Resolution with Symmetry Rule applied to Linear Equations
aa r X i v : . [ c s . CC ] J a n Resolution with Symmetry Rule applied to Linear Equations ∗ Pascal Schweitzer Constantin SeebachTU Kaiserslautern
Abstract
This paper considers the length of resolution proofs when using Krishnamurthy’s classic symmetryrules. We show that inconsistent linear equation systems of bounded width over a fixed finite field F p with p a prime have, in their standard encoding as CNFs, polynomial length resolutions when usingthe local symmetry rule (SRC-II).As a consequence it follows that the multipede instances for the graph isomorphism problemencoded as CNF formula have polynomial length resolution proofs. This contrasts exponential lowerbounds for individualization-refinement algorithms on these graphs.For the Cai-F¨urer-Immerman graphs, for which Tor´an showed exponential lower bounds for res-olution proofs (SAT 2013), we also show that already the global symmetry rule (SRC-I) suffices toallow for polynomial length proofs. ∗ The research leading to these results has received funding from the European Research Council (ERC) under theEuropean Union’s Horizon 2020 research and innovation programme (EngageS: grant agreement No. 820148).
Introduction
Refutation via logical resolution is one of the most basic and fundamental methods in theorem provingused to argue the validity of statements in propositional logic. It is famously sound and completefor proving that formulas in conjunctive normal form (CNF) are unsatisfiable. In automated theoremproving, resolution is in particular used for various primitive backtracking algorithms for the satisfiabilityproblem (SAT) such as the DPLL algorithm.However, resolution is primitive in that we know simple unsatisfiable CNF formulas that admit onlyresolution refutations of superpolynomial length. This was first proven by Haken [11] who showed thata canonical encoding of the pigeonhole principle into a CNF formula provides formulas whose shortestrefutations are superpolynomial in length. Other examples and exponential bounds were given by Chv´ataland Szemer´edi [5] as well as Urquhart who used formulas based on Tseitin tautologies [18]. Investigatingthe resolution complexity of the graph non-isomorphism problem, Tor´an [16] constructed CNF formulasfrom so-called CFI-graphs (see [4]) and showed the shortest resolution proofs of the arising formulas haveexponential length.As observed by Krishnamurthy, many simple examples without short resolution refutations exhibitsymmetries. This prompted the introduction of Krishnamurthy’s symmetry rule [12] which intuitivelyallows the deduction of a clause symmetric to a previously deduced clause in one step (formal definitionsare given in Section Section 2). For various formulas, Krishnamurthy argued polynomial bounds whenthe symmetry-rule is used, leading to exponential improvements. Further examples with this effect,including another analysis for pigeonhole principle formulas, were provided by Urquhart [19].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 touse symmetries of a subset of the clauses. These rules led to the proof systems SR-I ( symmetric resolu-tion ) and SR-II ( locally symmetric resolution ), respectively. Urquhart [19] introduced complementationsymmetries in addition to the variable permutations. This allows us to interchange literals with theirnegations and leads to the proof systems SRC-I and SRC-II. In [19] Urquhart also showed that there areexponential-to-polynomial improvements regarding proof length from the system SR-I to SRC-I. Araiand Urquhart [1] showed exponential-to-polynomial improvements from SR-I to SR-II and also providedexponential lower bounds for SRC-II.Szeider [15], who actually focuses on homomorphisms, describes another strengthening of the sym-metry 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 sym-metries and leads to the proof systems SR-III and SRC-III, depending on whether complementation isallowed. However, to date it remains an open problem to find superpolynomial lower bounds on prooflength in SR-III and SRC-III.
In this paper we are concerned with proof systems obtained by extending resolution with additional sym-metry rules. We prove that the CNF formulas arising from the CFI-graphs have refutations polynomiallybounded in length in the SR-I calculus. With Tor´an’s exponential lower bounds [16] mentioned above,this gives an exponential-to-polynomial improvement for the resolution complexity of non-isomorphismwhen introducing the symmetry rule. To those familiar with the details of the CFI-construction thismay not come as a surprise, since the CFI-graphs exhibit many global symmetries. However, this is notthe case for multipede graphs, these arise from a construction related to the CFI-graphs [10]. Cruciallythese graphs are asymmetric. That is, they have no symmetries at all. They provide exponential lowerbounds for all individualization-refinement algorithms for the graph isomorphism problem. This includesall tools currently viable in practice, such as nauty/traces [13]. The initial intuition might therefore bethat 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 localsymmetries rather than dynamic symmetries (i.e., in SCR-II rather than SCR-III) there are polynomialbounds on the respective formulas. In some sense this shows that the multipedes have substructureswith symmetries that allow them to be distinguished concisely.To prove this statement, we reduce the statement to one concerning linear equation systems. Itis known that isomorphism of CFI and multipede graphs are related to solvability of linear equation1one global local dynamicwithout complementation classical resol.[5, 18] SR-I [19] SR-II [1] SR-III (open)with complementation – SRC-I [19] SRC-II [1] SRC-III (open)Figure 1: Resolution calculi with symmetries rules of varying degree of generality and references withformulas proving exponential lower bounds on resolution length.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 isomorphisminstances to CNFs arising from linear equation systems. We then show our main theorem which saysthat inconsistent linear equation systems with equations of bounded width (i.e., the maximum numberof non-zero coefficients in an equation is bounded) have polynomial resolutions using the local symmetryrule.
Theorem 1.
Inconsistent linear equation systems of bounded width over a fixed finite field F p with p a prime have, in their standard encoding as CNFs, polynomial length resolutions when using the localsymmetry 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 proofsin SR-I. Section 4 shows that linear equation systems of bounded width have polynomial length proofsin SRC-II. Section 5 shows that the formulas arising from (bounded degree) multipede graphs can betransformed in the resolution calculus (without using symmetry) to linear equation systems of boundedwidth.
Figure 1 gives an overview of resolution calculi with symmetry and references to lower bound construc-tions. A proof system p-simulates another proof system if shortest proofs in the latter are polynomiallybounded in the length of shortest proofs in the former. We should remark that the extended resolutionsystem introduced by Tseitin [17] can p-simulate proof systems with symmetries [19]. See [2] for an im-plementation using Krishnamurthy’s symmetry rule. Symmetry rules have of course also been introducedfor other proof systems [3, 8]. See also [7] for another way to incorporate symmetries into resolution.
Connection to the graph isomorphism problem.
The results of our paper are connected tothe graph isomorphism problem in two conceptually very different ways. First, finding valid literalpermutations (with or without complementation) for the global symmetry rule is equivalent to the graphisomorphism problem itself (e.g., [4]). Therefore isomorphism solvers such as nauty/traces [13], whichare highly efficient in practice, can be used to find the symmetries (see [6]). Symmetry detection is oneof the standard applications of graph isomorphism solvers, for example there is a tool integrating nautyinto Prolog [9] for this purpose.Second, our results relate to the proof complexity of the graph isomorphism problem itself, whichexplains why we are interested in CNF formulas arising from non-isomorphism instances. Tor´an [16]describes a canonical way to encode the isomorphism problem as a CNF formula (see Subsection 2.2). Theresolution complexity of graph non-isomorphism is related to the complexity of the graph isomorphismproblem. 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 alreadydetected symmetries (i.e., automorphisms) of the underlying instances during run-time [13]. Vaguely, thistranslates into a symmetry rule that they apply already during the process of computing the symmetriesof the instance. Current tools basically only exploit local symmetries. Our new insights into the resolutioncomplexity of multipedes thus shows a combinatorial possibility to solve their isomorphism problem. Itbrings 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 ofthe proof systems with symmetry rule we have discussed.2
Preliminaries
We are interested in unsatisfiability proofs of Boolean formulas. The basic resolution proof system workswith formulas in conjunctive normal form.Let Γ be a finite set of variables. Lit(Γ) := Γ ∪ Γ is the set of literals, where Γ := { x | x ∈ Γ } . 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 setof clauses. ⊥ is the empty clause, i.e. the disjunction of the empty set, which is unsatisfiable. For setsof clauses C and C define C ⊑ C : ⇐⇒ ∀ c ∈ C ∃ c ∈ C : c ⊇ c . Since we will treat clauses assets of literals, we do not care for their order, i.e. we do not differentiate between x ∨ y and y ∨ x . Thesame applies to CNF formulas, which we interpret as sets of clauses. Definition 2.
