aa r X i v : . [ m a t h . L O ] A p r H-coloring Dichotomy in Proof Complexity
Azza Gaysin ∗ Department of Algebra, Faculty of Mathematics and PhysicsCharles University in Prague
Abstract
The H -coloring problem can be considered as an example of the com-putational problem from a huge class of the constraint satisfaction prob-lems (CSP): an H -coloring of a graph G is just a homomorphism from G to H and the problem is to decide for fixed H , given G , if a homomorphismexists or not.The dichotomy theorem for the H -coloring problem was proved byHell and Nešetřil [9] in 1990 (an analogous theorem for all CSPs was re-cently proved by Zhuk [14] and Bulatov [3]) and it says that for each H the problem is either p -time decidable or NP -complete. Since nega-tions of unsatisfiable instances of CSP can be expressed as propositionaltautologies, it seems to be natural to investigate the proof complexity ofCSP.We show that the decision algorithm in the p -time case of the H -coloring problem can be formalized in a relatively weak theory and thatthe tautologies expressing the negative instances for such H have shortproofs in propositional proof system R ∗ ( log ), a mild extension of reso-lution. In fact, when the formulas are expressed as unsatisfiable sets ofclauses they have p -size resolution proofs. To establish this we use a well-known connection between theories of bounded arithmetic and proposi-tional proof systems.We complement this result by a lower bound result that holds for manyweak proof systems for a special example of NP -complete case of the H -coloring problem, using the known results about proof complexity of thePigeonhole Principle. The constraint satisfaction problem (CSP) is a computational problem thatconsists in finding an assignment of values to a set of variables, such that thisassignment satisfies some specified feasibility conditions. If such assignmentexists, we call the instance of CSP satisfiable and unsatisfiable otherwise. One ∗ This work has been supported by Charles University Research Centre programNo.UNCE/SCI/022. H , denoted byCSP( H ) the question is, given a structure G over the same vocabulary, whetherthere exists a homomorphism from G to H . It turns out, that all CSPs canbe classified with only two complexity classes: there are either polynomial-timeCSPs, or N P -complete CSPs. This dichotomy was conjectured by Feder andVardi in 1998 [7] and recently proved by Zhuk [14] and Bulatov [3].The H -coloring problem is essentially CSP( H ) on relational structures thatare undirected graphs. Its computational complexity was investigated years agoand the Dichotomy theorem for the H -coloring problem was proved by Hell andNešetřil [9] in 1990. Theorem 1 (The Dichotomy theorem for the H -coloring problem, [9]) . If H isbipartite then the H -coloring problem is in P . Otherwise the H -coloring problemis N P -complete.
There is an easy H -colorability test when H is bipartite: Lemma 1 ([9]) . For all graphs G , H if H is bipartite, then G is H -colorable ifand only if G is bipartite graph. Instances of CSP( H ) can be expressed by propositional formulas: denoteby α ( G , H ) the propositional formula expressing that there is a homomorphismfrom G to H (see Definition 2). If the instance of CSP is unsatisfiable, then ¬ α ( G , H ) is a tautology (for the H -coloring problem we get a tautology everytime we consider bipartite graph H and non-bipartite graph G ). From this pointof view it is natural to ask about its proof complexity. Common way to do thisis to formalize the sentence in some weak theory of bounded arithmetic andfirst prove that this universal statement is valid in all finite structures. Thenit could be translated into a family of propositional tautologies, that will haveshort proofs in the corresponding proof system. The simpler the theory is, theweaker propositional proof system will be.In this paper we show, that the decision algorithm in the p -time case ofthe H -coloring problem (that is, the case where H is a bipartite graph) can beformalized in a relatively weak two-sorted theory V , which is quite convenientfor formalizing sets of vertices and relations between them, and proved by usingonly formulas of restricted complexity in the Induction scheme. The tautologiesexpressing the negative instances for such H hence have short proofs in propo-sitional proof system R ∗ ( log ), a mild extension of resolution. In fact, when theformulas are expressed as unsatisfiable sets of clauses they have p -size resolutionproofs.We shall complement this upper bound result by a lower bound results, bygiving examples of graphs H and G for which CSP( H ) is N P -complete andfor which any proof of the tautologies expressing that G / ∈ CSP( H ) must haveexponential size length in constant-depth Frege system (which contains R ∗ ( log ))and some other well-known proof systems. This is based on the proof complexityof the Pigeonhole Principle. 2 Preliminaries H -coloringproblem There are many equivalent definitions of the constraint satisfaction problem.Here we will use the definition in terms of homomorphisms.
