On Salum’s Algorithm for X3SAT

We give a short commentary on Latif Salum’s attempt at proving $\mbox{\rm P}=\mbox{\rm NP}$ in “Tractability of One-in-three 3SAT: $\mbox{\rm P}=\mbox{\rm NP}$” [Sal20], showing exactly where the argument breaks down. We first define the one-in-three 3SAT problem, also known as X3SAT, and its equivalence with Salum’s nonstandard formulation.

The X3SAT problem takes a 3CNF formula (a conjunction of disjunctions, each having at most 3 literals) as input. A 3CNF formula $\phi$ is a yes instance of X3SAT if and only if there exists an assignment that sets exactly one of the literals in every clause in $\phi$ to true. X3SAT is known to be NP-complete.

Salum’s paper introduces X3SAT formulas (not to be confused with the X3SAT problem) of the form $\phi=\psi\land\varphi$, where $\psi$, called the minterm, is a conjunction of literals, and $\varphi$ is a conjunction of X3SAT clauses of the form $(a\odot b\odot c)$.¹¹1Salum calls the $\odot$ symbol the “exactly-1 disjunction” which suggests it is an operator. Nevertheless, as an operator the semantics would be flawed as it is neither binary nor ternary. Instead, we use it in this critique as a purely notational element, which is essentially the role this symbol actually plays in Salum’s paper. An assignment of the variables in an X3SAT clause satisfies the clause if and only if it sets exactly one of the literals $a$, $b$, or $c$ to true. This holds similarly in the case of only two literals in $(a\odot b)$. The minterm $\psi$ is meant to hold a partial assignment of the literals in the formula and it is omitted when it is empty. An X3SAT formula $\phi$ is satisfiable if and only if there exists an assignment that makes $\psi$ true and satisfies every clause in $\varphi$.

It is easy to see that X3SAT formulas with empty minterms are equivalent to 3CNF formulas (by simply replacing the $\odot$ notational element with $\lor$), and a satisfying assignment for an X3SAT formula with an empty minterm is also a witness for the equivalent 3CNF being a yes instance of X3SAT. Thus a polynomial-time algorithm to determine the satisfiability of X3SAT formulas would yield a polynomial-time algorithm for X3SAT, thereby proving $\mbox{\rm P}=\mbox{\rm NP}$. Salum claims to provide such an algorithm.

Salum’s algorithm [Sal20] processes each variable in a given formula $\phi$ in turn, chooses a polarity, and simplifies the formula resulting from the chosen value. If a conflict is detected, the algorithm chooses the opposite polarity. We provide a more detailed summary in Section 2. The fatal flaw in the algorithm is that decisions are not recorded to be later revised when a conflict is found. We provide a counterexample exploiting this flaw in Section 3. We conclude in Section 4.

Salum’s paper [Sal20] introduces much notation to define four functions that ultimately comprise the algorithm. These functions can be easily stated in plain English and through the use of standard Boolean formula satisfiability (SAT) terminology. Instead of reintroducing Salum’s notation, we provide in Appendix A line-by-line reinterpretations of the four key functions of the proposed algorithm, but using standard SAT terminology.

Recall, from standard SAT terminology and the description of the DPLL [DLL62, DP60] family of algorithms, that a decision, during a solving process, corresponds to assigning an arbitrary truth value to a variable. The decision is then propagated through the formula, i.e., the formula is simplified over the assignment made. During this process, the truth values of some of the other variables are set as consequences of the decision made. The literals thus set are typically called implied literals. A solving process will typically simplify the formula iteratively until no further simplification can be achieved, at which point it will pick another variable to decide on and repeat. A decision should be recorded because the arbitrary nature of the assignment implies that it might be wrong, and the solving process would need to undo the decision and its consequences. Doing so is typically referred to as backtracking. Backtracking is triggered when the current decisions have led to a situation where a variable must be set to both true and false, a situation typically referred to as a conflict.

Salum’s functions resemble many of the components of a typical DPLL algorithm, extended of course to the context of X3SAT. $\mathtt{Reduce}(\phi_{s},r_{j})$ (Algorithm A.1) propagates the decision $r_{j}$ over the formula $\phi_{s}$ and performs some simplifications. $\mathtt{Scope}(r_{j},\phi_{s})$ (Algorithm A.2) iteratively calls $\mathtt{Reduce}$ to propagate a single decision and all of its implied literals. Both $\mathtt{Reduce}$ and $\mathtt{Scope}$ return NULL when a conflict is detected. $\mathtt{Scan(\phi_{s})}$ (Algorithm A.3), which is the entry point of Salum’s algorithm, performs some simplifications, then decides on a variable that is still unassigned in the simplified formula, and calls $\mathtt{Scope}$ on this decision. If $\mathtt{Scope}$ detects a conflict due to this decision, $\mathtt{Scan}$ calls $\mathtt{Remove}$ (Algorithm A.4) to assign the literal to the opposite polarity. $\mathtt{Remove}(r_{j},\phi_{s})$ calls $\mathtt{Reduce}(\phi_{s},\overline{r_{j}})$ but this time, if a conflict is detected, it returns UNSAT. If no conflict is detected, $\mathtt{Remove}(r_{j},\phi_{s})$ simplifies the current formula over the assignment $\overline{r_{j}}$ and calls $\mathtt{Scan}(\phi_{s+1})$ over the resulting formula $\phi_{s+1}$.

