~~~\mathid{x}~:=~8,~\mathid{y}~:=~4,~\mathid{z}~:=~\frac{8}{3}\\ \mdmathindent{2}\mbox{Move $x$ from $3$ to $6$:}~~&~~~\mathid{x}~:=~6,~\mathid{y}~:=~3,~\mathid{z}~:=~2 \end{mdmathpre}%mdk \]

Patching (N. Bjørner and Nachmanson, 2018)

CAV2024slide

Cubes (Bromberger and Weidenbach, 2016,Bromberger and Weidenbach (2017))

Proposition: If $Ax \leq b - \frac{1}{2} \onenorm{A}$ has a solution over the reals, then $Ax \leq b$ has an integer solution obtained by rounding.

Example: Given

\[3x + y \leq 9 \wedge - 3y \leq -2 \]

Solve instead

\[3x + y \leq \underbrace{9 - \frac{1}{2}(3 + 1)}_{7} \wedge -3y \leq \underbrace{-2 - \frac{1}{2}3}_{-3.5} \]

Real solution $y = \frac{7}{6}, x = \frac{35}{18}$. Then $y = 1, x = 2$ is an integer solution.

Cubes (N. Bjørner and Nachmanson, 2018)

Observation: One can often avoid strengthening inequalities.


bounds on variables $x \leq 4$.


differences between variables $x - y \leq 6$ need not be strengthened to $x - y \leq 5$.


octagon inequalities $x + y \leq 5$ can be strengthened to $x + y < 5$, or solutions can be patched.


unit horn inequalities $x \geq 5y + 6z + 3$

GCD Tests

Basic GCD: $5/6x + 3/6y + z + 5/6u = 0$ is infeasible, if

  • There are fixed variables: $2 \leq x \leq 2, -1 \leq u \leq -1$
  • After replacing fixed variables by values: $5 + 3(y + 2z) = 0$
  • $3 \not\uparrow 5$.

Extended GCD: $a x + b y + c = 0$ is infeasible if

  • $a < b$
  • $lb \leq x \leq ub$ is bounded

  • $\lfloor \frac{a \cdot ub}{\gcd(b,c)} \rfloor < \lceil \frac{a \cdot lb}{\gcd(b,c)} \rceil$

Example:

  • $3x + 7y + 14 = 0$, $1 \leq x \leq 2$
  • $0 = \lfloor \frac{3\cdot 2}{7} \rfloor < \lceil \frac{3 \cdot 1}{7} \rceil = 1$.

Gomory cuts

  • Z3's implementation is based on (Dutertre and Moura, 2006).
  • Applies generally to mixed integer problems.
  • Some value to finding cuts with small coefficients.

Bounds Tightening

\[\begin{mdmathpre}%mdk \mdmathindent{9}&~~2\mathid{x}~+~4\mathid{y}~+~\mathid{z}~\leq 5,~\mathid{z}~\leq 2~\\ \implies &~\mbox{by replacing $z$ with upper bound}\\ \mdmathindent{9}&~2\mathid{x}~+~4\mathid{y}~\leq 3~\\ \implies &~\mbox{by rounding upper bound}\\ \mdmathindent{9}&~2\mathid{x}~+~4\mathid{y}~\leq 2~\\ \implies &~\mbox{by adding back $z$}~\\ \mdmathindent{9}&~2\mathid{x}~+~4\mathid{y}~+~\mathid{z}~\leq 4 \end{mdmathpre}%mdk \]

Cuts from Hermite Matrices (Dillig et al., 2009; Christ and Hoenicke, 2015)

Constraints: $Ax \leq b$.


Assume $A$ is square and a tight non-integral solution $Ax_0 = b$.


Find Hermite $H$, unimodular $U$, s.t. $AU = H$. $\left(\begin{array}{rrrr}1 & 0 & 0 & 0 \\-1 & 2 & 0 & 0 \\-3 & -1 & 3 & 0 \\-1 & 0 & -2 & 4 \\\end{array}\right)$


Then $Ax \leq b \Leftrightarrow \exists y \ . \ Hy \leq b \land y = U^{-1}x$


Then $y_0 = U^{-1}x_0$ is not integral either.


Branch on some non-integral $y_{0i}$: $y_i \leq \lfloor y_{0i} \rfloor$.

