Characterizing Tseitin-formulas with short regular resolution refutations^††thanks: This work has been partly supported by the PING/ACK project of the French National Agency for Research (ANR-18-CE40-0011).

We assume the reader is familiar with the fundamentals of graph theory. For a graph $G$, we denote by $V(G)$ its vertices and by $E(G)$ its edges. For $v\in V(G)$, $E(v)$ denotes the edges incident to $v$ and $N(v)$ its neighbors ($v$ is not in $N(v)$). For a subset $V^{\prime}$ of $V(G)$ we denote by $G[V^{\prime}]$ the sub-graph of $G$ induced by $V^{\prime}$.

A binary tree whose leaves are in bijection with the edges of $G$ is called a branch decomposition¹¹1We remark that often branch decompositions are defined as unrooted trees. However, it is easy to see that our definition is equivalent, so we use it here since it is more convenient in our setting.. Each edge $e$ of a branch decomposition $T$ induces a partition of $E(G)$ into two parts as the edge sets that appear in the two connected components of $T$ after deletion of $e$. The number of vertices of $G$ that are incident to edges in both parts of this partition is the order of $e$, denoted by $order(e,T)$. The branchwidth of $G$, denoted by $bw(G)$, is defined as $bw(G)=\min_{T}\max_{e\in E(T)}order(e,T)$, where $\min_{T}$ is over all branch decompositions of $G$.

While it is convenient to work with branchwidth in our proofs, we state our main result with the more well-known treewidth $tw(G)$ of a graph $G$. This is justified by the following well-known connection between the two measures.

A separator $S$ in a connected graph $G$ is defined to be a vertex set such that $G\setminus S$ is non-empty and not connected. A graph $G$ is called $3$-connected if and only if it has at least $4$ vertices and, for every $S\subseteq V(G)$, $|S|\leq 2$, the graph $G\setminus S$ is connected.

Boolean variables can have value 0 ($false$) or 1 ($true$). The notation $\ell_{x}$ refers to a literal for a variable $x$, that is, $x$ or its negation $\overline{x}$. Given a set $X$ of Boolean variables, $lit(X)$ denotes its set of literals. A truth assignment to $X$ is a mapping $a:X\rightarrow\{0,1\}$. If $a_{X}$ and $a_{Y}$ are assignments to disjoint sets of variables $X$ and $Y$, then $a_{X}\cup a_{Y}$ denotes the combined assignment to $X\cup Y$. The set of assignments to $X$ is denoted by $\{0,1\}^{X}$. Let $f$ be a Boolean function, we denote by $var(f)$ its variables and by $sat(f)$ its set of models, i.e., assignments to $var(f)$ on which $f$ evaluates to $1$. A v-tree of $X$ is a binary tree $T$ whose leaves are labeled bijectively with the variables in $X$. A v-tree $T$ of $X$ induces a set of partitions $(X_{1},X_{2})$ of $X$ as follows: choose a vertex $v$ of $T$, setting $X_{1}$ to contain exactly the variables in $T$ that appear below $v$ and $X_{2}:=X\setminus X_{1}$.

Tseitin formulas are systems of parity constraints whose structure is determined by a graph. Let $G=(V,E)$ be a graph and let $c:V\rightarrow\{0,1\}$ be a labeling of its vertices called a charge function. The Tseitin-formula $T(G,c)$ has for each edge $e\in E$ a Boolean variable $x_{e}$ and for each vertex $v\in V$ a constraint $\chi_{v}:\sum_{e\in E(v)}x_{e}=c(v)\mod 2$. The Tseitin-formula $T(G,c)$ is then defined as $T(G,c):=\bigwedge_{v\in V}\chi_{v}$, i.e., the conjunction of the parity constraints for all $v\in V$. By $\overline{\chi_{v}}$ we denote the negation of $\chi_{v}$, i.e., the parity constraint on $(x_{e})_{e\in E(v)}$ with charge $1-c(v)$.

