Refutation via logical resolution is one of the most basic and fundamental methods in theorem proving used to argue the validity of statements in propositional logic. It is famously sound and complete for proving that formulas in conjunctive normal form (CNF) are unsatisfiable. In automated theorem proving, resolution is in particular used for various primitive backtracking algorithms for the satisfiability problem (SAT) such as the DPLL algorithm.
However, resolution is primitive in that we know simple unsatisfiable CNF formulas that admit only resolution refutations of superpolynomial length. This was first proven by Haken  who showed that a canonical encoding of the pigeonhole principle into a CNF formula provides formulas whose shortest refutations are superpolynomial in length. Other examples and exponential bounds were given by Chvátal and Szemerédi  as well as Urquhart who used formulas based on Tseitin tautologies . Investigating the resolution complexity of the graph non-isomorphism problem, Torán  constructed CNF formulas from so-called CFI-graphs (see ) and showed the shortest resolution proofs of the arising formulas have exponential length.
As observed by Krishnamurthy, many simple examples without short resolution refutations exhibit symmetries. This prompted the introduction of Krishnamurthy’s symmetry rule  which intuitively allows the deduction of a clause symmetric to a previously deduced clause in one step (formal definitions are given in Section Section 2). For various formulas, Krishnamurthy argued polynomial bounds when the symmetry-rule is used, leading to exponential improvements. Further examples with this effect, including another analysis for pigeonhole principle formulas, were provided by Urquhart .
Krishnamurthy in fact introduced two rules, each of them arises from permutations of the variables. The global rule allows only symmetries of the entire original formula, while the local one allows us to use symmetries of a subset of the clauses. These rules led to the proof systems SR-I (symmetric resolution) and SR-II (locally symmetric resolution), respectively. Urquhart  introduced complementation symmetries in addition to the variable permutations. This allows us to interchange literals with their negations and leads to the proof systems SRC-I and SRC-II. In  Urquhart also showed that there are exponential-to-polynomial improvements regarding proof length from the system SR-I to SRC-I. Arai and Urquhart  showed exponential-to-polynomial improvements from SR-I to SR-II and also provided exponential lower bounds for SRC-II.
Szeider , who actually focuses on homomorphisms, describes another strengthening of the symmetry rule. In his extension we are allowed the use of symmetries within clauses that have been resolved, rather than only allowing clauses of the original formula. This is called resolution with dynamic symmetries and leads to the proof systems SR-III and SRC-III, depending on whether complementation is allowed. However, to date it remains an open problem to find superpolynomial lower bounds on proof length in SR-III and SRC-III.
In this paper we are concerned with proof systems obtained by extending resolution with additional symmetry rules. We prove that the CNF formulas arising from the CFI-graphs have refutations polynomially bounded in length in the SR-I calculus. With Torán’s exponential lower bounds  mentioned above, this gives an exponential-to-polynomial improvement for the resolution complexity of non-isomorphism when introducing the symmetry rule. To those familiar with the details of the CFI-construction this may not come as a surprise, since the CFI-graphs exhibit many global symmetries. However, this is not the case for multipede graphs, these arise from a construction related to the CFI-graphs . Crucially these graphs are asymmetric. That is, they have no symmetries at all. They provide exponential lower bounds for all individualization-refinement algorithms for the graph isomorphism problem. This includes all tools currently viable in practice, such as nauty/traces . The initial intuition might therefore be that the CNF formulas arising from multipedes provide exponential lower bounds for SRC-III. However, this turns out not to be the case. In fact, maybe surprisingly, we show that even when using only local symmetries rather than dynamic symmetries (i.e., in SCR-II rather than SCR-III) there are polynomial bounds on the respective formulas. In some sense this shows that the multipedes have substructures with symmetries that allow them to be distinguished concisely.
To prove this statement, we reduce the statement to one concerning linear equation systems. It is known that isomorphism of CFI and multipede graphs are related to solvability of linear equation systems. (This is also the case for Tseitin tautologies.) We show that this relation can be exploited. Specifically, we show that there is a resolution transforming the CNFs arising from the graph isomorphism instances to CNFs arising from linear equation systems. We then show our main theorem which says that inconsistent linear equation systems with equations of bounded width (i.e., the maximum number of non-zero coefficients in an equation is bounded) have polynomial resolutions using the local symmetry rule.
Inconsistent linear equation systems of bounded width over a fixed finite field with a prime have, in their standard encoding as CNFs, polynomial length resolutions when using the local symmetry rule (i.e., in SRC-II).
Structure of the paper.
Section 3 shows that the CNF formulas arising from CFI-graph pairs have polynomial length proofs in SR-I. Section 4 shows that linear equation systems of bounded width have polynomial length proofs in SRC-II. Section 5 shows that the formulas arising from (bounded degree) multipede graphs can be transformed in the resolution calculus (without using symmetry) to linear equation systems of bounded width.