Resolution is a proof system in propositional logic. It operates on CNF formulas, em-ploying a single inference rule: x ∨ A, x ∨ BA ∨ B .
The clause produced by the resolution rule is called resolvent .Let A = { a , . . . , a m } and B be sets of clauses. We write A ⊢ n B if there exists a sequence of clauses a , . . . , a m , c , . . . , c n such that every c i is a resolvent of two earlier clauses and B ⊆ A ∪ { c , . . . , c n } .Such a sequence is called derivation of B from A . When the length of the sequence is irrelevant, we write A ⊢ B , meaning A ⊢ n B for some n . Given a clause b , we also write A ⊢ n b for A ⊢ n { b } .For a CNF formula F with F ⊢ n ⊥ , we say F has a resolution refutation of size n .We write A ⊢ wn B if there exists a set of clauses B ′ such that B ⊑ B ′ and A ⊢ n B ′ . This is a weakerrequirement than A ⊢ n B . Resolution is sound and complete, i.e. F ⊢ n ⊥ if and only if F is unsatisfiable. We examine the proofcomplexity of formulas in this proof system, i.e., the length of the shortest possible resolution refutationof a given formula, in relation to the formula size. There exist classes of formulas with exponential lowerbounds on the resolution proof complexity [5, 16, 18].In the following we define the symmetry rule, which is an extension to resolution, aiming to reducethe proof complexity of some of these hard formulas. Definition 3.
Let L be a finite set of literals. A bijection σ : L → L is called renaming if for every ℓ ∈ L we have σ ( ℓ ) = σ ( ℓ ) . A renaming is essentially a permutation of the variables that may also negate some of them. We canapply renamings to clauses (i.e., sets of literals) and CNF formulas (i.e., sets of clauses). In either casewe define σ ( C ) := { σ ( x ) | x ∈ C } . Definition 4 (The Symmetry Rule [1] [15]) . Consider a derivation S from a formula F and a subsequence S ′ of S which derives a clause C from a subset F ′ ⊆ F . If there exists a renaming σ with σ ( F ′ ) ⊆ F ,then the local symmetry rule allows derivation of σ ( C ) .With the restriction F = F ′ , we obtain the global symmetry rule . Adding the global or local symmetryrule to the resolution system yields the proof systems SRC-I and
SRC-II , respectively.
We write A ⊢ SRC-II n B to indicate that B can be derived from A using resolution and the localsymmetry rule, with a derivation of length at most n .Note that in order to apply σ via the local symmetry rule to some clause C in a derivation, we mustlook at the entire history of how C was derived, and find out which part F ′ ⊆ F of the original formulawas used. Then we need to check that σ ( F ′ ) ⊆ F .This means that in general we cannot chain derivations that use the symmetry rule together, becausesuch an operation changes the history for some of the clauses. Still, we can combine SRC-II derivationsin the following ways:
Lemma 5.
Let
A, B, C and D be sets of clauses and n, m ∈ N .(a) A ⊢ SRC-II n B and A ⊆ C implies C ⊢ SRC-II n B b) A ⊢ SRC-II n B and B ⊢ m C implies A ⊢ SRC-II n + m C (c) A ⊢ SRC-II n B and C ⊢ SRC-II m D implies A ∪ C ⊢ SRC-II n + m B ∪ D Lemma 6.
Let A and B be sets of clauses and d a clause.(a) A ⊢ ⊥ and { c ∨ d | c ∈ A } ⊑ B implies B ⊢ w d Our interest in the graph isomorphism problem is twofold: First, finding valid literal permutationsfor the symmetry rule is equivalent to finding certain graph isomorphisms. Secondly, we examine theproof complexity of the problem by translating it into propositional logic and applying resolution withsymmetry rule.A graph is a tuple (
V, E ) of a set of vertices V and edges E . Each edge is a two element subset of V .A colored graph is a graph ( V, E ) together with a function f : V → C , called coloring , assigning to everyvertex a color from some set C . Let G = ( V, E ) be a graph and v ∈ V . E G ( v ) := { e ∈ E | v ∈ e } are theedges incident with v . N G ( v ) := { u ∈ V | { u, v } ∈ E } is the neighborhood of v . deg G ( v ) := | N G ( v ) | = |E G ( v ) | is the degree of v .Given a colored graph G = ( V, E ) with coloring f and a vertex v ∈ V , we can individualize v bycreating a new coloring f ′ such that f ′ ( v ) := ( f ( v ) ,
1) and setting f ′ ( v ′ ) := ( f ( v ′ ) ,
0) for all v ′ ∈ V \ { v } .We write the individualized graph as G v . Definition 7.
Let G = ( V , E ) and G = ( V , E ) be graphs. • A graph isomorphism from G to G is a bijection ϕ : V → V such that for all v, v ′ ∈ V we have { v, v ′ } ∈ E if and only if { ϕ ( v ) , ϕ ( v ′ ) } ∈ E . • We say G and G are isomorphic , written G ∼ = G , if there exists a graph isomorphism from G to G . • An automorphism of a graph G is a graph isomorphism from G to itself. • Aut( G ) is the automorphism group of G. The automorphisms of a graph constitute its inherent combinatorial symmetries. We will use theterms automorphism and symmetry synonymously.Given two graphs G and G , one can construct a Boolean formula that is satisfiable if and only ifthere is an isomorphism between G and G [16]. This is commonly done by constraining variables ofthe form x u,v such that each satisfying assignment corresponds to an isomorphism: 1 is assigned to x u,v if and only if the isomorphism maps u to v . Definition 8.
For a pair of graphs G = ( V , E ) , G = ( V , E ) with | V | = | V | , define F ( G , G ) := T ∧ T ∧ T , where T := ^ v ∈ V _ v ∈ V x v ,v ,T := ^ v ∈ V ^ v ,v ′ ∈ V v = v ′ (cid:0) x v ,v ∨ x v ′ ,v (cid:1) ,T := ^ { u ,v }∈ E ⊕{ u ,v }∈ E ( x u ,u ∨ x v ,v ) . We refer to the clauses of this CNF formula as being “of Type i” , depending on which T i they comefrom. 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 V to V ; Type 3 clausesmake the function preserve edges.If the graphs G and G are colored by some functions l and l respectively, then an isomorphismbetween them should respect the colors. To represent this in the formula F ( G , G ), we simply assign 0to all variables x u,v for which l ( u ) = l ( v ). 4 ∅ m { , } m { , } m { , } a b a b a b Figure 2: The CFI-gadget X { , , } . In this section, we look at the graphs by Cai, F¨urer and Immerman [4], which were constructed toprove lower bounds for the Weisfeiler-Lehman method in isomorphism testing. These graphs are alsochallenging when we use resolution to decide isomorphism. They are built from gadget graphs which aredefined as follows (see Figure 2).
Definition 9 (CFI-gadget [4, 6]) . Given a finite set N , define: X N := ( V, E, γ ) , where V := A ∪ B ∪ M consists of A := { a w | w ∈ N } , B := { b w | w ∈ N } as well as M := { m S | S ⊆ N, | S | even } and E := {{ m S , a w } | w ∈ S } ∪ {{ m S , b w } | w ∈ N \ S } . Also define the coloring γ : V → C : v γ ( v ) := ( c w if v ∈ A ∪ B with v = a w or v = b w ,m if v ∈ M. The most important feature of the CFI-gadgets are their automorphisms:
Lemma 10. [4, 6.1] There are | N |− automorphisms of X N . Each is uniquely determined by inter-changing the vertices a w and b w for all w in some subset S ⊆ N of even cardinality. Definition 11 (CFI graph) . From a graph G = ( V, E ) construct X ( G ) by connecting the CFI gadgets { X v E G ( v ) | v ∈ V } with edges E ′ := {{ a ue , a ve } | e = { u, v } ∈ E } ∪ {{ b ue , b ve } | e = { u, v } ∈ E } . Definition 12.