When conditioning the formula $T(G,c)$ on a literal $\ell_{e}\in\{x_{e},\overline{x_{e}}\}$ for $e=ab$ in $E(G)$, the resulting function is another Tseitin formula $T(G,c)|\ell_{e}=T(G^{\prime},c^{\prime})$ where $G^{\prime}$ is the graph $G$ without the edge $e$ (so $G^{\prime}=G-e$) and $c^{\prime}$ depends on $\ell_{e}$. If $\ell_{e}=\overline{x_{e}}$ then $c^{\prime}$ equals $c$. If $\ell_{e}=x_{e}$ then $c^{\prime}=c+1_{a}+1_{b}\mod 2$, where $1_{v}$ denotes the charge function that assigns $1$ to $v$ and $0$ to all other variables.

Since we consider Tseitin-formulas in the setting of proof systems for CNF-formulas, we will assume in the following that they are encoded as CNF-formulas. In this encoding, every individual parity constraint $\chi_{v}$ is expressed as a CNF-formula $F_{v}$ and $T(G,c):=\bigwedge_{v\in V}F_{v}$. Since it takes $2^{|E(v)|-1}$ clauses to write the parity constraint $\chi_{v}$, each clause containing $E(v)$ literals, we make the standard assumption that $E(v)$ is bounded, i.e., there is a constant upper bound $\Delta$ on the degree of all vertices in $G$.

A circuit over $X$ in negation normal form (NNF) is a directed acyclic graph whose leaves are labeled with literals in $lit(X)$ or 0/1-constants, and whose internal nodes are labeled by $\lor$-gates or $\land$-gates. We use the usual semantics for the function computed by (gates of) Boolean circuits. Every NNF can be turned into an equivalent NNF whose nodes have at most two successors in polynomial time. So we assume that NNF in this paper have only binary gates and thus define the size $|D|$ as the number of gates, which is then at most half the number of wires. Given a gate $g$, we denote by $var(g)$ the variables for the literals appearing under $g$. When $g$ is a literal input $\ell_{x}$, we have $var(g)=\{x\}$, and when it is a 0/1-input, we define $var(g)=\emptyset$. A gate with two children $g_{l}$ and $g_{r}$ is called decomposable when $var(g_{l})\cap var(g_{r})=\emptyset$, and it is called complete (or smooth) when $var(g_{l})=var(g_{r})$. An NNF whose $\land$-gates are all decomposable is called a decomposable NNF (DNNF). We call a DNNF complete when all its $\lor$-gates are complete. Every DNNF can be made complete in polynomial time. For every Boolean function $f$ on finitely many variables, there exists a DNNF computing $f$.

When representing Tseitin-formulas by DNNF, we will use the following:

Proof trees of a DNNF $D$ are tree-like sub-circuits of $D$ constructed iteratively as follows: we start from the root gate and add it to the proof tree. Whenever an $\land$-gate is met, both its child gates are added to the proof tree. Whenever a $\lor$-gate is met, exactly one child is is added to the proof tree. Each proof tree of $D$ computes a conjunction of literals. By distributivity, the disjunction of the conjunctions computed by all proof trees of $D$ computes the same function as $D$. When $D$ is complete, every variable appears exactly once per proof tree, so every proof tree of a complete DNNF encodes a single model.