Diophantine Solutions and Cuts (Griggio, 2012)

Bring tableau into integer solved form $x := \vec{a}\vec{y}$ with $\vec{a}$ integer (also used by Pugh's Omega test):

\[\begin{mdmathpre}%mdk \mdmathindent{7}&~3\mathid{x}~+~14\mathid{y}~-~7~=~0~\\ \equiv &~\\ \mdmathindent{7}&~3~(\mathid{x}~+~(14~\idiv 3)~\mathid{y}~-~(7~\idiv 3))\\ \mdmathindent{7}&~+~(14~\;\mbox{mod}\;~3)\mathid{y}~-~(7~\;\mbox{mod}\;~3)~=~0\\ \equiv &~\\ \mdmathindent{7}&~3~(\mathid{x}~+~4~\mathid{y}~-~2)~+~2~\mathid{y}~-~1~=~0\\ \simeq &~\\ \mdmathindent{7}&~3~\mathid{z}~+~2~\mathid{y}~-~1~=~0~~~~~~~~~~~~~~~~~~&~\mathid{fresh}\;~\mathid{z}\\ \mdmathindent{7}&~\mathid{x}~:=~\mathid{z}~-~4~\mathid{y}~-~2 \end{mdmathpre}%mdk \]
  • Detect GCD conflicts during normalization.
  • Cuts and bounds tightening on solved form.

Branching

Example 1.
LRA solver assigns $val(x) := \frac{1}{2}$, for integer $x$. Create branch atom $x \leq 0$.

When $x \leq 0$ is assigned, upper bound $x \leq 0$ is added to tableau.

When $\neg(x \leq 0)$ is assigned, then lower bound $1 \leq x$ is added to tableau.

LIA - next

  • Opportunities for micro-tuning are endless.

  • Smart scheduling for LIA (and NRA/NIA) end-game solvers?

  • LIA solver for small integers.

  • IntSAT/CutSAT not part of Z3.

NRA - Non-linear Real Arithmetic

NRAArch

NRA

What: Variables range over Reals. Addition, unrestricted multiplication.

How: Incremental linearization, CAD.

Why: A basis for solving also NIA and interesting self-contained uses pop up, such as Economy.

\[\begin{mdmathpre}%mdk \mdmathindent{2}\mathid{x},~\mathid{y},~\mathid{z}~\in \mathit{R}\\ \mdmathindent{2}\mathid{x}~>~1~\land \mathid{y}~>~1~\land \mathid{x}\cdot \mathid{y}~<~1~\leadsto \mathid{unsat} \end{mdmathpre}%mdk \]

Tableau with monomial definitions

Subset of variables are monomials $\mulvars$:

\[\begin{array}{ll} x_i = \sum_{x_j \in \nonbasicvars} a_{ij} x_j & x_i \in \basicvars \\ m_i := \prod_{x_{ij}} x_{ij} & m_i \in \mulvars \subset \nonbasicvars \cup \basicvars \\ l_j \leq x_j \leq u_j & x_j \in \nonbasicvars \cup \basicvars \\ val : x_j \mapsto {\mathcal R} & \forall x_j \in \nonbasicvars \\ nval : x_i \mapsto \sum_{x_j \in \nonbasicvars} a_{ij} val(x_j) & \forall x_i \in \basicvars \end{array} \]

Z3 tracks some equalities: If $m := xy$ and $x = -z$ is known, then $m := -zy$ is also used for lemmas.

NL Patching

Suppose $ m := x \cdot x \cdot y$ is in tableau, but $val(m) \neq val(x)^2 val(y)$.

  • Try patch $val(m) := val(x) \cdot val(x) \cdot val(y)$,
  • Try patch $val(y) := val(m) / (val(x) \cdot val(x))$,
  • Try patch $val(x) := \pm r / val(y)$, when $val(m) = r^2$ for a rational $r$.

Patching uses LP patch lookahead.

NL Bounds propagation using Interval arithmetic

Monomial propagation: $m := x\cdot x\cdot y$

\[\begin{mdmathpre}%mdk \mdmathindent{2}\mathid{y}~\geq 3~\implies \mathid{m}~\geq 0~~~~~~~~~~~~~~~~~&~\mbox{by}~&~\mathid{m}~\in [-\infty,\infty]^2~\cdot [3,~\infty]\\ \\ \mdmathindent{2}\mathid{x}~\geq 1,~\mathid{y}~\geq 3~\implies \mathid{m}~\geq 3~~~~~~~&~\mbox{by}~&~\mathid{m}~\in [1,\infty]^2[3,\infty]\\ \\ \mdmathindent{2}\mathid{m}~\leq 12,~\mathid{y}~\geq 3~\implies -2~\leq \mathid{x}~\leq 2~&~\mbox{by}~&~\mathid{x}~\in \pm\sqrt{(-\infty,12]~/~[3,\infty)}. \end{mdmathpre}%mdk \]

NL Bounds propagation on terms

Recall: A term is a slack variable introduced as short-hand for arithmetic expression.