Given a graph G = ( V, E ) with E = ∅ , construct ˜ X ( G ) from X ( G ) by choosing someedge e = { u, v } ∈ E and replacing the edges { a ue , a ve } , { b ue , b ve } with the edges { a ue , b ve } , { b ue , a ve } . We saythat the edges corresponding to e have been twisted. Note that Theorem 12 does not specify how to choose the edge which is to be twisted, so there are infact multiple graphs that we could call ˜ X ( G ). If G is connected however, these graphs are isomorphic.On the other hand, for any graph G with at least one edge, ˜ X ( G ) and X ( G ) are not isomorphic [[4, seeLemma 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 G = ( G n ) n ∈ N such that for every n , G n has n vertices and the resolution refutation of the formula F ( X ( G n ) , ˜ X ( G n )) requires size exp(Ω( n )) .The graphs X ( G n ) and ˜ X ( G n ) 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 isexpected to reduce the proof length significantly. With symmetry rule, short proofs exist, as we show inSection 3.In order to obtain examples for which the symmetry rule is not able to produce short proofs, it is anatural idea to consider asymmetric graphs instead.
In [10], the so-called Multipedes were defined - a method to construct asymmetric structures. Combiningthis construction with CFI-gadgets, one obtains a family of asymmetric graphs which provide exponentiallower bounds for individualization-refinement algorithms [14].5 efinition 14 (Multipede graph [14]) . From a bipartite graph G = ( V, W, E ) , we construct the Multipedegraph
M P ( G ) as follows: For every w ∈ W create a pair of vertices a w , b w , colored with c w . We callthese pairs feet . Then for every v ∈ V take a CFI-gadget X vN G ( v ) and identify the vertices a vw and b vw with a w and b w respectively. Theorem 15 ([14]) . There exists a family of bipartite graphs G = ( G n ) n ∈ N such that for each n thegraph G n has O ( n ) vertices, M P ( G n ) is asymmetric and individualization-refinement algorithms take exp(Ω( n )) steps to verify M P ( G n ) a ω ≇ M P ( G n ) b ω . 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. Hencethe global symmetry rule is insufficient to get short proofs concerning Multipedes. However, as we willprove, 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 equationsystem. As a consequence of Theorem 10, any automorphism ϕ of a Multipede can be uniquely specifiedby the set of feet Y := { w ∈ W | ϕ ( a w ) = b w } for which the a - b -pairs are swapped. The set Y representsa valid automorphism exactly if for every CFI-gadget in the graph, an even number of incident feet isswapped.Using linear algebra, we can encode a subset Y ⊆ W = { w , . . . , w n } uniquely as a vector y ∈ F n ,by setting y i = 1 if and only if w i ∈ Y for all i . Then the evenness-condition, which the CFI-gadgetsrequire, can be expressed as a set of linear equations:for all v ∈ V : X w i ∈ N G ( v ) y i = 0 . We can write the equations in matrix form: Let G = ( { v , . . . , v m } , { w , . . . , w n } , E ) be a bipartitegraph. Define M ( G ) ∈ F m × n as follows: M ( G ) i,j := ( { v i , w j } ∈ E M ( G ) y = correspond to the automorphisms of M P ( G ).We will show how to apply resolution and the symmetry rule to linear equations, and extend our resultsto Multipedes. 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 F , andare hard for resolution [18]. In the next section we will show that by adding the symmetry rule toresolution, we get short proofs for linear equations.To work with linear equations, some basic definitions and notations from linear algebra are needed.Let K be a field and n, m ∈ N . We write K m × n for the set of all m by n matrices over K . Symbols formatrices will be written in boldface. Given a matrix A ∈ K m × n and numbers i ∈ [1 , m ] and j ∈ [1 , n ],we write A i,j for the element at the i -th row and j -th column of A . We write K n for the set of all n -element vectors. For our purposes they can be treated like single-column matrices, i.e., K n = K n × .We write for a vector consisting of zeros, where its size is clear from context. Similarly is a vectorfilled with ones.Applying resolution to a linear equation system means providing a refutation certifying that thesystem cannot be solved, if that is indeed the case. The following lemma is essential in proving anequation system unsolvable: Lemma 16.
Let A ∈ F m × np and b ∈ F mp . If the equation system Ax = b does not have a solution x ∈ F np , then there exists some v ∈ F mp such that vA = and v · b = 1 .Proof. Since the equation system does not have a solution, applying the Gaussian elimination algorithmyields the equation 0 = 1. Writing the row operations used by the algorithm as a vector, we get thesought-after v .We require some notation for standard operations from linear algebra. Let r = ( r , . . . , r n ) ∈ K n and A ∈ K m × n . supp( r ) := { i ∈ { , . . . , n } | r i = 0 } is the support of r . A i, ∗ := ( A i, , . . . , A i,n ) ∈ K n
6s the i-th row of A . diag( r ) ∈ K n × n is the diagonal matrix with diagonal entries equal to r . ΣA := P mi =1 A i, ∗ = · A is the row sum of A . Let v = ( v , . . . , v n ) ∈ K n . Then r | v ∈ K n is the restriction of r to the support of v ,defined by ( r | v ) i := ( r i if v i = 00 if v i = 0 for i ∈ { , . . . , n } .To encode linear equations as CNF formulas, we first introduce variables which correspond to thesolution vector of the linear equation system: Vars := { ξ i,k | i ∈ [1 , n ] and k ∈ F p } . For a given vector x ∈ F np , the corresponding assignment to the variables would set ξ i,k to true if and only if x i = k .Our CNF formula has a clause for every x with Ax = b , ensuring the forumla is false under theassignment corresponding to x . For every row ( a , b ) of the equation system, we consider all x with a · x = b . We can restrict x to the components for which a is nonzero. P ( a , b ) := { x ∈ F np | a · x = b and supp( x ) ⊆ supp( a ) } . The formula for the row ( a , b ) is then defined as follows: F ( a , b ) := ^ x ∈ P ( a ,b ) C a ( x ) , where C a ( x ) := _ i ∈ supp( a ) ξ i, x i . We extend this definition to whole systems of equations: F ( A , b ) := V mi =1 F ( A i, ∗ , b i ). Notice thatassigning false to every variable satisfies F ( A , b ), but this assignment does not represent a vector. Forthis reason, we additionally need the clauses V := V ni =1 W k ∈ F p ξ i,k . Lemma 17.
Let A ∈ F m × np and b ∈ F mp . There exists an x ∈ F np with Ax = b if and only if F ( A , b ) ∧ V is satisfiable.Proof. = ⇒ : Assume Ax = b . Define an assignment ϕ : Vars → B such that ϕ ( ξ i,k ) = 1 if and only if x i = k . It is easy to see that ϕ ( V ) = 1. Let j ∈ [1 , m ] and x ′ ∈ P ( A j, ∗ , b j ). Then A j, ∗ · x ′ = b j = A j, ∗ · x .Hence there exists i ∈ supp( A j, ∗ ) such that x i = x ′ i . Therefore ϕ ( ξ i, x ′ i ) = 0, so ϕ ( C A j, ∗ ( x ′ )) = 1. Then ϕ ( F ( A j, ∗ , b j )) = 1 and thus F ( A , b ) is satisfied by ϕ . ⇐ = : Assume that we have an assignment ϕ with ϕ ( F ( A , b )) = 1 and ϕ ( V ) = 1. For all i ∈ [1 , n ]there exists a k ∈ F p such that ϕ ( ξ i,k ) = 1. Define x i := k . Towards a contradiction, assume there exists j ∈ [1 , m ] with b j = A j, ∗ · x . Then x | A j, ∗ ∈ P ( A j, ∗ , b j ) and ϕ ( C A j, ∗ ( x )) = 1. Hence there must exist an i ∈ supp( A j, ∗ ) such that ϕ ( ξ i, x i ) = 0, which contradicts our construction of x . Therefore Ax = b . Due to the symmetric nature of the CFI graphs, using the symmetry rule gives us linear-sized resolutionproofs of non-isomorphism for a pair of these graphs.