A branching program (BP) $B$ is a directed acyclic graph with a single source, sinks that uniquely correspond to the values of a finite set $Y$, and whose inner nodes, called decision nodes are each labeled by a Boolean variable $x\in X$ and have exactly two output wires called the 0- and 1-wire pointing to two nodes respectively called its 0- and the 1-child. The variable $x$ appears on a path in $B$ if there is a decision node $v$ labeled by $x$ on that path. A truth assignment $a$ to $X$ induces a path in $B$ which starts at the source and, when encountering a decision node for a variable $x$, follows the 0-wire (resp. the 1-wire) if $a(x)=0$ (resp. $a(x)=1$). The BP $B$ is defined to compute the value $y\in Y$ on an assignment $a$ if and only if the path of $a$ leads to the sink labeled with $y$. We denote this value $y$ as $B(a)$. Let $f:X\rightarrow Y$ be a function where $X$ is a finite set of Boolean variables and $Y$ any finite set. Then we say that $B$ computes $f$ if for every assignment $a\in\{0,1\}^{X}$ we have $B(a)=f(a)$. We say that a node $v$ in $B$ computes a function $g$ if the BP we get from $B$ by deleting all nodes that are not reachable from $v$ computes $g$.

Let $R\subseteq\{0,1\}^{X}\times Y$ be a relation where $Y$ is again finite. Then we say that a BP $B$ computes $R$ if for every assignment $a$ we have that $(a,B(a))\in R$. Let $T(G,c)$ be an unsatisfiable Tseitin-formula for a graph $G=(V,E)$. Then we define the two following relations: $\textup{Search}_{T(G,c)}$ consists of the pairs $(a,C)$ such that $a$ is an assignment to $T(G,c)$ that does not satisfy the clause $C$ of $T(G,c)$. The relation $\textup{SearchVertex}(G,c)$ consists of the pairs $(a,v)$ such that $a$ does not satisfy the parity constraint $\chi_{v}$ of a vertex $v\in V$. Note that $\textup{Search}_{T(G,c)}$ and $\textup{SearchVertex}(G,c)$ both give a reason why an assignment $a$ does not satisfy $T(G,c)$ but the latter is more coarse: $\textup{SearchVertex}(G,c)$ only gives a constraint that is violated while $\textup{Search}_{T(G,c)}$ gives an exact clause that is not satisfied.

We only introduce some minimal notions of proof complexity here; for more details and references the reader is referred to the recent survey [9]. Let $C_{1}=x\lor D_{1}$ and $C_{2}=\overline{x}\lor D_{2}$ be two clauses such that $D_{1},D_{2}$ contain neither $x$ nor $\overline{x}$. Then the clause $D_{1}\lor D_{2}$ is inferred by resolution of $C_{1}$ and $C_{2}$ on $x$. A resolution refutation of length $s$ of a CNF-formula $F$ is defined to be a sequence $C_{1},\ldots,C_{s}$ such that $C_{s}$ is the empty clause and for every $i\in[s]$ we have that $C_{i}$ is a clause of $F$ or it is inferred by resolution of two clauses $C_{j},C_{\ell}$ such that $j,\ell<i$. It is well-known that $F$ has a resolution refutation if and only if $F$ is unsatisfiable.

To every resolution refutation $C_{1},\ldots,C_{s}$ we assign a directed acyclic graph $G$ as follows: the vertices of $G$ are the clauses $\{C_{i}\mid i\in[s]\}$. Moreover, there is an edge $C_{j}C_{i}$ in $G$ if and only if $C_{i}$ is inferred by resolution of $C_{j}$ and some other clause $C_{\ell}$ on a variable $x$ in the refutation. We also label the edge $C_{j}C_{i}$ with the variable $x$. Note that there might be two pairs of clauses $C_{j},C_{\ell}$ and $C_{j^{\prime}},C_{\ell^{\prime}}$ such that resolution on both pairs leads to the same clause $C_{i}$. If this is the case, we simply choose one of them to make sure that all vertices in $G$ have indegree at most $2$. A resolution refutation is called regular if on every directed path in $G$ every variable $x$ appears at most once as a label of an edge. It is known that there is a resolution refutation of $F$ if and only if a regular resolution refutation of $F$ exists [13], but the latter are in general longer [1, 25].

In this paper, we will not directly deal with regular resolution proofs thanks to the following well-known result.

Since in our setting, from an unsatisfied clause we can directly inferred an unsatisfied parity constraint, we can use the following simple consequence.