1.2 Related Work
Figure 1 gives an overview of resolution calculi with symmetry and references to lower bound constructions. A proof system p-simulates another proof system if shortest proofs in the latter are polynomially bounded in the length of shortest proofs in the former. We should remark that the extended resolution system introduced by Tseitin  can p-simulate proof systems with symmetries . See  for an implementation using Krishnamurthy’s symmetry rule. Symmetry rules have of course also been introduced for other proof systems [3, 8]. See also  for another way to incorporate symmetries into resolution.
|without complementation||classical resol.[5, 18]||SR-I ||SR-II ||SR-III (open)|
|with complementation||–||SRC-I ||SRC-II ||SRC-III (open)|
Connection to the graph isomorphism problem. The results of our paper are connected to the graph isomorphism problem in two conceptually very different ways. First, finding valid literal permutations (with or without complementation) for the global symmetry rule is equivalent to the graph isomorphism problem itself (e.g., ). Therefore isomorphism solvers such as nauty/traces , which are highly efficient in practice, can be used to find the symmetries (see ). Symmetry detection is one of the standard applications of graph isomorphism solvers, for example there is a tool integrating nauty into Prolog  for this purpose.
Second, our results relate to the proof complexity of the graph isomorphism problem itself, which explains why we are interested in CNF formulas arising from non-isomorphism instances. Torán  describes a canonical way to encode the isomorphism problem as a CNF formula (see Subsection 2.2). The resolution complexity of graph non-isomorphism is related to the complexity of the graph isomorphism problem. After all isomorphism solvers need to prove, some way or another, that the inputs are non-isomorphic, if they are. A crucial feature of isomorphism solvers is that they are able to exploit already detected symmetries (i.e., automorphisms) of the underlying instances during run-time . Vaguely, this translates into a symmetry rule that they apply already during the process of computing the symmetries of the instance. Current tools basically only exploit local symmetries. Our new insights into the resolution complexity of multipedes thus shows a combinatorial possibility to solve their isomorphism problem. It brings up the question how to exploit local symmetries in graph isomorphism solvers.
It remains unknown whether graph non-isomorphism has polynomial resolution complexity in any of the proof systems with symmetry rule we have discussed.
2.1 Resolution and the Symmetry Rule
We are interested in unsatisfiability proofs of Boolean formulas. The basic resolution proof system works with formulas in conjunctive normal form.
Let be a finite set of variables. is the set of literals, where . A clause is a disjunction of literals. We also represent clauses as sets of literals. A Boolean formula is in conjunctive normal form (CNF) if it is a conjunction of clauses. We may treat such a formula as a set of clauses. is the empty clause, i.e. the disjunction of the empty set, which is unsatisfiable. For sets of clauses and define . Since we will treat clauses as sets of literals, we do not care for their order, i.e. we do not differentiate between and . The same applies to CNF formulas, which we interpret as sets of clauses.
Resolution is a proof system in propositional logic. It operates on CNF formulas, employing a single inference rule:
The clause produced by the resolution rule is called resolvent.
Let and be sets of clauses. We write if there exists a sequence of clauses such that every is a resolvent of two earlier clauses and . Such a sequence is called derivation of from . When the length of the sequence is irrelevant, we write , meaning for some . Given a clause , we also write for .
For a CNF formula with , we say has a resolution refutation of size .
We write if there exists a set of clauses such that and . This is a weaker requirement than .
Resolution is sound and complete, i.e. if and only if is unsatisfiable. We examine the proof complexity of formulas in this proof system, i.e., the length of the shortest possible resolution refutation of a given formula, in relation to the formula size. There exist classes of formulas with exponential lower bounds on the resolution proof complexity [5, 16, 18].
In the following we define the symmetry rule, which is an extension to resolution, aiming to reduce the proof complexity of some of these hard formulas.
Let be a finite set of literals. A bijection is called renaming if for every we have .
A renaming is essentially a permutation of the variables that may also negate some of them. We can apply renamings to clauses (i.e., sets of literals) and CNF formulas (i.e., sets of clauses). In either case we define .
Consider a derivation from a formula and a subsequence of which derives a clause from a subset . If there exists a renaming with , then the local symmetry rule allows derivation of .
With the restriction , we obtain the global symmetry rule. Adding the global or local symmetry rule to the resolution system yields the proof systems SRC-I and SRC-II, respectively.
We write to indicate that can be derived from using resolution and the local symmetry rule, with a derivation of length at most .
Note that in order to apply via the local symmetry rule to some clause in a derivation, we must look at the entire history of how was derived, and find out which part of the original formula was used. Then we need to check that .