Theorem 18.
Let G be a graph with at least one edge. Then F ( X ( G ) , ˜ X ( G )) ⊢ SRC-I O ( | F ( X ( G ) , ˜ X ( G )) | ) ⊥ . Proof.
Notation: When writing clauses of the isomorphism formula F ( X ( G ) , ˜ X ( G )) for the CFI pair,we give different names to the variables x u,v , depending on the kind of vertices, to increase readability.This notation is borrowed from [16] and is as follows: • For middle vertices m uS , m uS ′ we define: z uS,S ′ := x m uS ,m uS ′ • For a/b vertices a ue , b ue we define: y a ue ,b ue := x a ue ,b ue Definition: For any Graph G = ( V, E ), we define Φ ( G ) := P v ∈ V deg G ( v ) . This measure will helpto describe the size of the formula F ( X ( G ) , ˜ X ( G )). For v ∈ V and e ∈ E define G − e := ( V, E \ { e } )and G − v := ( V \ { v } , { e ′ ∈ E | v / ∈ e ′ } ). 7et G = ( V, E ) be a graph with E = ∅ . We assume that G is connected without loss of gener-ality, otherwise we can apply this proof to the connected component with the twist. We will show, forsome constant α : F ( X ( G ) , ˜ X ( G )) ⊢ SRC-I α | E | + α Φ ( G ) ⊥ and then α | E | + α Φ ( G ) = O ( | F ( X ( G ) , ˜ X ( G )) | ). Proceed by induction over the number of edges.Induction basis: | E | = 1: The graphs are shown in Figure 3. Then F ( X ( G ) , ˜ X ( G )) admits the following m u ∅ a ue b ue a ve b ve m v ∅ m u ∅ a ue b ue a ve b ve m v ∅ Figure 3: X ( G ) and ˜ X ( G ) for the connected graph G containing exactly one edgeresolution refutation:(Notation: “Type n: C” means that clause C is contained in F ( X ( G ) , ˜ X ( G )). “ = ⇒ C” signifiesthat C is resolved from preceding clauses.)Type 1: z u ∅ , ∅ Type 3: z u ∅ , ∅ ∨ y b ue ,a ue = ⇒ y b ue ,a ue Type 1: y b ue ,a ue ∨ y b ue ,b ue = ⇒ y b ue ,b ue Type 3: y b ue ,b ue ∨ y a ve ,a ve = ⇒ y a ve ,a ve Type 1: y a ve ,a ve ∨ y a ve ,b ve = ⇒ y a ve ,b ve Type 3: z v ∅ , ∅ ∨ y a ve ,b ve = ⇒ z v ∅ , ∅ Type 1: z v ∅ , ∅ = ⇒ ⊥ Here we took 6 steps to derive ⊥ . We want the following to hold:6 ≤ α | E | + α Φ ( G )= α · α · (4 + 4 ) = 9 α An appropriate α will be determined later.Induction step:Case 1: G is a tree.Pick a vertex u of G with exactly one incident edge e = { u, v } , where the corresponding edges in˜ X ( G ) are not twisted. This is possible since G is a tree by assumption, so it has at least two verticesof degree 1. Note that X ( G ) and ˜ X ( G ) are locally identical in this case. The situation is shown inFigure 4. Define S v := { A ⊆ E G ( v ) | | A | even } . We have |S v | = 2 |E G ( v ) |− = 2 deg G ( v ) − . Then resolve8he following clauses: Type 1: z u ∅ , ∅ Type 3: z u ∅ , ∅ ∨ y a ue ,b ue = ⇒ y a ue ,b ue Type 1: y a ue ,a ue ∨ y a ue ,b ue = ⇒ y a ue ,a ue Type 3: y a ue ,a ue ∨ y a ve ,b ve = ⇒ y a ve ,b ve Type 1: y a ve ,a ve ∨ y a ve ,b ve = ⇒ y a ve ,a ve ∀ S, S ′ ∈ S v with e / ∈ S, e ∈ S ′ :Type 3: z vS,S ′ ∨ y a ve ,a ve ∀ S, S ′ ∈ S v with e / ∈ S, e ∈ S ′ : = ⇒ z vS,S ′ ∀ S ∈ S v , e / ∈ S :Type 1: _ S ′ ∈S v z vS,S ′ ≡ _ S ′ ∈S v ,e/ ∈ S ′ z vS,S ′ ∨ _ S ′ ∈S v ,e ∈ S ′ z vS,S ′ ∀ S ∈ S v , e / ∈ S : = ⇒ _ S ′ ∈S v ,e/ ∈ S ′ z vS,S ′ This resolution takes 4 + |S v | + |S v | steps. We obtain the Type 1 clauses of F ( X ( G − u ) , ˜ X ( G − u )).The Type 2 and 3 clauses of F ( X ( G − u ) , ˜ X ( G − u )) are already present in F ( X ( G ) , ˜ X ( G )).By induction, F ( X ( G − u ) , ˜ X ( G − u )) can be resolved to ⊥ in α | E \ { e }| + α Φ ( G − u ) steps. Itholds: Φ ( G − u ) = X µ ∈ V \{ u } deg G − u ( µ ) = 4 deg G − u ( v ) + X µ ∈ V \{ u,v } deg G − u ( µ ) = 4 deg G ( v ) − + X µ ∈ V \{ u,v } deg G ( µ ) = deg G ( v ) + Φ ( G ) − deg G ( u ) − deg G ( v ) = Φ ( G ) − − deg G ( v ) . We calculate the total number of resolution steps T for this case:= ⇒ T = 4 + |S v | + |S v | + α | E \ { e }| + α Φ ( G − u )= 4 + (2 deg G ( v ) − ) + α | E | − α + α Φ ( G − u )= 4 + deg G ( v ) + α | E | − α + α (Φ ( G ) − − deg G ( v ) )Then our bound for T must hold: T ≤ α | E | + α Φ ( G ) ⇐⇒ deg G ( v ) + α | E | − α + α (Φ ( G ) − − deg G ( v ) ) ≤ α | E | + α Φ ( G ) ⇐⇒ deg G ( v ) − α + α ( − − deg G ( v ) ) ≤ ⇐⇒ deg G ( v ) ≤ α + α deg G ( v ) We can satisfy this by requiring 4 ≤ α and ≤ α. u ∅ a ue b ue a ve b ve m vS Figure 4: Vertex of degree 1 in X ( G )Case 2: G has a cycle.We choose a simple cycle, i.e. one which does not repeating vertices. Pick an arbitrary edge e = { u, v } along the cycle, such that the corresponding edge in ˜ X ( G ) is not twisted (see Figure 5), and resolve theclauses: Type 3: y a ue ,a ue ∨ y a ve ,b ve Type 1: y a ve ,a ve ∨ y a ve ,b ve = ⇒ y a ue ,a ue ∨ y a ve ,a ve ∀ µ ∈ e : ∀ S, S ′ ∈ S µ , e / ∈ S, e ∈ S ′ :Type 3: z µS,S ′ ∨ y a µe ,a µe ∀ µ ∈ e : ∀ S, S ′ ∈ S µ , e / ∈ S, e ∈ S ′ : = ⇒ z µS,S ′ ∨ y a ue ,a ue ∀ µ ∈ e : ∀ S ∈ S µ , e / ∈ S :Type 1: _ S ′ ∈S µ z µS,S ′ ∀ µ ∈ e : ∀ S ∈ S µ , e / ∈ S : = ⇒ y a ue ,a ue ∨ _ S ′ ∈S µ ,e/ ∈ S ′ z µS,S ′ We obtain the Type 1 clauses of F ( X ( G − e ) , ˜ X ( G − e )), but with some extra literals y a ue ,a ue . Since e is part of a cycle, G − e is still connected. By induction, F ( X ( G − e ) , ˜ X ( G − e )) can be resolved to ⊥ ;consequently, F ( X ( G ) , ˜ X ( G )) can be resolved to y a ue ,a ue . Together, this takes1 + |S v | + ( |S u | + |S v | ) + α | E \ { e }| + α Φ ( G − e )resolution steps.By deriving y a ue ,a ue we showed that there is no isomorphism from X ( G ) to ˜ X ( G ) which maps a ue to a ue .Hence the only way an isomorphism could exist is by mapping b ue to a ue . But we will see that X ( G ) hasan automorphism swapping a ue with b ue , so we can apply the same reasoning to show that an isomorphismalso cannot map b ue to a ue .This is how we exploit the symmetries of the CFI graphs: Let C be the set of vertices of G that lie onthe cycle. Then for every x ∈ C , exactly two of its neighbors are in C . By Theorem 10, the CFI-gadgetcorresponding to x has an automorphism exchanging the two a - b -pairs related to these neighbors. Weassemble these automorphisms together for every x ∈ C to obtain an automorphism ϕ of X ( G ) thatswaps all a - b -pairs along the cycle.Then ϕ ( a ue ) = b ue . This automorphism of the graph induces a symmetry ψ on the formula F ( X ( G ) , ˜ X ( G )) with ψ ( y a ue ,a ue ) = y b ue ,a ue . Apply the global symmetry rule on the resolution of y a ue ,a ue to10 uS a ue b ue a ve b ve m vS Figure 5: Vertices of higher degree in X ( G )obtain y b ue ,a ue . Resolve: Previously: y a ue ,a ue Symmetry rule = ⇒ y b ue ,a ue Type 1: y a ue ,a ue ∨ y a ue ,b ue Type 2: y a ue ,b ue ∨ y b ue ,b ue = ⇒ y a ue ,a ue ∨ y b ue ,b ue Type 1: y b ue ,a ue ∨ y b ue ,b ue = ⇒ y b ue ,a ue ∨ y a ue ,a ue = ⇒ y b ue ,a ue = ⇒ ⊥ Then the total number of steps is T = 1 + |S v | + ( |S u | + |S v | ) + α | E \ { e }| + α Φ ( G − e ) + 5= 6 + deg G ( v ) + deg G ( u ) + α | E | − α + α Φ ( G − e ) , with Φ ( G − e ) = X µ ∈ V deg G − e ( µ ) = X µ ∈ V \{ u,v } deg G − e ( µ ) + 4 deg G − e ( u ) + 4 deg G − e ( v ) = X µ ∈ V \{ u,v } deg G ( µ ) + 4 deg G ( u ) − + 4 deg G ( v ) − = Φ ( G ) − deg G ( u ) − deg G ( v ) . Then T ≤ α | E | + α Φ ( G ) ⇐⇒ deg G ( v ) + deg G ( u ) − α + α ( − deg G ( u ) − deg G ( v ) ) ≤ ⇐⇒ deg G ( v ) + deg G ( u ) ≤ α + α ( deg G ( u ) + deg G ( v ) )This is satisfied by the constraint 6 ≤ α ∧ ≤ α ∧ ≤ α. Using α = 6 will suffice. Lastly, we need to show that α | E | + α Φ ( G ) is linearly bounded by the size ofthe formula F ( X ( G ) , ˜ X ( G )). We have: | E | = X v ∈ V deg G ( v ) ≤ X v ∈ V deg G ( v ) = O (Φ ( G ))Now count the clauses of F ( X ( G ) , ˜ X ( G )): For every v ∈ V we have 2 deg G ( v ) − middle vertices in both X ( G ) and ˜ X ( G ). This results in X v ∈ V (cid:18) deg G ( v ) − (cid:19) = X v ∈ V deg G ( v ) − (2 deg G ( v ) − − X v ∈ V deg G ( v ) − clauses of Type 1. Thus | F ( X ( G ) , ˜ X ( G )) | ≥ X v ∈ V deg G ( v ) − (2 deg G ( v ) − −
1) + X v ∈ V deg G ( v ) − = X v ∈ V deg G ( v ) − deg G ( v ) − − X v ∈ V deg G ( v ) − + X v ∈ V deg G ( v ) − = X v ∈ V deg G ( v ) − + X v ∈ V deg G ( v ) − ≥ Φ ( G )= ⇒ α | E | + α Φ ( G ) = O (Φ ( G )) = O ( | F ( X ( G ) , ˜ X ( G )) | )This result stands in contrast to the exponential lower bound of Theorem 13. The usual approach to showing that a system of linear equations is inconsistent, is to build a linearcombination of the equations to derive the obvious contradiction 0 = 1. This method is complete byTheorem 16. We recreate this process in the resolution proof system. However we need to ensure thatthe support of equations we create along the way is not excessively large. We will do so by using the thesymmetry rule.Note that the formula F ( a , b ) is invariant under linear scaling of the inputs: For any k ∈ F p \ { } wehave supp( a ) = supp( k a ) and P ( a , b ) = P ( k a , kb ). Hence F ( a , b ) = F ( k a , kb ).For the computation of linear combinations, we use the following definition. For θ ⊆ [1 , n ] defineΩ( θ ) := (cid:8)W i ∈ θ ξ i, x i | x ∈ F np with supp( x ) ⊆ θ (cid:9) . Together, the clauses in Ω( θ ) forbid all possible assign-ments to the components in range θ . In a sense, Ω( θ ) is our basic building block for contradictions. Lemma 19.