Definition 1 (Constraint satisfaction problem) . • A vocabulary is a finite set of relational symbols R ,..., R n each of whichhas a fixed arity. • A relational structure over the vocabulary R ,..., R n is the tuple H =( H, R H , ..., R H n ) s.t. H is non-empty set , called the universe of H , andeach R H i is a relation on H having the same arity as the symbol R i . • For G , H being relational structures over the same vocabulary R ,..., R n a homomorphism from G to H is a mapping φ : G → H from the universe G to H s.t., for every m -ary relation R G and every tuple ( a , ..., a m ) ∈ R G we have ( φ ( a ) , ..., φ ( a m )) ∈ R H .Let H be a relational structure over a vocabulary R ,..., R n . In the constraintsatisfaction problem associated with H , denoted by CSP( H ) the question is,given a structure G over the same vocabulary, whether there exists a homo-morphism from G to H . If the answer is positive, then we call the instance G satisfiable and unsatisfiable otherwise [2].The H -coloring problem could be described as follows: let H = ( V H , E H ) bea simple undirected graph without loops, whose vertices we consider as differentcolors. An H -coloring of a graph G = ( V G , E G ) is an assignment of colors tothe vertices of G such that adjacent vertices of G obtain adjacent colors. Sincea graph homomorphism h : G → H is a mapping of V G to V H such that if g, g ′ are adjacent vertices of G , then so are h ( g ) , h ( g ′ ), it is easy to see that an H -coloring of G is just a homomorphism G → H . A simple undirected graph H can be considered as a relational structure H = ( V H , E H ) with only one binarysymmetric relation E H ( i, j ) (to i, j be adjacent vertices). Thus, the problem of H -coloring of a graph G is equivalent to CSP( H ).To express an instance of CSP( H ) by propositional formula we use the fol-lowing construction [1]. For any sets V G and V H by V ( V G , V H ) we denote a setof propositional variables: for every v ∈ V G and every u ∈ V H there is a variable x v,u in the set V ( V G , V H ). A variable x v,u is assigned the truth value 1 if andonly if the vertex v is mapped to vertex u . To every graph G = ( V G , E G ) weassign a set of clauses CN F ( G , H ) over the variables in V ( V G , V H ) in such away that there is a one-to-one correspondence between the truth valuations ofthe variables in V ( V G , V H ) satisfying this set and the homomorphisms from G to H : 3 efinition 2. For any two graphs G = ( V G , E G ), H = ( V H , E H ) by CN F ( G , H )we denote the following set of clauses: • a clause W u ∈ V H x v,u for each v ∈ V G ; • a clause ¬ x v,u ∨ ¬ x v,u for each v ∈ V G and u , u ∈ V H with u = u ; • a clause ¬ x v ,u ∨ ¬ x v ,u for every adjacent vertices v , v ∈ V G and non-adjacent vertices u , u ∈ V H .It is easy to see that if we exchange the last item with more general definition: • a clause W i ∈ [ r ] ¬ x v i ,u i for each natural number r , each relation R of arity r , each ( v , v , ..., v r ) ∈ R G , and each u .u , ..., u r / ∈ R H ,we get the set of clauses CN F ( G , H ) for common CSP on any relational struc-ture. Some definitions, examples and results are adapted from [5]. In our work weuse two-sorted first-order (sometimes called second-order) set-up as a frameworkfor the theory. Here there are two kinds of variables: the variables x, y, z, ... ofthe first sort are called number variables and range over the natural numbers,and the variables
X, Y, Z, ... of the second sort are called set (or also strings)variables and range over finite subsets of natural numbers (which representbinary strings). Functions and predicate symbols may involve both sorts andthere are two kinds of functions: the number-valued functions (or just numberfunctions ) and the string-valued functions (or just string functions ).The usual language of arithmetic for two-sorted first-order theories is theextension of standard language for Peano Arithmetic L PA . Definition 3 ( L PA ) . L PA = { , , + , · , | | ; = , = , ≤ , ∈} Here the symbols 0 , , + , · , = and ≤ are well-known and are from L PA :they are function and predicate symbols over the first sort. The function | X | ( the length of X ) is a number-valued function and is intended to denote theleast upper bound of the set X (the length of the corresponding string). Thebinary predicate ∈ for a number and a set denotes set membership, and = is the equality predicate for sets. The defining properties of all symbols fromlanguage L PA are described by a set of basic axioms denoted as 2- BASIC [5],which we do not present here.
Notation 1.
We will use the abbreviation: X ( t ) = def t ∈ X where t is a number term. Thus we think of X ( i ) as the i -th bit of binary string X of length | X | .
4o define the theory V , in which we will formalize the H -coloring problem,we need the following definitions: Definition 4 (Bounded formulas) . Let L be a two-sorted vocabulary. If x is anumber variable, X is a string variable that do not occur in the L -number term t , then ∃ x ≤ tφ stands for ∃ x ( x ≤ t ∧ φ ), ∀ x ≤ tφ stands for ∀ x ( x ≤ t → φ ), ∃ X ≤ tφ stands for ∃ X ( | X | ≤ t ∧ φ ) and ∀ X ≤ tφ stands for ∀ X ( | X | ≤ t → φ ).Quantifiers that occur in this form are said to be bounded , and a bounded formula is one in which every quantifier is bounded. Definition 5 (Σ Bi and Π Bi formulas in L PA ) . We will define Σ Bi and Π Bi formulas recursively as follows: • Σ B = Π B is the set of L PA -formulas whose only quantifiers are boundednumber quantifiers (there can be free string variables); • For i ≥