Interval arithmetic isn't distributive:

Let $x \geq 2, y \geq -1, z \geq 2$

Then $xy + xz \in [2,\infty)\cdot[-1,\infty) + [2,\infty)\cdot[2,\infty) = (-\infty,\infty)$

But $x(y + z) \in [2,\infty)([-1,\infty) + [2,\infty)) = [2,\infty)$

So z3 explores different variants of distributing multiplication over terms.

Gröbner basis saturation

  • $V := \{ m_i, x_{ij} \mid m_i := \prod_{ij} x_{ij}, val(m_i) \neq \prod_{ij} val(x_{ij}) \}$.
  • $\mathcal{R} := \emptyset$
  • Ignore $r_i : x + by = 0$, $x$ is unbounded.
  • Add $r_i : ax + by = 0$ to $\mathcal{R}$, and add $y$ to $V$, if $x \in V$.
  • Run bounded Gröbner saturation based on $\mathcal{R}$.
  • Propagate new equations, detect infeasible equations
    • using interval bounds propagation
    • GCD tests
    • Euclidian tests

Multi-terminal ZDDs for polynomials

Design goals:

  • single instance store
  • polynomial simplification is built-in

PDD

Incremental Linearization (Cimatti et al., 2018a)

\[\begin{mdmathpre}%mdk \mdmathindent{2}\mathid{x}~=~0~~~~~&~\implies &~\mathid{xy}~=~0~\\ \mdmathindent{2}\mathid{x}~=~\pm 1~&~\implies &~\mathid{xy}~=~\pm \mathid{y}\\ \mdmathindent{2}\mathid{x}~\geq 1~~&~\implies &~\mathid{xy}~\geq \mathid{y}\\ \mdmathindent{2}\mathid{x}~\geq 0,~\mathid{y}~\geq 0~&~\implies &~\mathid{xy}~\geq 0 \end{mdmathpre}%mdk \]

NLSAT (Jovanovic and Moura, 2012)

nlsat

NRA - next

  • Integration of linearization with NLSAT (Promies et al., 2025)

  • $\delta$-satisfiability to leverage numeric methods.

  • Postitiv Stellensatz (ground and quantifiers)

NIA - Non-linear Integer Arithmetic

NRAArch

NIA

What: Variables range over Integers. Addition, unrestricted multiplication.

How: Methods for NRA + LIA + $\epsilon$

Why: The tougher spot in verification tools, such as F* and Certora.

\[\begin{mdmathpre}%mdk \mdmathindent{2}\mathid{x},~\mathid{y},~\mathid{z}~\in \mathit{Z}\\ \mdmathindent{2}\mathid{x}~>~1~\land \mathid{y}~>~1~\land \mathid{x}\cdot \mathid{y}~<~3~\leadsto \mathid{unsat} \end{mdmathpre}%mdk \]

NIA - Theoretical Difficulty and Practical Scope

  • QF_NIA is undecidable (Davis et al., 1961; Matiyasevich, 1970): there is no complete proof system for UNSAT.

  • QF_NIA used by ZK/Smart Contract auditors for bounded integers $[0, 2^{256}($.

  • Partial solution by $NIA := LIA + NRA$

NIA := LIA + NRA

NLSAT Branch and Bound (Jovanović, 2017)

  • Idea: Augment NLSAT (complete for NRA) post-check for integers.

  • If solution to $x \in \mathcal{Z}$ is not integer, but in interval $(k, k+1), k \in \mathcal{Z}$, then

  • Add axiom $x \leq k \lor x \geq k + 1$.

  • Run NLSAT again.

Incremental Linearization for Integers (Cimatti et al., 2018a)

  • Makes the biggest difference for solving NIA in Z3.
  • Creates constraints for violated monomial assignments $val(xy) \neq val(x)val(y)$.