This means that in general we cannot chain derivations that use the symmetry rule together, because such an operation changes the history for some of the clauses. Still, we can combine SRC-II derivations in the following ways:
Let and be sets of clauses and .
Let and be sets of clauses and a clause.
2.2 Encoding Graph Isomorphism
Our interest in the graph isomorphism problem is twofold: First, finding valid literal permutations for the symmetry rule is equivalent to finding certain graph isomorphisms. Secondly, we examine the proof complexity of the problem by translating it into propositional logic and applying resolution with symmetry rule.
A graph is a tuple of a set of vertices and edges . Each edge is a two element subset of . A colored graph is a graph together with a function , called coloring, assigning to every vertex a color from some set . Let be a graph and . are the edges incident with . is the neighborhood of . is the degree of .
Given a colored graph with coloring and a vertex , we can individualize by creating a new coloring such that and setting for all . We write the individualized graph as .
Let and be graphs.
A graph isomorphism from to is a bijection such that for all we have if and only if .
We say and are isomorphic, written , if there exists a graph isomorphism from to .
An automorphism of a graph is a graph isomorphism from to itself.
is the automorphism group of G.
The automorphisms of a graph constitute its inherent combinatorial symmetries. We will use the terms automorphism and symmetry synonymously.
Given two graphs and , one can construct a Boolean formula that is satisfiable if and only if there is an isomorphism between and . This is commonly done by constraining variables of the form such that each satisfying assignment corresponds to an isomorphism: is assigned to if and only if the isomorphism maps to .
For a pair of graphs with , define
We refer to the clauses of this CNF formula as being “of Type i”, depending on which they come from. The clause types naturally encode the concept of a graph isomorphism in propositional logic. Specifically, Type 1 and Type 2 clauses ensure that we have a bijection from to ; Type 3 clauses make the function preserve edges.
If the graphs and are colored by some functions and respectively, then an isomorphism between them should respect the colors. To represent this in the formula , we simply assign to all variables for which .
2.3 The CFI Graphs
In this section, we look at the graphs by Cai, Fürer and Immerman , which were constructed to prove lower bounds for the Weisfeiler-Lehman method in isomorphism testing. These graphs are also challenging when we use resolution to decide isomorphism. They are built from gadget graphs which are defined as follows (see Figure 2).
Definition 9 (CFI-gadget [4, 6]).
Given a finite set , define: , where consists of , as well as and . Also define the coloring
The most important feature of the CFI-gadgets are their automorphisms:
[4, 6.1] There are automorphisms of . Each is uniquely determined by interchanging the vertices and for all in some subset of even cardinality.
Definition 11 (CFI graph).
From a graph construct by connecting the CFI gadgets with edges .
Given a graph with , construct from by choosing some edge and replacing the edges with the edges We say that the edges corresponding to have been twisted.
Note that 12 does not specify how to choose the edge which is to be twisted, so there are in fact multiple graphs that we could call . If is connected however, these graphs are isomorphic. On the other hand, for any graph with at least one edge, and are not isomorphic [[4, see Lemma 6.2]]. The CFI graphs have been used to prove the following lower bound for resolution:
Theorem 13 ([16, Corollary 5.2]).
There exists a family of graphs such that for every , has vertices and the resolution refutation of the formula requires size . The graphs and have color multiplicity at most 4.
This exponential lower bound motivates the use of a more efficient proof system to prove non-isomorphism of CFI graphs. Because of the symmetric nature of the CFI-gadgets, the symmetry rule is expected to reduce the proof length significantly. With symmetry rule, short proofs exist, as we show in Section 3.
In order to obtain examples for which the symmetry rule is not able to produce short proofs, it is a natural idea to consider asymmetric graphs instead.
2.4 Multipede Graphs
In , the so-called Multipedes were defined - a method to construct asymmetric structures. Combining this construction with CFI-gadgets, one obtains a family of asymmetric graphs which provide exponential lower bounds for individualization-refinement algorithms .
Definition 14 (Multipede graph ).
From a bipartite graph , we construct the Multipede graph as follows: For every create a pair of vertices , colored with . We call these pairs feet. Then for every take a CFI-gadget and identify the vertices and with and respectively.
Theorem 15 ().
There exists a family of bipartite graphs such that for each the graph has vertices, is asymmetric and individualization-refinement algorithms take steps to verify .
The Multipede graphs are of particular interest, because they are a generalization of the CFI graphs, and thus also hard for resolution, and additionally they can be constructed to be asymmetric. Hence the global symmetry rule is insufficient to get short proofs concerning Multipedes. However, as we will prove, the local symmetry of the CFI-gadgets can be used by the local symmetry rule.