Let θ ⊆ [1 , n ] . Then Ω( θ ) ∧ V ⊢ p | θ | +1 − pp − ⊥ .Proof. Induction over | θ | . If θ = ∅ then Ω( θ ) = { W ∅} = {⊥} , so Ω( θ ) ⊢ ⊥ .Induction step: θ = θ ′ ∪ { j } . It holds:Ω( θ ) = (_ i ∈ θ ξ i, x i | x ∈ F np with supp( x ) ⊆ θ ) = ( ξ j, x j ∨ _ i ∈ θ ′ ξ i, x i | x ∈ F np with supp( x ) ⊆ θ ) = ( ξ j,k ∨ _ i ∈ θ ′ ξ i, x i | x ∈ F np with supp( x ) ⊆ θ ′ and k ∈ F p ) = (cid:8) ξ j,k ∨ c ′ | c ′ ∈ Ω( θ ′ ) , k ∈ F p (cid:9) For each c ′ ∈ Ω( θ ′ ), we can derive the clause c ′ by resolving W k ∈ F p ξ j,k from V with the clauses fromΩ( θ ). Doing this for all c ′ ∈ Ω( θ ′ ) takes p · | Ω( θ ′ ) | = | Ω( θ ) | = p | θ | resolution steps. By induction, wecan then derive ⊥ from Ω( θ ′ ) and V in p | θ ′| +1 − pp − = p | θ | − pp − steps. The total number of steps taken is p | θ | − pp − + p | θ | = p | θ | +1 − pp − . 12 yx + y ϑ ( x , y )Figure 6: A visualization of ϑ ( x , y ).When we sum two vectors x and y , some components may become zero which were nonzero before.The following definition captures this phenomenon: ϑ ( x , y ) := (supp( x ) ∪ supp( y )) \ supp( x + y ). Ifa coefficient vanishes in a sum, it has to appear in both summands: ϑ ( x , y ) ⊆ supp( x ) ∩ supp( y ) (seeFigure 6).With these ingredients, we can finally explain the process of building sums using resolution. Theorem 20 (Sum Resolution) . Let a ∈ F × np and b ∈ F p . Define θ := ϑ ( a , a ) , where a i is the i -throw of a . For all c ∈ F ( Σa , Σb ) it holds: F ( a , b ) ∪ F ( a , b ) ∪ V ⊢ w p | θ | − c .Proof. Let c ∈ F ( Σa , Σb ). By definition of F , there exists some x ∈ P ( Σa , Σb ) such that c = W i ∈ supp( Σa ) ξ i, x i . The resolution derivation will have to get rid of all variables corresponding to compo-nents in θ . For κ ∈ { , } define R κ := W i ∈ supp( a κ ) \ θ ξ i, x i . From this we will build the desired clause c .It holds: (supp( a ) ∪ supp( a )) \ θ = (supp( a ) ∪ supp( a )) ∩ supp( Σa ) = supp( Σa ). Thus R ∨ R = _ i ∈ supp( a ) \ θ ξ i, x i ∨ _ i ∈ supp( a ) \ θ ξ i, x i = _ i ∈ (supp( a ) ∪ supp( a )) \ θ ξ i, x i = _ i ∈ supp( Σa ) ξ i, x i = c Consider an arbitrary y ∈ F np with supp( y ) ⊆ θ . Since supp( y ) ∩ supp( Σa ) = ∅ , we have Σa · y = 0.There exists κ with a κ · ( x + y ) = b κ , because otherwise we would have Σb = b + b = a · ( x + y ) + a · ( x + y ) = ( a + a ) · ( x + y ) = Σa · x + Σa · y = Σa · x , which contradicts x ∈ P ( Σa , Σb ).Hence ( x + y ) | a κ ∈ P ( a κ , b κ ) and we have a clause c ′ ( y ) := W i ∈ supp( a κ ) ξ i, ( x + y ) i ∈ F ( a κ , b κ ). Itholds: c ′ ( y ) = _ i ∈ supp( a κ ) ∩ θ ξ i, ( x + y ) i ∨ _ i ∈ supp( a κ ) \ θ ξ i, ( x + y ) i = _ i ∈ supp( a κ ) ∩ θ ξ i, y i ∨ _ i ∈ supp( a κ ) \ θ ξ i, x i = _ i ∈ θ ξ i, y i ∨ R κ Now, looking at the set C := { c ′ ( y ) | y ∈ F np } ⊆ F ( a , b ) ∪ F ( a , b ), note that for every clause d ∈ Ω( θ ), we have d ∨ R ∈ C or d ∨ R ∈ C . By Theorem 19 and Theorem 6 we can resolve theclauses in C together with V to obtain R ∨ R = c or stronger. Since p ≥
2, this takes at most p | θ | +1 − pp − ≤ p | θ | +1 − pp/ = 2( p | θ | −
1) resolution steps.By applying Theorem 20 iteratively, we can construct the formulas for linear combinations with anarbitrary number of summands. This method, however, is inefficient since the produced intermediateequations may accumulate more and more variables, leading to an exponential growth of the number ofrequired clauses.We solve this problem by deriving only a single representative clause for intermediate results, andusing the local symmetry rule to derive more clauses as necessary.13 .2 Local Symmetry in Equations
We want to understand which symmetries the formulas corresponding to linear equations have. For d ∈ F np define ∆ d : Vars → Vars : ξ i,k ∆ d ( ξ i,k ) := ξ i,k + d i . This bijective map is a translation by d ofthe vector corresponding to the variables. Lemma 21.
Let b ∈ F p and a , d ∈ F np . Then ∆ d ∈ Sym( F ( a , b )) if and only if a · d = 0 .Proof. ⇐ = : Assume a · d = 0. Let c = C a ( x ) = W i ∈ supp( a ) ξ i, x i ∈ F ( a , b ) for some x ∈ P ( a , b ). We have a · ( x + d ) = a · x = b . Hence ( x + d ) | a ∈ P ( a , b ) and thus ∆ d ( c ) = W i ∈ supp( a ) ξ i, x i + d i = C a ( x + d ) = C a (( x + d ) | a ) ∈ F ( a , b ).= ⇒ : Assume a · d = 0. Then a = and hence there exists a vector x ∈ P ( a , b ) with a · x = b − a · d = b . But then a · ( x + d ) = b and thus ∆ d ( C a ( x )) = C a ( x + d ) / ∈ F ( a , b ). Hence ∆ d / ∈ Sym( F ( a , b )). Corollary 22.
Let A ∈ F m × np , b ∈ F mp and d ∈ F np . If A · d = , then ∆ d ∈ Sym( F ( A , b )) . Note the following: If d , d ′ ∈ F np such that d | a = d ′ | a , then for all c ∈ F ( a , b ) it holds: ∆ d ( c ) = ∆ d ′ ( c ).In particular, ∆ d ∈ Sym( F ( a , b )) implies ∆ d ′ ∈ Sym( F ( a , b )). The condition d | a = d ′ | a can equivalentlybe expressed using matrix algebra: diag( a ) d = diag( a ) d ′ .To make use of the symmetry rule, we want to apply the symmetries of F ( A , b ) to derive clausesof F ( ΣA , Σb ). From the statements and Theorem 22, we conclude the following relation betweenSym( F ( A , b )) and Sym( F ( ΣA , Σb )). Lemma 23.
Let A ∈ F m × np , b ∈ F mp and d ∈ F np with ∆ d ∈ Sym( F ( ΣA , Σb )) If there exists d ′ ∈ F np such that Ad ′ = and diag( ΣA ) d ′ = diag( ΣA ) d , then ∆ d ′ ∈ Sym( F ( A , b )) and ∆ d ′ ( c ) = ∆ d ( c ) forall c ∈ F ( ΣA , Σb ) . Concerning V , the symmetries are simpler: For any d ∈ F np we have ∆ d ∈ Sym( V ).We will assume that the coefficient matrices A in the following have at most L nonzero entries ineach row. In other words, the width of A is at most L . Theorem 24.