0, Σ Bi +1 (resp. Π Bi +1 ) is the set of formulas of the form ∃ ~X ≤ ~tφ ( ~X )(resp. ∀ ~X ≤ ~tφ ( ~X )), where φ is a Π Bi formula (resp. Σ Bi formula), and ~t is a sequence of L PA -terms not involving any variable in ~X . Definition 6 (Comprehension Axiom) . If Φ is a set of formulas, then the comprehension axiom scheme for Φ, denoted by Φ-
COM P , is the set of formulas ∃ X ≤ y ∀ z < y ( X ( z ) ←→ φ ( z )) (1)where φ ( z ) is any formula in Φ, X does not occur free in φ ( z ), and φ ( z ) mayhave free variables of both sorts, in addition to z . Definition 7 ( V ) . The theory V has the vocabulary L PA and is axiomatizedby 2- BASIC and Σ B - COM P .There is no explicit Induction axiom scheme in V , but it is known [4] that V ⊢ Σ B - IN D , where Φ-
IN D is:
Definition 8 (Number Induction Axiom) . If Φ is a set of two-sorted formulas,then Φ-
IN D axioms are the formulas( φ (0) ∧ ∀ x ( φ ( x ) → φ ( x + 1))) → ∀ zφ ( z ) (2)where φ is a formula in Φ. In this section we define propositional proof systems R , R ( log ) and their tree-likeversions. Some definitions and result are adopted from [10],[12]. Definition 9 (Propositional proof system, [6]) . A propositional proof system is a polynomial time function P whose range is set T AU T . For a tautology τ ∈ T AU T , any string w such that P ( ω ) = τ is called a P -proof of τ .5roof systems are usually defined by a finite number of inference rules of aparticular form and the proof is created by applying them step be step. Thecomplexity of proof is measured by its size and number of steps.The resolution system R operates with atoms and there negations and hasno other logical connectives. The basic object is a clause , a disjunction of afinite set of literals. The resolution rule allows us to derive new clause C ∪ C from two clauses C ∪ { p } and C ∪ {¬ p } : C ∪ { p } C ∪ {¬ p } C ∪ C (3)If we manage to derive the empty clause ∅ from the initial set of clauses C ,the clauses in the set C are not simultaneously satisfiable. Thus, the resolutionsystem can be interpreted as a refutation proof system : instead of proving thata formula is a tautology, it proves that a set of clauses C = { C , C , ..., C n } isnot satisfiable, and therefore the formula α = W ni =1 ¬ C i is a tautology. Definition 10 (An R -proof) . Let C be a set of clauses, an R -refutation of C isa sequence of clauses D , ..., D k such that: • For each i ≤ k , either D i ∈ C or there are u, v < i such that D i followsfrom D u , D v by the resolution rule, • D k = ∅ .The number of steps in the refutation is k .The DNF-resolution (denoted by DNF- R ) is a proof system extending R byallowing in clauses not only literals but their conjunctions as well [12]. DNF- R has the following inference rules: C ∪ { V j l j } D ∪ {¬ l ′ , ..., ¬ l ′ t } C ∪ D (4)if t = 1 and all l ′ i occur among l j , and C ∪ { V j ≤ s l j } D ∪ { V s Definition 12 (DNF -formula) . A basic formula is an atomic formula or thenegation of an atomic formula. A DNF -formula is a formula that is built frombasic formulas by: • first apply any number of conjunctions and bounded universal quantifiers, • then apply any number of disjunctions and bounded existential quantifiers. H -coloring problem in V In this section we define all the notions we need to formalize the decision algo-rithm in the p -time case of the H -coloring problem, i.e. the notions of a graph,bipartite and non-bipartite graphs and a homomorphism between graphs, in thevocabulary L PA and using only basic axioms of V . To do this we extend ourtheory with new predicate and function symbols and for each of them we adddefining axioms which ensure that they receive their standard interpretations ina model of V . Definition 13 (Representable/Definable relations) . Let L ⊇ L PA be a two-sorted vocabulary, and let φ be a L -formula. Then we say that φ ( ~x, ~X ) repre-sents (or defines) a relation R ( ~x, ~X ) if R ( ~x, ~X ) ←→ φ ( ~x, ~X ) . (6)If Φ is a set of L -formulas, then we say that R ( ~x, ~X ) is Φ -representable (orΦ-definable) if it is represented by some φ ∈ Φ. Definition 14 (Definable number functions) . Let T be a theory with two-sortedvocabulary L ⊇ L PA , and let Φ be a set of L -formulas. A number function f is Φ -definable in T if there is a formula φ ( ~x, y, ~X ) in Φ such that T ⊢ ∀ ~x ∀ ~X ∃ ! y φ ( ~x, y, ~X ) (7)and y = f ( ~x, ~X ) ←→ φ ( ~x, y, ~X ) . (8)7uxiliary predicate and function symbols, which we will use further to definedifferent notions in V , are the following: Definition 15 (Divisibility) . The relation of divisibility is defined by: x | y ←→ ∃ z ≤ y ( x · z = y ) . (9) Definition 16 (Pairing function) . If x, y ∈ N we define the pairing function h x, y i to be the following term in V : h x, y i = ( x + y )( x + y + 1) + 2 y (10)Since the formula for pairing function is just a term in standard vocabulary forthe theory V , it is obvious that V proves the condition (7). It is also easy toprove in V that pairing function is a one-one function, that is: V ⊢ ∀ x , x , y , y h x , y i = h x , y i → x = x ∧ y = y (11)Using pairing function we can code pair of numbers x, y by one number h x, y i , and the sequence of pairs by a subset of numbers. To define a graphon n vertices, consider a string V G where | V G | = n and ∀ i < n V G ( i ). We saythat V G is the set of n vertices of graph G . Then we define string E G of length | E G | < n to be the set of edges of the graph G as following: if there is anedge between vertices i, j then, using the pairing function, set E G ( h i, j i ) and ¬ E G ( h i, j i ) otherwise. Notation 2. Instead of E G ( h i, j i ) we will write just E G ( i, j ) to denote that thereis an edge between i and j , and sometimes instead of ( V G , E G ) we will write G . Definition 17 (Undirected graph G without loops) . A pair of sets G = ( V G , E G )with | V G | = n denotes an undirected graph without loops if it satisfies thefollowing relation: GRAP H ( V G , E G ) ←→ ∀ i < n ( V G ( i )) ∧ ∀ i < j < n ( E G ( i, j ) ←→ E G ( j, i )) ∧ ∀ i < n ¬ ( E G ( i, i )) (12)Further, talking about graphs we will consider only pairs of strings G =( V G , E G ) that satisfy the above relation. Since we formalize the H -coloringproblem we need to define the homomorphism on graphs in the vocabulary L PA . Consider two graphs G = ( V G , E G ) and H = ( V H , E H ), where | V G | = n , | V H | = m . Firstly we define a map between two sets of vertices V G , V H , thatis between sets [0 , n − 1] and [0 , m − Z < h n − , m − i + 1, where Z ( h i, j i ) means that i -th vertexis mapped to j -th vertex. For Z to be a well-defined map it should satisfy thefollowing Σ B -definable relation M AP ( n, m, Z ): Definition 18 (Map between two sets) . We say that a set Z is a well-definedmap between two sets [0 , n − 1] and [0 , m − 1] if it satisfy the relation: M AP ( n, m, Z ) ←→ ∀ i < n ∃ j < m Z ( h i, j i ) ∧∀ i < n ∀ j , j < m ( Z ( h i, j i ) ∧ Z ( h i, j i ) → j = j ) (13)8ow we can formalize the standard notion of existence of a homomorphismbetween two graphs G and H (here the homomorphism is formalized by a set Z with certain properties): Definition 19 (The existence of a homomorphism between graphs G and H ) . There is a homomorphism between two graphs G = ( V G , E G ) and H = ( V H , E H )with | V G | = n , | V H | = m , if they satisfy the relation: HOM ( G , H ) ←→ ∃ Z ≤ h n − , m − i (cid:0) M AP ( n, m, Z ) ∧∀ i , i < n, ∀ j , j < m ( E G ( i , i ) ∧ Z ( h i , j i ) ∧ Z ( h i , j i ) → E H ( j , j )) (cid:1) (14)Note that the relation HOM ( G , H ) is a Σ B -definable relation. Finally, weneed to formalize what does it means to be a bipartite or non-bipartite graph.The notion of being bipartite is Σ B -definable in L PA : Definition 20 (Bipartite graph H ) . A graph H = ( V H , E H ) with | V H | = m is bipartite if it satisfy the relation: BIP ( H ) ←→ ∃ W H , U H ≤ m (cid:0) ∀ i < m ( W H ( i ) ↔ ¬ U H ( i )) ∧∀ i < j < m ( E H ( i, j ) → ( W H ( i ) ∧ U H ( j )) ∨ ( W H ( j ) ∧ U H ( i ))) (cid:1) (15)To define a non-bipartite graph we use commonly-known characterizationof non-bipartite graphs (to contain an odd cycle, or, more generally, to allow ahomomoprhism from an odd cycle). The reason here is to get a Σ B -definablerelation for a non-bipartite graph. This makes the formula in the main statementfrom the next section be Π B , and hence translatable into propositional logic.First we define a cycle. Definition 21 (Cycle C k ) . A graph C k = ( V C k , E C k ) with V C k = { , , ..., k − } is a cycle of length k if it satisfies the relation: CY CLE ( C k ) ←→ E C k (0 , k − ∧ ∀ i < ( k − E C k ( i, i + 1) ∧∀ i, j < ( k − j = i + 1 → ¬ E C k ( i, j )) (16) Definition 22 (Non-bipartite graph G ) . A graph G = ( V G , E G ) with | V G | = n is non-bipartite if it satisfies the following Σ B -definable relation: N ON BIP ( G ) ←→ ∃ k ≤ n (2 | ( k − ∃ V C k = k, ∃ E C k < k CY CLE ( V C k , E C k ) ∧ HOM ( C k , G ) (17) V To prove in the theory V the universal statement about p -decidable case ofthe H -coloring problem, that is there is no homomorphism from a non-bipartitegraph to a bipartite graph, we firstly need to establish that the homomorphismrelation, as we defined it, has the property of transitivity.9 emma 3 (Homomorphism transitivity) . For all graphs G , H , S V ⊢ ∀G , H , S ( HOM ( G , H ) ∧ HOM ( H , S ) → HOM ( G , S )) (18) Proof. Consider graphs G = ( V G , E G )), H = ( V H , E H ) and S = ( V S , E S ), where | V G | = n , | V H | = m and | V S | = t . Since HOM ( G , H ) and HOM ( H , S ), thereexist two sets Z ≤ h n − , m − i and Z ′ ≤ h m − , t − i which satisfy thehomomorphism definition. We need to prove that there exists a set Z ′′ ≤h n − , t − i , such that: M AP ( n, t, Z ′′ ) ∧ ∀ i , i < n, ∀ k , k < t ( E G ( i , i ) ∧ Z ′′ ( h i , k i ) ∧ Z ′′ ( h i , k i ) → E S ( k , k ))Consider the set Z ′′ ≤ h n − , t − i which we define by the formula: Z ′′ ( h i, k i ) ←→ ∃ j < m ( Z ( h i, j i ) ∧ Z ′ ( h j, k i )) . (19)This set should exist due to Comprehension Axiom Σ B - COM P , since the for-mula φ ( h i, k i ) = ∃ j < m ( Z ( h i, j i ) ∧ Z ′ ( h j, k i )) ∈ Σ B . It is easy to check thatthe set Z ′′ satisfies the homomorphism relation between graphs G and S . Notation 3. K will denote the complete graph on two vertices. In the following two lemmas we prove that there is always a homomorphismfrom a bipartite graph to K and there is no homomorphism from a non-bipartitegraph to K . Lemma 4. For all bipartite graphs H , V proves the existence of a homomor-phism from H to K : V ⊢ ∀H ( BIP ( H ) → HOM ( H , K )) (20) Proof. Consider a bipartite graph H = ( V H , E H ) with | V H | = n . We need toshow that there exists a homomorphism from H to K , that is an appropriateset Z ≤ h n − , i . Since H is bipartite, then there exist two subsets W H and U H , such that ( W H ( i ) ↔ ¬ U H ( i )). Consider a set Z ≤ h n − , i , such that: (cid:26) Z ( h i, i ) ←→ W H ( i ) Z ( h i, i ) ←→ U H ( i )This set also exists due to Comprehension Axiom Σ B - COM P , since the formula φ ( h i, j i ) = ( j = 0 ∧ W H ( i )) ∨ ( j = 1 ∧ U H ( i )) ∈ Σ B . Obviously, since ( W H ( i ) ↔¬ U H ( i )), by the definition of Z we have M AP ( n, , Z ). Consider any i , i < n ,such that E H ( i , i ). Then ( W H ( i ) ∧ U H ( i )) or ( W H ( i ) ∧ U H ( i )). In the firstcase we have Z ( h i , i ) ∧ Z ( h i , i ), in the second case Z ( h i , i ) ∧ Z ( h i , i ), andin both cases E K (0 , Z is a homomorphism from H to K . Lemma 5. For all non-bipartite graphs G , V proves that there is no homo-morphism from G to K : V ⊢ ∀G ( N ON BIP ( G ) → ¬ HOM ( G , K )) (21)10 roof. Suppose that a graph G = ( V G , E G ), | V G | = n is non-bipartite, thatis there exist k ≤ n , C k = ( V C k , H C k ) with | V C k | = k , such that 2 | ( k − CY CLE ( C k ) and HOM ( C k , G ).Assume that there exists a homomorphism form G to K . Due Lemma 3 bytransitivity there also exists a homomorphism Z ≤ h k − , i from C k to K .Since it is a homomorphism from C k to K then for every 0 ≤ i ≤ ( k − 1) either Z ( h i, i ) or Z ( h i, i ).Without loss of generality suppose that Z ( h , i ) and lets prove that Z ( h k − , i ) too. Since 2 | ( k − k > 2. Due CY CLE ( C k ), E C k (0 , E C k (1 , i < k , if 2 | i then Z ( h i, i ) and Z ( h i, i )otherwise. Consider the formula: φ ( i, k, Z ) = (2 | i → Z ( h i, i )) ∧ (2 ∤ i → Z ( h i, i )) (22)Since φ ( i, n, Z ) ∈ Σ B , we can prove this claim by induction on i , because V proves Σ B - IN D :( φ (0 , n, Z ) ∧ ∀ i < k ( φ ( i, n, Z ) → φ ( i + 1 , n, Z )) → ∀ j < k φ ( j, n, Z ) (23)The base case is considered above. For step of induction suppose that it istrue for ( i − 1) and consider i . We have two options. If 2 | ( i − 1) then by theinduction hypothesis Z ( h i − , i ). Thus, since for ( i − 1) by CY CLE ( C k ) wehave E C k ( i − , i ), by the definition of homomorphism Z ( h i, i ). Analogically, if2 ∤ ( i − 1) then Z ( h i, i ).Hence Z ( h , i ) and Z ( h k − , i ). But since there is an edge between vertices0 and ( k − 1) in the graph C k , Z cannot be a homomorphism between C k and K .Therefore, our assumption leads to contradiction and there is no homomorphismfrom G to K .The main result of this paper is an immediate conclusion from the previouslemmas. Theorem 2 (The main universal statement) . For all non-bipartite graphs G and bipartite graphs H , V proves that there is no homomorphism from G to H : V ⊢ ∀G , H ( BIP ( H ) ∧ N ON BIP ( G ) → ¬ HOM ( G , H )) (24) Proof. Suppose that there exists a homomorphism from G to H . According toLemma 4, since H is bipartite then there exists a homomorphism from H to K .Thus due to Lemma 3 by the transitivity there exists a homomorphism from G to K . But this is the contradiction with Lemma 5. In this section we proceed with translation of the main universal statement inthe theory V into propositional tautologies. There is well-known translation of11 B formulas into propositional calculus formulas: we can translate each formula φ ( ~x, ~X ) ∈ Σ B into a family of propositional formulas [5]: || φ ( ~x, ~X ) || = { φ ( ~x, ~X )[ ~m, ~n ] : ~m, ~n ∈ N } (25) Lemma 6 ([5]) . For every Σ B ( L PA ) formula φ ( ~x, ~X ) , there is a constant d ∈ N and a polynomial p ( ~m, ~n ) such that for all ~m, ~n ∈ N , the propositionalformula φ ( ~x, ~X )[ ~m, ~n ] has depth at most d and size at most p ( ~m, ~n ) [5]. There is a theorem that establish a connection between Σ B -fragment of thetheory V and constant-depth Frege proof system: Theorem 3 ( V Translation, [5]) . Suppose that φ ( ~x, ~X ) is a Σ B formula suchthat V ⊢ ∀ ~x ∀ ~Xφ ( ~x, ~X ) . Then the propositional family || φ ( ~x, ~X ) || has polyno-mial size bounded depth Frege proofs. That is, there are a constant d and apolynomial p ( ~m, ~n ) such that for all ≤ ~m, ~n ∈ N , φ ( ~x, ~X )[ ~m, ~n ] has a d -Fregeproof of size at most p ( ~m, ~n ) . Further there is an algorithm which finds a d -Fregeproof of φ ( ~x, ~X )[ ~m, ~n ] in time bounded by a polynomial in ( ~m, ~n ) [5]. Consider the Π B -formula φ ( G , H ) from Theorem 2 which expresses that thereis no homomorphism from a non-bipartite graph G to a bipartite graph H : φ ( G , H ) = ¬ GRAP H ( G ) ∨ ¬ GRAP H ( H ) ∨¬ BIP ( H ) ∨ ¬ N ON BIP ( G ) ∨ ¬ HOM ( G , H ) (26)For the graphs G = ( V G , E G ) with | V G | = n and H = ( V H , E H ) with | V H | = m we can rewrite this formula as follows: φ ( V G , E G , V H , E H ) = ∃ i < n ¬ V G ( i ) ∨ ∃ i < j < n (( ¬ E G ( i, j ) ∨ ¬ E G ( j, i )) (I) ∧ ( E G ( i, j ) ∨ E G ( j, i ))) ∨ ∃ i < n E G ( i, i ) ∨∃ i < m ¬ V H ( i ) ∨ ∃ i < j < m (( ¬ E H ( i, j ) ∨ ¬ E H ( j, i )) (II) ∧ ( E H ( i, j ) ∨ E H ( j, i ))) ∨ ∃ i < n E H ( i, i ) ∨∀ W H , U H ≤ m (cid:0) ∃ i < m (( ¬ W H ( i ) ∨ U H ( i )) ∧ ( W H ( i ) ∨ ¬ U H ( i ))) ∨ (III) ∃ i < j < m ( E H ( i, j ) ∧ ( ¬ W H ( i ) ∨ ¬ U H ( j )) ∧ ( ¬ W H ( j ) ∨ ¬ U H ( i ))) (cid:1) ∨∀ k ≤ n (2 | ( k − ∀ V C k = k, ∀ E C k < k (cid:0) ( ∃ i < k ¬ V C k ( i ) ∨∃ i < j < k (( ¬ E C k ( i, j ) ∨ ¬ E C k ( j, i )) ∧ ( E C k ( i, j ) ∨ E C k ( j, i ))) ∨∃ i < k E C k ( i, i )) ∨ ( ¬ E C k (0 , k − ∨ ∃ i < ( k − 1) (IV) ¬ E C k ( i, i + 1) ∨ ∃ i, j < ( k − 1) ( j = i + 1 ∧ E C k ( i, j ))) ∨ ∀ Z ≤ h k − , n − i ( ¬ M AP ( k, n, Z ) ∨ ∃ i , i < k ∃ j , j < nE C k ( i , i ) ∧ Z ( h i , j i ) ∧ Z ( h i , j i ) ∧ ¬ E G ( j , j ))) (cid:1) ∨∀ Z ′ ≤ h n − , m − i (cid:0) ¬ M AP ( n, m, Z ′ ) ∨ ∃ i , i < n, ∃ j , j < m (V)( E G ( i , i ) ∧ Z ′ ( h i , j i ) ∧ Z ′ ( h i , j i ) ∧ ¬ E H ( j , j )) (cid:1) In strict form (with all string quantifiers occur in front) the formula φ ( V G , E G , V H , E H )looks like: φ ( V G , E G , V H , E H ) = ∀ W H , U H ≤ m, ∀ V C k ≤ n, ∀ E C k ≤ n , ∀ Z ≤ h k − , n − i , ∀ Z ′ ≤ h n − , m − i [ ψ ( n, m, V G , V H , W H , U H , V C k , E G , E H , E C k , Z, Z ′ )] , (27)where ψ ( n, m, V G , V H , W H , U H , V C k , E G , E H , E C k , Z, Z ′ ) is the Σ B -formula. Thus,by Lemma 6 one can translate it into a family of short propositional formulas.For every free string variable X , | X | = n X in the formula ψ we introduce propo-sitional variables p X , p X , ..., p Xn X − where p Xi is intended to mean X ( i ). The firsttwo parts (I),(II) of the formula φ ( V G , E G , V H , E H ) say that G , H are not graphs.Free number variables here are n, m , free string variables are V G , V H , E G , E H .For graph G , (I) translates into: (cid:2) n − _ i =0 ( ¬ p V G i ) (cid:3) ∨ (cid:2) n − _ j =0 j − _ i =0 ( ¬ p E G h i,j i ∨ ¬ p E G h j,i i ) ∧ ( p E G h i,j i ∨ p E G h j,i i ) (cid:3) ∨ (cid:2) n − _ i =0 ( p E G h i,i i ) (cid:3) (28)And for graph H , (II) translates into: (cid:2) m − _ i =0 ( ¬ p V H i ) (cid:3) ∨ (cid:2) m − _ j =0 j − _ i =0 ( ¬ p E H h i,j i ∨ ¬ p E H h j,i i ) ∧ ( p E H h i,j i ∨ p E H h j,i i ) (cid:3) ∨ (cid:2) m − _ i =0 ( p E H h i,i i ) (cid:3) (29)The third part (III) of the formula φ ( V G , E G , V H , E H ) is about the graph H not being bipartite, free number variable here is m , free string variables are W H , U H , E H . The translation of (III) is: (cid:2) m − _ i =0 ( ¬ p W H i ∨ p U H i ) ∧ ( p W H i ∨ ¬ p U H i ) (cid:3) ∨ (cid:2) m − _ j =0 j − _ i =0 p E H h i,j i ∧ ( ¬ p W H i ∨ ¬ p U H j ) ∧ ( ¬ p W H j ∨ ¬ p U H i ) (cid:3) (30)13he fourth part (IV) of the formula φ ( V G , E G , V H , E H ) expresses that G is nota non-bipartite graph. Free number variable here is n , free string variables are V C k , E C k , Z . This complex subformula we split into parts. Firstly, the part ofsubformula saying that C k is not a graph is translated into: (cid:2) k − _ i =0 ( ¬ p V C k i ) (cid:3) ∨ (cid:2) k − _ j =0 j − _ i =0 ( ¬ p E C k h i,j i ∨ ¬ p E C k h j,i i ) ∧ ( p E C k h i,j i ∨ p E C k h j,i i ) (cid:3) ∨ (cid:2) n − _ i =0 ( p E C k h i,i i ) (cid:3) (31)Then the part saying that C k is not a cycle translates into: (cid:2) ¬ p E C k h ,k − i (cid:3) ∨ (cid:2) k − _ i =0 ¬ p E C k h i,i +1 i (cid:3) ∨ (cid:2) k − _ i =0 k − _ j =0 , j = i +1 p E C k h j,i i (cid:3) (32)And the part, saying that Z is not a map or not a homomorphism between C k and G , is translated into: (cid:2) k − _ i =0 n − ^ j =0 ( ¬ p Z h i,j i ) (cid:3) ∨ (cid:2) k − _ i =0 n − _ j =0 n − _ j =0 , j = j ( p Z h i,j i ∧ p Z h i,j i ) (cid:3) ∨ (cid:2) k − _ i ,i =0 n − _ j ,j =0 ( p E C k h i ,i i ∧ p Z h i ,j i ∧ p Z h i ,j i ∧ ¬ p E G h j ,j i ) (cid:3) (33)Finally, to get the translation of the whole subformula we need first to make adisjunction of all formulas (31)-(33) and then make a conjunction on k : n − ^ k =3 , | ( k − "(cid:2) k − _ i =0 ( ¬ p V C k i ) (cid:3) ∨ (cid:2) k − _ j =0 j − _ i =0 ( ¬ p E C k h i,j i ∨ ¬ p E C k h j,i i ) ∧ ( p E C k h i,j i ∨ p E C k h j,i i ) (cid:3) ∨ (cid:2) n − _ i =0 ( p E C k h i,i i ) (cid:3) ∨ (cid:2) ¬ p E C k h ,k − i (cid:3) ∨ (cid:2) k − _ i =0 ¬ p E C k h i,i +1 i (cid:3) ∨ (cid:2) k − _ i =0 k − _ j =0 , j = i +1 p E C k h j,i i (cid:3) ∨ (cid:2) k − _ i =0 n − ^ j =0 ( ¬ p Z h i,j i ) (cid:3) ∨ (cid:2) k − _ i =0 n − _ j =0 n − _ j =0 , j = j ( p Z h i,j i ∧ p Z h i,j i ) (cid:3) ∨ (cid:2) k − _ i ,i =0 n − _ j ,j =0 ( p E C k h i ,i i ∧ p Z h i ,j i ∧ p Z h i ,j i ∧ ¬ p E G h j ,j i ) (cid:3) (34)And the fifth part (V) of the formula φ ( V G , E G , V H , E H ) saying that there isno homomorphism from G to H , with free number variables n, m , free string14ariables Z ′ , E G , E H , is translated into: (cid:2) n − _ i =0 m − ^ j =0 ( ¬ p Z ′ h i,j i ) (cid:3) ∨ (cid:2) n − _ i =0 m − _ j =0 m − _ j =0 , j = j ( p Z ′ h i,j i ∧ p Z ′ h i,j i ) (cid:3) ∨ (cid:2) n − _ i ,i =0 m − _ j ,j =0 ( p E G h i ,i i ∧ p Z ′ h i ,j i ∧ p Z ′ h i ,j i ∧ ¬ p E H h j ,j i ) (cid:3) (35)The family of propositional formulas || ψ ( n, m, V G , V H , W H , U H , V C k , E G , E H , E C k ,Z, Z ′ ) || is therefore the disjunction of formulas (28)-(35) for all possible n , m , n V G , n V H , n W H , n U H , n V C k , n E G , n E H , n E C k , n Z , n Z ′ ∈ N . By Theorem 3 thisfamily of tautologies has polynomial size bounded depth Frege proof.We are now ready to prove our main goal, to show that the formulas ¬ HOM ( G , H ), for any non-bipartite graph G and bipartite graph H , have shortpropositional proofs. Theorem 4 (The main result) . For any non-bipartite graph G and bipartitegraph H the propositional family ||¬ HOM ( G , H ) || has polynomial size boundeddepth Frege proofs.Proof. By the construction above and Theorem 3 the translation of formula(26) has p -size constant-depth Frege proof. If G and H are graphs, then thetranslations of the first two disjuncts in (26) are propositional sentences thatevaluate to 0 and this can be computed in the proof system.Further, because H is bipartite, we can find its two parts W H , U H and evalu-ate accordingly the atoms in the translation of ¬ BIP ( H ) corresponding to W H and U H such that the whole translation of the disjuct ¬ BIP ( H ) becomes false.That is, as before it is a propositional sentence that evaluates to 0. Analogousargument removes the translation of the disjunct ¬ N ON BIP ( G ): substitutefor the atoms corresponding to a homomorphism from an odd cycle for some k values determined by an actual homomorphism from C k into G . This will turnthe translation of the fourth disjunct ¬ N ON BIP ( G ) into a sentence equal to 0as well.To summarize: after these substitutions the first four disjucts in the trans-lation of the formula (26) become propositional sentences evaluating to 0 andthus the whole translation of the formula (26) is equivalent to the translationof ¬ HOM ( G , H ). That is, we obtained polynomial size constant-depth Fregeproof of ||¬ HOM ( G, H ) || . Actually, we can improve a little bit our upper bound result from the Sec. 3.3.1.While reasoning about graph we used convenient for this purpose set-up of two-sorted theory V , including the Comprehension axiom. But actually we canavoid using it in both proofs of Lemmas 3 and 4. For example, in the proofof Lemma 3 instead of declaring the existence of the set Z ′′ ( h i, k i ) ←→ ∃ j IN D scheme for Σ b -formulas. Then there is a theorem: Theorem 5 ([12]) . Suppose that φ ( ~x, ~X ) is a Σ B , DNF -formula such that T ( α ) ⊢ ∀ ~x ∀ ~Xφ ( ~x, ~X ) . Then the propositional family || φ ( ~x, ~X ) || has polynomialsize R ∗ ( log ) -proofs. That is, there is a polynomial p ( ~m, ~n ) such that for all ≤ ~m, ~n ∈ N , φ ( ~x, ~X )[ ~m, ~n ] has a R ∗ ( log ) -refutation of size at most p ( ~m, ~n ) .Further there is an algorithm which finds a R ∗ ( log ) -refutation of φ ( ~x, ~X )[ ~m, ~n ] in time bounded by a polynomial in ( ~m, ~n ) . It is obvious that we can modify a little the formula (27) to become DNF .Thus, the negations of the family of tautologies, expressing that there is nohomomorphism from a non-bipartite graph G to a bipartite graph H have short R ∗ ( log )-refutation in R ∗ ( log ) system, which is essentially a constant-depth Fregesystem with depth 2 and narrow logical terms.Another note, that one of our auxiliary lemmas, Lemma 5, gives us a collat-eral result. The Π B -formula (21): φ ( G ) = ¬ N ON BIP ( G ) ∨ ¬ HOM ( G , K ) , expressing that there is no homomorphism from non-bipartite graph G to com-plete graph K , also could be rewritten in strict form as the universal statementof the Σ B -fragment of V . Thus, the family of tautologies into which one cantranslate this universal statement also has polynomial size R ∗ ( log )-proofs. Es-sentially, the formula (21) means that the sets of bipartite and non-bipartitegraphs are disjoint, since we can define a bipartite graph H as: BIP ( H ) ←→ HOM ( H , K ) . (36)We know that resolution R p -simulates R ∗ ( log ) system (see Lemma 2). Thus,due to the Feasible interpolation theorem 6, there is a p -time algorithm separat-ing bipartite and non-bipartite graphs. Of course, this is well-known but herewe obtain the algorithm as a consequence of the existence of short resolutionproofs. Theorem 6 (The feasible interpolation theorem, [12]) . Assume that the set ofclauses { A , ..., A m , B , ..., B l } for all i ≤ m, j ≤ l satisfies A i ⊆ { p , ¬ p , ..., p n , ¬ p n , q , ¬ q , ..., q s , ¬ q s } ;16 j ⊆ { p , ¬ p , ..., p n , ¬ p n , r , ¬ r , ..., r t , ¬ r t } , and has a resolution refutation with k clauses. Then the implication ^ i ≤ m ( _ A i ) → ¬ ^ j ≤ l ( _ B j ) has an interpolating circuit I ( ~p ) whose size is O ( kn ) . If the refutation is tree-like, I is a formula. Moreover, if all atoms ~p occur only positively in all A i ,then there is a monotone interpolating circuit (or a formula in the tree-like case)whose size is O ( kn ) . In this section we consider another side of the Dichotomy of the H -coloringproblem, namely, N P -complete case for non-bipartite graphs H . Well-studiedexample of the H -coloring problem is the K n -coloring problem, which is essen-tially the n -coloring problem, where K n is a complete graph on n > K n ) is the graph K n +1 : it is im-possible to n -color complete graph with n + 1 vertices. Propositional formula,expressing that there is no homomorphism from K n +1 to K n , can be reduced tothe Pigeonhole Principle formula PHP n +1 n , because essentially trying to find ahomomorphism from K n +1 to K n is trying to map injectively the set [0 , n + 1]to the set [0 , n ]. The PHP n +1 n formula is: ¬ [ ^ i _ j p ij ∧ ^ i ^ j = j ′ ( ¬ p ij ∨ ¬ p ij ′ ) ∧ ^ i = i ′ ^ j ( ¬ p ij ∨ ¬ p i ′ j )] , (37)where ( n + 1) n atoms p ij with i ∈ [ n + 1] and j ∈ [ n ] expressing that i is mappedto j . For PHP n +1 n there is a lot of known lower bounds in different weak proofsystems: Theorem 7 ([8]) . There exist a constant c , c > , so that, for suffisiently large n , every resolution refutation of ¬ PHP n +1 n contains al teast c n different clauses. Theorem 8 (Ajtai 1988, Beame et al. 1992, [10]) . Assume that F is a Fregeproof system and d a constant, and let n > . Then in every depth d F -proof of the formula PHP n +1 n at least n (1 / d different formulas must occur. Inparticular, each depth d F -proof of PHP n must have size at least n (1 / d andmust have at least Ω(2 n (1 / d ) proof steps. We also can consider weak variants of PHP principle, PHP mn , where thenumber m of pigeons is larger then n + 1 (which will be equivalent to non-existence of homomorphism from K m to K n ). Theorem 9 ([13]) . For m > n PHP mn has no polynomial calculus refutation ofdegree d ≤ ⌈ n/ ⌉ . heorem 10 ([11]) . Let c, d and a prime p be fixed, and let q be a number notdivisible by p . Then there is δ > such that for all n large enough it holds:there is m ≤ n such that in every tree-like F cd ( M OD p ) -proof of PHP n + mn atleast exp ( n δ ) different formulas must occur. Thus, we see that even for such an elementary negative instance of N P -complete case of the H -coloring problem, CSP( K n ), the tautology, expressingthat there is no homomorphism from K m to K n , m ≥ n + 1, has no short proofsin many weak proof systems. We have constructed in Sec. 3.3 short proofs of propositional statements ex-pressing that G / ∈ CSP ( H ) for non-bipartite graphs G and bipartite graphs H by translating into propositional logic a suitable formalization of the algorithmfor the p -time case of the H -coloring pronlem. Note that while this algorithmis very simple, it is not AC -computable (parity is easily AC -reducible to thequestion whether or not a graph is bipartite) while our propositional proofs op-erate only with clauses and are thus, in this respect, more rudimentary that thedecision algorithm is.The condition for the p -time case of the H -coloring problem (and the algo-rithm) are so simple that one could perhaps directly construct short proposi-tional proofs and the use of bounded arithmetic may seem redundant. However,we think of this work as a stepping block towards proving analogous result for thefull Dichotomy theorem. Its known proofs rely on universal algebra and formal-izing it in a suitable bounded arithmetic theory ought to accessible while directpropositional formalization looks unlikely. For this reason we used boundedarithmetic here as a common framework. Moreover, this framework generallyallows to obtain some collateral results that help to compose a complete pictureof the problem.An interesting issue which we left out is to prove a lower bound not justfor suitable H (as we did in Sec.4) but for all H which fall under the N P -complete case of the Dichotomy theorem. If CSP( H ) is N P -complete then,unless N P = coN P , no proof system can prove in p -size all valid statements G / ∈ CSP( H ). In addition, if the N P -completeness of the class can be formalizedin a theory T and we have a lower bound for the proof system corresponding to T (see [12] for this topic) then one can use it to construct G for which the lowerbound holds. This uses well-known part of proof complexity but we do feelthat it adds to our understanding of the proof complexity of CSP; it is rather atransposition of known results via known techniques. For this reason we do notpursue here this avenue of research. Acknowledgements: I would like to thank my supervisor Jan Krajíček forhelpful comments that resulted in many improvements to this paper. Also, I’mgrateful to Pavel Pudlák, Neil Thapen, and others for the opportunity to presentthis work in the Institute of Mathematics of the Czech Academy of Sciences and18or further discussion. Finally, I would like to thank Albert Atserias and JoannaOchremiak, whose paper [1] inspired this direction of research. References [1] Albert Atserias and Joanna Ochremiak. Proof complexity meets algebra,2017. Available at https://arxiv.org/abs/1711.07320.[2] Andrei A. Bulatov. H-coloring dichotomy revisited. Theoretical ComputerScience , 349(1):31 – 39, 2005. Graph Colorings.[3] Andrei A. Bulatov. A dichotomy theorem for nonuniform csps, 2017. Avail-able at https://arxiv.org/abs/1703.03021.[4] Stephen A. Cook. Feasibly constructive proofs and the propositional calcu-lus. 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