Our analysis reveals that the proposed algorithm is missing one key component of DPLL algorithms: it does not backtrack. Without backtracking, it is usually easy to show that a solving algorithm is incorrect by providing a formula and an ordering of the literals that trigger a bad decision. We do that in Section 3.

We provide a counterexample that “tricks” Scope, namely,

This formula is quite clearly satisfiable, for instance, by the assignment $(a,b,c,x,y)=(0,0,1,0,1)$.²²2As is standard, 1 represents true and 0 represents false. Salum claims in [Sal20, Section 3.1] that if $\mathtt{Scan}(\phi)$ does not find a satisfying assignment (in Salum’s words, it is “interrupted”), then $\phi$ is unsatisfiable, so we run $\mathtt{Scan}(\phi)$ and show it does not find a satisfying assignment. The line numbers mentioned in the explanation below correspond to the line numbers of the algorithms in the original paper [Sal20], and not to those of our reinterpretations in Appendix A.

The minterm in $\phi$ is empty, so we skip to line 4 of $\mathtt{Scan}$ to process the literals in turn. Salum claims in [Sal20, Theorem 36] that the order in which the literals are processed is arbitrary. Accordingly, we choose to process $a$ first, invoking $\mathtt{Scope}(a,\phi)$ in line 6. From line 4 of $\mathtt{Scope}(a,\phi)$, we call $\mathtt{Reduce}(\phi,a)$. Given that $a$ was chosen to be true, the first clause in $\phi$ propagates to the minterm $a\land\bar{b}\land\bar{c}$. The second and third clauses contain neither $a$ nor $\bar{a}$ so they are not touched. Thus at the end of $\mathtt{Reduce}(\phi,a)$, we are left with

Back in $\mathtt{Scope}(a,\phi)$, line 2 will call $\mathtt{Reduce}(\phi,\bar{b})$ and $\mathtt{Reduce}(\phi,\bar{c})$ in turn. $\mathtt{Reduce}(\phi,\bar{b})$ will first propagate the decision $\bar{b}$. Since there are no clauses containing $\bar{b}$ it moves on to line 8, where it will delete $\overline{\overline{b}}=b$ from all clauses containing it. The only such clause is $(b\odot x\odot y)$, and by removing $b$ we obtain $(x\odot y)$. When we return to line 3 of $\mathtt{Scope}(a,\phi)$, the current formula is

Analogously, in $\mathtt{Reduce}(\phi,\bar{c})$ we reduce $(c\odot x\odot\bar{y})$ to $(x\odot\bar{y})$, thus obtaining the formula

upon finishing $\mathtt{Scope}(a,\phi)$.

At this point the algorithm is already doomed as, having chosen the positive polarity of $a$, it has no way to satisfy the remainder of the formula, namely $(x\odot y)\land(x\odot\bar{y})$. For completeness, however, we continue with the remainder of the execution.

Having returned to line 6 of $\mathtt{Scan}(\phi)$, we now run $\mathtt{Scope}(\bar{a},\phi)$. Line 5 in $\mathtt{Scope}(\bar{a},\phi)$ detects the conflict with $a$ already in the minterm, and returns NULL. Thus $\mathtt{Scan}(\phi)$ calls $\mathtt{Remove}(\bar{a},\phi)$, which leaves the formula unaffected, as neither $a$ nor $\bar{a}$ is contained in the remainder of the formula $(x\odot y)\land(x\odot\bar{y})$.

The execution returns to line 4 of $\mathtt{Scan}(\phi)$, and we process the variable $x$ next. Now in $\mathtt{Scope}(x,\phi)$, we simplify the formula $\phi$ due to $x$ being assigned to true, and obtain

which is detected in line 6 of $\mathtt{Reduce}(\phi,x)$ as a conflict. Thus $\mathtt{Scan}(\phi)$ calls $\mathtt{Remove}(\phi,x)$ which in turn calls $\mathtt{Reduce}(\phi,\bar{x})$. $\mathtt{Reduce}$ then detects the conflict $y\land\bar{y}$ and returns NULL. $\mathtt{Remove}(\phi,x)$ thus concludes that the formula is unsatisfiable. In the next subsection, we provide reasoning for why this counterexample “tricks” Salum’s algorithm.

We have studied Latif Salum’s paper titled “Tractability of One-in-three 3SAT: $\mbox{\rm P}=\mbox{\rm NP}$” [Sal20]. We showed that this algorithm is flawed by providing a counterexample: an X3SAT formula that is satisfiable, but for which the algorithm returns UNSAT. Thus Salum’s proposed algorithm does not settle the long-standing question of P vs. NP. In fact, an algorithm as straightforward as the one proposed will very likely fail to show $\mbox{\rm P}=\mbox{\rm NP}$.

We thank Michael C. Chavrimootoo, Lane A. Hemaspaandra, and Mandar Juvekar for their helpful feedback on an earlier version of our critique. Any remaining errors are the responsibility of the authors.