Let A ∈ F m × np and b ∈ F mp . For any H ⊆ F ( ΣA , Σb ) it holds: F ( A , b ) ∧ V ⊢ SRC-II O ( m Θ( p ) p L +1 )+ | H | H .Proof. Define λ := log(2)log( p/ ( p − and f ( x ) := Cp L +1 x λ for some constant C chosen later. Regarding therelationship between λ and p we have λ ∼ log(2) p (i.e., lim n →∞ λ/ log(2) p = 1)).We prove the following by induction over the number of equations m : For any H ⊆ F ( ΣA , Σb ) itholds: F ( A , b ) ∧ V ⊢ SRC-II f ( m )+ | H | H .Induction basis: m = 0. In this case, we have ΣA = and Σb = 0. Then F ( ΣA , Σb ) = ∅ ; hence H = ∅ and we have nothing to prove.Induction step: m − → m . Here we have two cases:Case 1: Symmetric sum : For this case we assume that for all d with ∆ d ∈ Sym( F ( ΣA , Σb )) we havea d ′ with d | ΣA = d ′ | ΣA and ∆ d ′ ∈ Sym( F ( A , b )). Thanks to this property, all the symmetries of F ( ΣA , Σb ) are already present in F ( A , b ) and can be used by the symmetry rule. So we only need toderive a few clauses of F ( ΣA , Σb ) to obtain a set allowing us to generate all clauses via symmetries.This which can be done by inductively applying Theorem 20 as follows.Define A ′ and b ′ to be the first m − A and b respectively. Define θ := ϑ ( ΣA ′ , A m, ∗ ).We have | θ | ≤ | supp( A m, ∗ ) | ≤ L . If ΣA = , then for each k = Σb there exists a vector z k withsupp( z k ) ⊆ supp( ΣA ) such that ΣA · z k = k . Define G := { C ΣA ( z k ) | k ∈ F p \ { Σb }} ⊆ F ( ΣA , Σb ).Then | G | = p − F ( ΣA ′ , Σb ′ ) ∧ F ( A m, ∗ , b m ) ∧ V ⊢ w | G |·O ( p L ) G . This derivation onlyuses a subset H ′ ⊆ F ( ΣA ′ , Σb ′ ) of at most | H ′ | ≤ O ( p L +1 ) clauses. By induction, it holds that F ( A ′ , b ′ ) ∧ V ⊢ SRC-II f ( m − | H ′ | H ′ . We can combine these derivations by Theorem 5 to obtain F ( A , b ) ∧ V ⊢ SRC-II f ( m − O ( p L +1 ) G . Using λ ≥
1, we take in total f ( m −
1) + Cp L +1 = Cp L +1 ( m − λ + Cp L +1 ≤ Cp L +1 m λ = f ( m ) steps, for some constant C .Now we show that G is a generator for H : Let c ∈ F ( ΣA , Σb ). Then c = C ΣA ( x ) for some x ∈ P ( ΣA , Σb ). Define d := x − z ΣA · x . Then ΣA · d = 0, so ∆ d ∈ Sym( F ( ΣA , Σb )). Hence there14xists a ϕ ∈ Sym( F ( A , b )) such that ϕ ( C ΣA ( z ΣA · x )) = ∆ d ( C ΣA ( z ΣA · x )) = C ΣA ( x ) = c . We can applythe local symmetry rule to derive c from G in a single step, using the symmetries of F ( A , b ). Repeatingthis for every c ∈ H yields F ( A , b ) ∧ V ⊢ SRC-II f ( m )+ | H | H .If ΣA = and Σb = 0 then F ( ΣA , Σb ) = { C ( ) } = {⊥} =: G , which can be derived in at most Cp L +1 steps, again using Theorem 20.If ΣA = and Σb = 0 then F ( ΣA , Σb ) = ∅ . We treat this the same way as the case m = 0.Case 2: Composite : If Case 1 does not apply, the following must hold by Theorem 23: For some d with ΣA · d = 0, the equations Ad ′ = and diag( ΣA ) d ′ = diag( ΣA ) d have no common solution d ′ .Applying Theorem 16 to the combined inconsistent equations, we have v , w such that vA + w diag( ΣA ) = and w diag( ΣA ) d = 0. We will use the vector v to decompose A intotwo smaller matrices, each contributing independently to the derivation of H . First we show that v hasspecial properties which make this divide and conquer approach work. Then we need to ensure that thesub-problems are not too large for our proof length bound f .It holds: vA = − w diag( ΣA ); thus vAd = − w diag( ΣA ) d = 0. For all i ∈ [1 , n ], if ( ΣA ) i = 0, then( vA ) i = ( − w diag( ΣA )) i = − w i ( ΣA ) i = 0. Hence supp( vA ) ⊆ supp( ΣA ). We show that vA and ΣA are linearly independent: Let α , α ∈ F p such that α vA + α ΣA = . Then 0 = α vAd + α ΣAd = α vAd , which implies α = 0. Since ΣA = , we also have α = 0.Let k ∈ arg max k ∈ F p |{ i | v i = k }| be the most common component of v . Let k ∈ arg max k ∈ F p , k = k |{ i | v i = k }| be the second most common component of v . Since vA is linearly independent from ΣA , we have v = k · for all k ∈ F p , so there are at least two different components in v . Hence k and k exist. Define m i to be the number of times k i occurs in v . We have m i ≥ i ∈ { , } . Furthermore m ≥ m/p and m ≥ ( m − m ) / ( p − v := v − k and v := k − v . It holds: v + v = ( k − k ) . By subtracting k i fromevery component, we get exactly m i zeros in v i , i.e. | supp( v i ) | = m − m i .Towards a contradiction, assume there is some j ∈ ϑ ( v A , v A ). Then 0 = ( v A + v A ) j =(( k − k ) ΣA ) j , so 0 = ( ΣA ) j . Thus 0 = − w j ( ΣA ) j = ( vA ) j = ( v A + k ΣA ) j = ( v A ) j + k ( ΣA ) j = ( v A ) j . This contradicts the assumption. Hence ϑ ( v A , v A ) = ∅ . By Theorem 20 we canderive the sum clauses of v A + v A in 0 steps, so they are already implied by the summand clauses: F (( v + v ) A , ( v + v ) b ) ⊑ F ( v A , v b ) ∪ F ( v A , v b ). It follows that F ( ΣA , Σb ) = F (( k − k ) ΣA , ( k − k ) Σb )= F (( v + v ) A , ( v + v ) b ) ⊑ F ( v A , v b ) ∪ F ( v A , v b ) . Hence we can partition H ⊆ F ( ΣA , Σb ) into H and H such that H i ⊑ F ( v i A , v i b ) for i ∈ { , } .Note that F ( v i A , v i b ) = F ( Σ diag( v i ) A , Σ diag( v i ) b ). Since | supp( v i ) | ≤ m −
1, we have at leastone zero row each in diag( v ) A and diag( v ) A . This makes it possible to apply the induction hypothesis,yielding F (diag( v i ) A , diag( v i ) b ) ∧ V ⊢ SRC-II f ( | supp( v i ) | )+ | H i | H i .Scaling the equations does not produce different clauses, so we have F (diag( v ) A , diag( v ) b ) ∪ F (diag( v ) A , diag( v ) b ) ⊆ F ( A , b ). Then we can combine the deriva-tions of H and H to obtain F ( A , b ) ∧ V ⊢ SRC-II f ( | supp( v ) | )+ f ( | supp( v ) | )+ | H | H . It holds: f ( | supp( v ) | ) + f ( | supp( v ) | ) = f ( m − m ) + f ( m − m ) ≤ f ( m − m ) + f ( m − ( m − m ) / ( p − T ( m ). By applyingstandard calculus techniques to the function T , we find that T ( m ) ≤ f ( m ) for all possible values of m . Corollary 25.
Let A ∈ F m × np and b ∈ F mp , such that there is no x ∈ F np satisfying Ax = b . Then thereexists a resolution refutation of F ( A , b ) ∧ V using the local symmetry rule, with its length bounded by O ( m Θ( p ) p L +1 ) . We can use the result on linear equations to show that there are short resolution proofs for the non-isomorphism of Multipede graphs. 15 heorem 26.