  • Example, tangent lemma:
\[\begin{mdmathpre}%mdk \mathid{x}~>~\mathid{a},~\mathid{y}~>~\mathid{b}~&~\implies &~(\mathid{x}~-~\mathid{a})~\cdot (\mathid{y}~-~\mathid{b})~>~0\\ \mdmathindent{13}&~\implies &~\mathid{xy}~-~\mathid{ay}~-~\mathid{bx}~+~\mathid{ab}~>~0 \end{mdmathpre}%mdk \]

Suppose $val(xy) < val(x)\cdot val(y)$. Then setting $a := val(x) - 1$, $b := val(y) -1$ ensures that $val(xy) - a\cdot val(y) - b\cdot val(x) - a\cdot b \leq 0$.

Pseudo-linear monomials

\[\begin{mdmathpre}%mdk \mdmathindent{4}&~\mathid{x}~=~\mathid{val}(\mathid{x})~\implies \mathid{x}\cdot \mathid{y}~=~\mathid{val}(\mathid{x})\cdot \mathid{y}\\ \mathid{if}~~&~\\ \mdmathindent{4}&~\mathid{lo}~\leq \mathid{x}~\leq \mathid{hi},~\mathid{z}~\in \mathcal{Z}\\ \mdmathindent{4}&~\mathid{hi}~-~\mathid{lo}~\leq 4~~~ \end{mdmathpre}%mdk \]

But NIA > LIA + NRA

Euclidian Lemmas (I)

Consistency checks on equations produced by Gröbner saturation.

\[\begin{mdmathpre}%mdk \\ \mdmathindent{1}\mathid{x}~\mathid{y}~-~\mathid{z}~=~0~&~\implies &~\mathid{x}~\uparrow \mathid{z}~~~&~~\mathid{x},~\mathid{y},~\mathid{z}~\in \mathid{Z}\\ \mdmathindent{13}&~\implies &~(\mathid{x}~>~\mathid{z}~\geq 0~\implies \mathid{z}~=~0)~&~\clubsuit\\ \mdmathindent{13}&~\implies &~(\mathid{x}~<~\mathid{z}~\leq 0~\implies \mathid{z}~=~0)~&~\spadesuit\\ \mdmathindent{13}&~\implies &~(2\mathid{x}~>~\mathid{z}~\geq \mathid{x}~>~0~\implies \mathid{z}~=~\mathid{x})~&~\heartsuit\\ \mdmathindent{13}&~\implies &~\ldots \end{mdmathpre}%mdk \]

Create lemma if current assignment to $x, y, z$ violates $\clubsuit$ or $\spadesuit$. To not introduce divergence, z3 skips lemmas of the form $\heartsuit$.

Euclidian Lemmas (II)

NIAParity

NIAParitySolution

NIA - next

  • Identifying practical inference algorithms $X$ where $NIA = NRA + LIA + X + Y$.

  • Local search

  • Still very fragile, some can be mitigated by parallelism

NIAscatter

Extra Credit

LIA*

LIAstarIntro

LIA* - non-starter approach

LIAstarNonstarter RuzicaMunch

LIA* - mile-high

LIAstarMileHigh

LIA* - under approxiations

Grow under-approximation: $U^* \implies F_2^*$:

Initially $U := \emptyset, U^* := (0,0)$

Maintain, $U = \bigcup_i a_i + \lambda B_i$ under-approximates $F_2$ and set $U^* := \exists \mu\lambda (x = \sum_i \mu_i a_i + \lambda_i B_i) \land \bigwedge_i (\mu_i = 0 \implies \lambda_i = 0)$.

  • If $U^* \land F_1$ is SAT done. Otherwise,
  • Let $x_1, y_1: U^*(x_0, y_0) \land F_2(x_1, y_1) \land \neg U^*(x_0 + x_1, y_0 + y_1)$.
  • Add $(x_1, y_1)$ to $U$.
  • Reduce vectors $a_i, B_i$ using new element.

LIA* over-approximation (I)

LIAstar

LIA* over-approximation (II)

Suppose

\[\begin{mdmathpre}%mdk \mdmathindent{6}\mathid{U}^*(\mathid{x}_0,~\mathid{y}_0)~\land \mathid{F}_2(\mathid{x}_1,~\mathid{y}_1)~\land \mathid{F}_2(\mathid{x}_2,~\mathid{y}_2)~&~\\ \land \mathid{F}_1(\mathid{x}_0~+~\mathid{x}_1~+~\mathid{x}_2,~\mathid{y}_0~+~\mathid{y}_1~+~\mathid{y}_2)~~~~~~~~~~~~~~~~~&~\mbox{is unsat} \end{mdmathpre}%mdk \]

Compute interpolant $I$:

\[\begin{mdmathpre}%mdk \mathid{U}^*(\mathid{x}_0,~\mathid{y}_0)~\land \mathid{F}_2(\mathid{x}_1,~\mathid{y}_1)~&~\implies &~\mathid{I}(\mathid{x}_0~+~\mathid{x}_1,~\mathid{y}_0~+~\mathid{y}_1)~\\ \mathid{I}(\mathid{x},~\mathid{y})~&~\implies &~(\mathid{F}_2(\mathid{x}_2,~\mathid{y}_2)~\implies \neg \mathid{F}_1(\mathid{x}~+~\mathid{x}_2,~\mathid{y}~+~\mathid{y}_2)) \end{mdmathpre}%mdk \]

Initially $O^* := \top$

Add conjunctions from I to $O^*$ that are inductive, that is:

\[\begin{mdmathpre}%mdk \mathid{O}^*(\mathid{x},~\mathid{y})~\land \mathid{F}_2(\mathid{x}_1,~\mathid{y}_1)~\implies \mathid{O}^*(\mathid{x}~+~\mathid{x}_1,~\mathid{y}~+~\mathid{y}_1) \end{mdmathpre}%mdk \]

Warm Start

You can set the initial value of variables. 

 (declare-const x Int)
 (set-initial-value x 10)
 (push)
 (assert (> x 0))
 (check-sat)
 (eval x)
 ; 10

It is not always possible to enforce: If a benchmark is translated to SAT, initialization is partial.

References

Marc Bezem, Robert Nieuwenhuis, and Enric Rodríguez-Carbonell. “The Max-Atom Problem and Its Relevance.” In Logic for Programming, Artificial Intelligence, and Reasoning, 15th   International Conference, LPAR 2008, Doha, Qatar, November 22-27, 2008. Proceedings, 47–61. 2008. doi:10.1007/978-3-540-89439-1_4. 🔎
Nikolaj Bjørner, and Lev Nachmanson. “Theorem Recycling for Theorem Proving.” In Vampire 2017. Proceedings of the 4th Vampire Workshop, edited by Laura Kovács and Andrei Voronkov}, 53:1–8. EPiC Series in Computing. EasyChair. 2018. doi:10.29007/r58f. 🔎
Nikolaj S. Bjørner, and Lev Nachmanson. “Arithmetic Solving in Z3.” In Computer Aided Verification - 36th International Conference, CAV   2024, Montreal, QC, Canada, July 24-27, 2024, Proceedings, Part I, edited by Arie Gurfinkel and Vijay Ganesh, 14681:26–41. Lecture Notes in Computer Science. Springer. 2024. doi:10.1007/978-3-031-65627-9_2. 🔎
Martin Bromberger, and Christoph Weidenbach. “Fast Cube Tests for LIA Constraint Solving.” In IJCAR. 2016. doi:10.1007/978-3-319-40229-1_9. 🔎
Martin Bromberger, and Christoph Weidenbach. “New Techniques for Linear Arithmetic: Cubes and Equalities.” Formal Methods in System Design 51 (3): 433–461. 2017. doi:10.1007/s10703-017-0278-7. 🔎
Jürgen Christ, and Jochen Hoenicke. “Cutting the Mix.” In CAV. 2015. doi:10.1007/978-3-319-21668-3_3. 🔎
Alessandro Cimatti, Alberto Griggio, Ahmed Irfan, Marco Roveri, and Roberto Sebastiani. “Experimenting on Solving Nonlinear Integer Arithmetic with Incremental   Linearization.” In SAT. 2018. doi:10.1007/978-3-319-94144-8_23. 🔎
Alessandro Cimatti, Alberto Griggio, Ahmed Irfan, Marco Roveri, and Roberto Sebastiani. “Experimenting on Solving Nonlinear Integer Arithmetic with Incremental Linearization.” In Theory and Applications of Satisfiability Testing – SAT 2018, 10929:383–398. Lecture Notes in Computer Science. Springer. 2018. doi:10.1007/978-3-319-94144-8_23. 🔎
