refinement

1 files changed, 5 insertions, 6 deletions

diff --git a/chapters/background/03-satisfiability-modulo-theories.tex b/chapters/background/03-satisfiability-modulo-theories.tex
index a55d54d..80e9932 100644
--- a/chapters/background/03-satisfiability-modulo-theories.tex
+++ b/chapters/background/03-satisfiability-modulo-theories.tex
@@ -1,7 +1,7 @@
 \section{Satisfiability Modulo Theories}
 \label{sec:satisfiability-modulo-theories}
 
-Satisfiability Modulo Theories (SMT) is the problem of determining whether a mathematical formula is
+\textit{Satisfiability Modulo Theories} (SMT) is the problem of determining whether a mathematical formula is
 satisfiable within a certain formal theory in first-order logic. Barrett et al.~\cite{barrett2016smtlib}
 defined the SMT-LIB standard for input format and theory definitions.
 
@@ -11,14 +11,13 @@ multiplication by constants, and equality/inequality relations:
 \begin{equation}
     \phi ::= t_1 \sim t_2 \mid \phi_1 \lor \phi_2 \mid \phi_1 \land \phi_2 \mid \neg \phi_1
 \end{equation}
-where $t_1, t_2$ are linear terms of the form $c_1 x_1 + \dots + c_k x_k + c_0$ (with $c_i \in \mathbb{R}$ and $x_i$ being real
+where $t_1, t_2$ are linear terms of the form $c_1 \cdot x_1 + \dots + c_k \cdot x_k + c_0$ (with $c_i \in \mathbb{R}$ and $x_i$ being real
 variables), and $\sim \;\in \{=, \le, <, \ge, >\}$.
 
 To represent a ReLU activation $y = \max(0, x)$ in LRA, we must introduce a disjunction:
 \begin{equation}
     (x > 0 \land y = x) \lor (x \le 0 \land y = 0)
 \end{equation}
-For a network with $N$ ReLU neurons, there are up to $2^N$ possible activation patterns. Solving the
-verification property requires the SMT solver, such as Z3 presented by De Moura et al.~\cite{demoura2008z3},
-to implicitly explore this branching search space.
-
+For a network with $n$ ReLU neurons, there are up to $2^n$ possible activation patterns. Solving the
+verification problem requires the SMT solver, such as Z3 presented by De Moura et
+al.~\cite{demoura2008z3}, to implicitly explore this branching search space.