Let G = ( V, W, E ) be a connected bipartite graph such that M P ( G ) is asymmetric, and ω ∈ W . Then F ( M P ( G ) a ω , M P ( G ) b ω ) has a linear-sized resolution refutation using the local symmetryrule.Proof. Let G = ( { v , . . . , v m } , { w , . . . , w n } , E ) be a connected bipartite graph and ω := w k for some k .Our goal is to apply the techniques of the previous section to the formula F := F ( M P ( G ) a ω , M P ( G ) b ω ).We first inspect the simpler formula F := F ( M P ( G ) , M P ( G )). The solutions of this formula cor-respond to the automorphisms of M P ( G ). By applying resolution to F , we can derive the formula F ( M ( G ) , ): Let i ∈ [1 , m ]. Then F ( M ( G ) i, ∗ ,
0) = ^ x ∈ P ( M ( G ) i, ∗ , _ j ∈ supp( M ( G ) i, ∗ ) ξ j, x j = ^ x ∈ P ( M ( G ) i, ∗ , _ { v i ,w j }∈ E ξ j, x j = ^ B ⊆ N G ( v i ) | B | odd _ w j ∈ B ξ j, ∨ _ w j ∈ N G ( v i ) \ B ξ j, Define N := N G ( v i ). We define P even ( N ) to be the subsets of N with even cardinality. For all B ⊆ N with odd | B | there exists a surjective function γ : P even ( N ) → N such that for all S ∈ P even ( N ) : γ ( S ) ∈ S \ B ∪ B \ S . We can make the following resolution derivation from F :Type 1: _ S ∈P even ( N ) z v ∅ ,S ∀ S ∈ P even ( N ) with w := γ ( S ) ∈ B \ S : Type 3: z v ∅ ,S ∨ y a w ,b w ∀ S ∈ P even ( N ) with w := γ ( S ) ∈ S \ B : Type 3: z v ∅ ,S ∨ y a w ,a w = ⇒ _ w ∈ B y a w ,b w ∨ _ w ∈ N \ B y a w ,a w , taking |P even ( N ) | ≤ | N | steps. Repeating this process for every B and i takes P v ∈ V |P odd ( N G ( v )) | · | N G ( v ) | = O ( | F | ) resolution steps. Define a variable renaming r on F ( M ( G ) , ) as follows: r ( ξ j,κ ) := ( y a wj ,a wj if κ = 0 y a wj ,b wj if κ = 1Then we have the derivation F ⊢ O ( | F | ) r ( F ( M ( G ) , )). As a consequence, | F ( M ( G ) , ) | = O ( | F | ).The clauses of r ( V ) are simply the Type 1 clauses of F .To apply Theorem 24, we need to translate the symmetries ∆ d ∈ Sym( F ( M ( G ) , )) into symmetriesof F . Let d ∈ F n such that M ( G ) d = . Define D := { w i ∈ W | d i = 1 } . Then the followingmap ψ d is a symmetry of F : for w ∈ W set ψ d ( y a w ,a w ) to be y a w ,b w if w ∈ D and y a w ,a w otherwise.Similarly ψ d ( y a w ,b w ) is y a w ,a w if w ∈ D and y a w ,b w . We also set ψ d ( y b w ,a w ) to be y b w ,b w if w ∈ D and y b w ,a w and we set ψ d ( y b w ,b w ) to be y b w ,a w if w ∈ D and y b w ,b w . Finally for v ∈ V we define ψ d ( z vS,T ) := z vS,T △ D and have the property ψ d ( r ( c )) = r (∆ d ( c )) for all clauses c ∈ F ( M ( G ) , ).Now, if the graph M P ( G ) is asymmetric, the only solution of M ( G ) y = is y = . Then we candeduce y k = 0 from the equation system by combining rows. Applying Theorem 24, we get F ( M ( G ) , ) ∧ V ⊢ SRC-II O ( m L +1 ) ξ k, . Renaming variables yields r ( F ( M ( G ) , )) ∧ r ( V ) ⊢ SRC-II O ( m L +1 ) y a ω ,a ω . As we have seen, r ( F ( M ( G ) , )) and r ( V ) can be derived from F and the symmetries are preserved; hence F ⊢ SRC-II O ( m L +1 ) y a ω ,a ω .Note that F is obtained from F simply by replacing y a ω ,a ω and y b ω ,b ω with 0. From this we concludethat, F ⊢ SRC-II O ( m L +1 ) ⊥ . 16 eferences [1] Noriko H. Arai and Alasdair Urquhart. Local symmetries in propositional logic. In RoyDyckhoff, editor, Automated Reasoning with Analytic Tableaux and Related Methods, Interna-tional Conference, TABLEAUX 2000, St Andrews, Scotland, UK, July 3-7, 2000, Proceed-ings , volume 1847 of
Lecture Notes in Computer Science , pages 40–51. Springer, 2000. URL: https://doi.org/10.1007/10722086_3 , doi:10.1007/10722086\_3 .[2] Belaid Benhamou and Lakhdar Sais. Tractability through symmetries in propositional calculus. J.Autom. Reasoning , 12(1):89–102, 1994. doi:10.1007/BF00881844 .[3] Joshua Blinkhorn and Olaf Beyersdorff. Proof complexity of QBF symmetry recomputation.In Mikol´as Janota and Inˆes Lynce, editors,
Theory and Applications of Satisfiability Testing -SAT 2019 - 22nd International Conference, SAT 2019, Lisbon, Portugal, July 9-12, 2019, Pro-ceedings , volume 11628 of
Lecture Notes in Computer Science , pages 36–52. Springer, 2019. doi:10.1007/978-3-030-24258-9\_3 .[4] Jin-yi Cai, Martin F¨urer, and Neil Immerman. An optimal lower bound on the number of variablesfor graph identifications.
Combinatorica , 12(4):389–410, 1992. doi:10.1007/BF01305232 .[5] Vasek Chv´atal and Endre Szemer´edi. Many hard examples for resolution.
J. ACM , 35(4):759–768,1988. doi:10.1145/48014.48016 .[6] Thierry Boy de la Tour and St´ephane Demri. On the complexity of extending ground resolutionwith symmetry rules. In
Proceedings of the Fourteenth International Joint Conference on ArtificialIntelligence, IJCAI 95, Montr´eal Qu´ebec, Canada, August 20-25 1995, 2 Volumes , pages 289–297.Morgan Kaufmann, 1995. URL: http://ijcai.org/Proceedings/95-1/Papers/038.pdf .[7] Heidi E. Dixon, Matthew L. Ginsberg, David K. Hofer, Eugene M. Luks, and Andrew J.Parkes. Implementing a generalized version of resolution. In Deborah L. McGuinness andGeorge Ferguson, editors,
Proceedings of the Nineteenth National Conference on Artificial In-telligence, Sixteenth Conference on Innovative Applications of Artificial Intelligence, July 25-29,2004, San Jose, California, USA , pages 55–60. AAAI Press / The MIT Press, 2004. URL: .[8] Uwe Egly. A first order resolution calculus with symmetries. In Andrei Voronkov, editor,
LogicProgramming and Automated Reasoning,4th International Conference, LPAR’93, St. Petersburg,Russia, July 13-20, 1993, Proceedings , volume 698 of
Lecture Notes in Computer Science , pages110–121. Springer, 1993. doi:10.1007/3-540-56944-8\_46 .[9] Michael Frank and Michael Codish. Logic programming with graph automorphism:Integrating nauty with prolog (tool description).
TPLP , 16(5-6):688–702, 2016. doi:10.1017/S1471068416000223 .[10] Yuri Gurevich and Saharon Shelah. On finite rigid structures.
J. Symb. Log. , 61(2):549–562, 1996. doi:10.2307/2275675 .[11] Armin Haken. The intractability of resolution.
Theor. Comput. Sci. , 39:297–308, 1985. doi:10.1016/0304-3975(85)90144-6 .[12] Balakrishnan Krishnamurthy. Short proofs for tricky formulas.
Acta Informatica , 22(3):253–275,August 1985. doi:10.1007/BF00265682 .[13] Brendan D. McKay and Adolfo Piperno. Practical graph isomorphism, II.
J. Symb. Comput. ,60:94–112, 2014. doi:10.1016/j.jsc.2013.09.003 .[14] Daniel Neuen and Pascal Schweitzer. An exponential lower bound for individualization-refinementalgorithms for graph isomorphism. In Ilias Diakonikolas, David Kempe, and Monika Hen-zinger, editors,
Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Com-puting, STOC 2018, Los Angeles, CA, USA, June 25-29, 2018 , pages 138–150. ACM, 2018. doi:10.1145/3188745.3188900 . 1715] Stefan Szeider. The complexity of resolution with generalized symmetry rules.
Theory Comput.Syst. , 38(2):171–188, 2005. doi:10.1007/s00224-004-1192-0 .[16] Jacobo Tor´an. On the resolution complexity of graph non-isomorphism. In Matti J¨arvisaloand Allen Van Gelder, editors,
Theory and Applications of Satisfiability Testing - SAT2013 - 16th International Conference, Helsinki, Finland, July 8-12, 2013. Proceedings , vol-ume 7962 of
Lecture Notes in Computer Science , pages 52–66. Springer, 2013. URL: https://doi.org/10.1007/978-3-642-39071-5_6 , doi:10.1007/978-3-642-39071-5\_6 .[17] G. S. Tseitin. On the Complexity of Derivation in Propositional Calculus , pages 466–483. SpringerBerlin Heidelberg, Berlin, Heidelberg, 1983. doi:10.1007/978-3-642-81955-1_28 .[18] Alasdair Urquhart. Hard examples for resolution.
J. ACM , 34(1):209–219, January 1987. URL: http://doi.acm.org/10.1145/7531.8928 , doi:10.1145/7531.8928 .[19] Alasdair Urquhart. The symmetry rule in propositional logic. Discrete Appl. Math. , 96-97:177– 193, 1999. URL: , doi:https://doi.org/10.1016/S0166-218X(99)00039-6doi:https://doi.org/10.1016/S0166-218X(99)00039-6