Alexandru Costan, Stephane Gaubert, Eric Goubault, Matthieu Martel, and Sylvie Putot. “A Policy Iteration Algorithm for Computing Fixed Points in Static   Analysis of Programs.” In Computer Aided Verification, 17th International Conference, CAV   2005, Edinburgh, Scotland, UK, July 6-10, 2005, Proceedings, 462–475. 2005. doi:10.1007/11513988_46. 🔎
Martin Davis, Hilary Putnam, and Julia Robinson. “The Decision Problem for Exponential Diophantine Equations.” Annals of Mathematics 74 (3). Princeton University Press: 425–436. 1961. doi:10.2307/1970295. 🔎
Isil Dillig, Thomas Dillig, and Alex Aiken. “Cuts from Proofs: A Complete and Practical Technique for Solving   Linear Inequalities over Integers.” In CAV. 2009. doi:10.1007/978-3-642-02658-4_20. 🔎
B. Dutertre, and L. de Moura. “A Fast Linear-Arithmetic Solver for DPLL(T).” In CAV. 2006. 🔎
Thomas Gawlitza, and Helmut Seidl. “Precise Relational Invariants Through Strategy Iteration,” in @DBLP:conf/csl/2007, 4646:23–40. 2007. doi:10.1007/978-3-540-74915-8_6. 🔎
Alberto Griggio. “A Practical Approach to Satisability Modulo Linear Integer Arithmetic.” J. Satisf. Boolean Model. Comput. 8 (1/2): 1–27. 2012. doi:10.3233/SAT190086. 🔎
Dejan Jovanović. “Solving Nonlinear Integer Arithmetic with MCSAT.” In Proceedings of the 18th International Conference on Verification, Model Checking, and Abstract Interpretation (VMCAI), 1–20. Springer. 2017. https://​yices.​csl.​sri.​com/​papers/​vmcai2017.​pdf. 🔎
Dejan Jovanovic, and Leonardo Mendonça de Moura. “Solving Non-Linear Arithmetic.” In Automated Reasoning - 6th International Joint Conference, IJCAR   2012, Manchester, UK, June 26-29, 2012. Proceedings, 339–354. 2012. doi:10.1007/978-3-642-31365-3_27. 🔎
Egor George Karpenkov. “Finding Inductive Invariants Using Satisfiability modulo Theories   and Convex Optimization. (Recherche D’invariants Inductifs Par Satisfiabilité modulo Théorie et Optimisation Convexe).” Phdthesis, Grenoble Alpes University, France. 2017. https://​tel.​archives-​ouvertes.​fr/​tel-​01681555. 🔎
Egor George Karpenkov, David Monniaux, and Philipp Wendler. “Program Analysis with Local Policy Iteration,” in @DBLP:conf/vmcai/2016, 9583:127–146. 2016. doi:10.1007/978-3-662-49122-5_6. 🔎
Yuri V. Matiyasevich. “Enumerable Sets Are Diophantine.” Doklady Akademii Nauk SSSR 191: 279–282. 1970. Translated in Soviet Math. Dokl. 11 (1970), 354–358. 🔎
Leonardo Mendonça de Moura, and Nikolaj Bjørner. “Model-Based Theory Combination.” Electron. Notes Theor. Comput. Sci. 198 (2): 37–49. 2008. doi:10.1016/j.entcs.2008.04.079. 🔎
Valentin Promies, Jasper Nalbach, Erika, and Paul Wagner. “More Is Less: Adding Polynomials for Faster Explanations in NLSAT.” In Automated Deduction - CADE 30 - 30th International Conference on   Automated Deduction, Stuttgart, Germany, July 28-31, 2025, Proceedings, edited by Clark W. Barrett and Uwe Waldmann, 15943:116–135. Lecture Notes in Computer Science. Springer. 2025. doi:10.1007/978-3-031-99984-0_7. 🔎
Yoni Zohar, Ahmed Irfan, Makai Mann, Aina Niemetz, Andres Nötzli, Mathias Preiner, Andrew Reynolds, Clark W. Barrett, and Cesare Tinelli. “Bit-Precise Reasoning via Int-Blasting.” In Verification, Model Checking, and Abstract Interpretation - 23rd International   Conference, VMCAI 2022, Philadelphia, PA, USA, January 16-18, 2022, Proceedings, edited by Bernd Finkbeiner and Thomas Wies, 13182:496–518. Lecture Notes in Computer Science. Springer. 2022. doi:10.1007/978-3-030-94583-1_24. 🔎