The automorphism group of a Multipede graph is closely related to the solution set of a linear equation system. As a consequence of 10, any automorphism of a Multipede can be uniquely specified by the set of feet for which the --pairs are swapped. The set represents a valid automorphism exactly if for every CFI-gadget in the graph, an even number of incident feet is swapped.
Using linear algebra, we can encode a subset
uniquely as a vector, by setting if and only if for all . Then the evenness-condition, which the CFI-gadgets require, can be expressed as a set of linear equations:
We can write the equations in matrix form: Let be a bipartite graph. Define as follows:
The solutions of the linear equation system correspond to the automorphisms of . We will show how to apply resolution and the symmetry rule to linear equations, and extend our results to Multipedes.
2.5 Encoding Linear Equations
Linear equations over finite fields have been used to show lower bounds in Proof Complexity. For example, the Tseitin formulas are constructed from graphs, representing a system of linear equations over , and are hard for resolution . In the next section we will show that by adding the symmetry rule to resolution, we get short proofs for linear equations.
To work with linear equations, some basic definitions and notations from linear algebra are needed. Let be a field and . We write for the set of all by matrices over . Symbols for matrices will be written in boldface. Given a matrix and numbers and , we write for the element at the -th row and -th column of . We write for the set of all -element vectors. For our purposes they can be treated like single-column matrices, i.e., .
We write for a vector consisting of zeros, where its size is clear from context. Similarly is a vector filled with ones.
Applying resolution to a linear equation system means providing a refutation certifying that the system cannot be solved, if that is indeed the case. The following lemma is essential in proving an equation system unsolvable:
Let and . If the equation system does not have a solution , then there exists some such that and .
Since the equation system does not have a solution, applying the Gaussian elimination algorithm yields the equation . Writing the row operations used by the algorithm as a vector, we get the sought-after . ∎
We require some notation for standard operations from linear algebra.
Let and .
is the support of .
is the i-th row of .
is the diagonal matrix with diagonal entries equal to .
is the row sum of .
Then is the restriction of to the support of ,
defined by for .
To encode linear equations as CNF formulas, we first introduce variables which correspond to the solution vector of the linear equation system: . For a given vector , the corresponding assignment to the variables would set to true if and only if .
Our CNF formula has a clause for every with , ensuring the forumla is false under the assignment corresponding to . For every row of the equation system, we consider all with . We can restrict to the components for which is nonzero.
The formula for the row is then defined as follows:
We extend this definition to whole systems of equations: . Notice that assigning false to every variable satisfies , but this assignment does not represent a vector. For this reason, we additionally need the clauses .
Let and . There exists an with if and only if is satisfiable.
: Assume . Define an assignment such that if and only if . It is easy to see that . Let and . Then . Hence there exists such that . Therefore , so . Then and thus is satisfied by .
: Assume that we have an assignment with and . For all there exists a such that . Define . Towards a contradiction, assume there exists with . Then and . Hence there must exist an such that , which contradicts our construction of . Therefore . ∎
3 Linear-sized Refutations for Non-Isomorphism of CFI graphs
Due to the symmetric nature of the CFI graphs, using the symmetry rule gives us linear-sized resolution proofs of non-isomorphism for a pair of these graphs.
Let be a graph with at least one edge. Then
Notation: When writing clauses of the isomorphism formula for the CFI pair, we give different names to the variables , depending on the kind of vertices, to increase readability. This notation is borrowed from  and is as follows:
For middle vertices we define:
For a/b vertices we define:
Definition: For any Graph , we define . This measure will help to describe the size of the formula .
For and define and .
Let be a graph with . We assume that is connected without loss of generality, otherwise we can apply this proof to the connected component with the twist. We will show, for some constant :
and then .
Proceed by induction over the number of edges.
Induction basis: : The graphs are shown in Figure 3.
Then admits the following resolution refutation:
(Notation: “Type n: C” means that clause C is contained in . “ C” signifies that C is resolved from preceding clauses.)
Here we took steps to derive . We want the following to hold:
An appropriate will be determined later.
Case 1: is a tree.
Pick a vertex of with exactly one incident edge , where the corresponding edges in are not twisted. This is possible since is a tree by assumption, so it has at least two vertices of degree 1. Note that and are locally identical in this case. The situation is shown in Figure 4. Define . We have . Then resolve the following clauses:
This resolution takes steps. We obtain the Type 1 clauses of . The Type 2 and 3 clauses of are already present in .
By induction, can be resolved to in steps. It holds:
We calculate the total number of resolution steps for this case:
Then our bound for must hold:
We can satisfy this by requiring
Case 2: has a cycle.
We choose a simple cycle, i.e. one which does not repeating vertices. Pick an arbitrary edge along the cycle, such that the corresponding edge in is not twisted (see Figure 5), and resolve the clauses: