Resolution with Counting: Lower Bounds over Different Moduli

06/25/2018 ∙ by Fedor Part, et al. ∙ Royal Holloway, University of London 0

Resolution over linear equations (introduced in [RT08]) emerged recently as an important object of study. This refutation system, denoted Res(lin_R), operates with disjunction of linear equations over a ring R. On the one hand, the system captures a natural "minimal" extension of resolution in which efficient counting can be achieved; while on the other hand, as observed by, e.g., Krajicek [Kra16] (cf. [IS14,KO18,GK17]), when considered over prime fields, and specifically F_2, super-polynomial lower bounds on Res(lin_F_2) is a first step towards the long-standing open problem of establishing constant-depth Frege with counting gates (AC^0[2]-Frege) lower bounds. In this work we develop new lower bound techniques for resolution over linear equations and extend existing ones to work over different rings. We obtain a host of new lower bounds, separations and upper bounds, while calibrating the relative strength of different sub-systems. We first establish, over fields of characteristic zero, exponential-size lower bounds against resolution over linear equations refutations of instances with large coefficients. Specifically, we demonstrate that the subset sum principle α_1 x_1 +... +α_n x_n = β, for β not in the image of the linear form, requires refutations proportional to the size of the image. Moreover, for instances with small coefficients, we separate the tree and dag-like versions of Res(lin_F), when F is of characteristic zero, by employing the notion of immunity from Alekhnovich-Razborov [AR01], among other techniques. (Abstract continued in the full paper.)

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1 Introduction

1.1 Background

The resolution refutation system is among the most prominent and well-studied propositional proof systems, and for good reasons: it is a natural and simple refutation system, that, at least in practice, is capable of being easily automatized. Furthermore, while being non-trivial, it is simple enough to succumb to many lower bound techniques.

Formally, a resolution refutation of an unsatisfiable CNF formula is a sequence of clauses , where is the empty clause, such that each is either a clause of the CNF or is derived from previous clauses by means of applying the following resolution rule: from the clauses and derive .

The tree-like version of resolution, where every occurrence of a clause in the refutation is used at most once as a premise of a rule, is of particular importance, since it helps us to understand certain kind of satisfiability algorithms known as DPLL algorithms (cf. [Nor15]

). DPLL algorithms are simple recursive algorithms for solving SAT that are the basis of successful contemporary SAT-solvers. The transcript of a run of DPLL on an unsatisfiable formula is a decision tree, which can be interpreted as a tree-like resolution refutation. Thus, lower bounds on the size of tree-like resolution refutations imply lower bounds on the run-time of DPLL algorithms.

In contrast to the apparent practical success of SAT-solvers, a variety of hard instances that require exponential-size refutations have been found for resolution during the years. Many classes of such hard instances are based on principles expressing some sort of counting. One famous example is the pigeonhole principle, denoted , expressing that there is no (total) injective map from a set with cardinality to a set with cardinality if [Hak85]. Another important example is Tseitin tautologies, denoted , expressing that the sum of the degrees of vertices in a graph must be even [Tse68].

Since such counting tautologies are a source of hard instances for resolution, it is useful to study extensions of resolution that can efficiently count, so to speak. This is important firstly, because such systems may become the basis of more efficient SAT-solvers and secondly, in order to extend the frontiers of lower bound techniques against stronger and stronger propositional proof systems. Indeed, there are quite a few works dedicated to the study of weak systems operating with De Morgan formulas with counting connectives; these are variations of resolution that operate with disjunctions of certain arithmetic expressions.

One such extension of resolution was introduced by Raz and Tzameret [RT08] under the name resolution over linear equations in which literals are replaced by linear equations. Specifically, the system R(lin), which operates with disjunctions of linear equations over was studied in [RT08]. This work demonstrated the power of resolution with counting over the integers, and specifically provided polynomial upper bounds for the pigeonhole principle and the Tseitin formulas, as well as other basic counting formulas. It also established exponential lower bounds for a subsystem of R(lin), denoted . Subsequently, Itsykson and Sokolov [IS14] studied resolution over linear equations over , denoted Res(). They demonstrated the power of resolution with counting mod 2 as well as its limitations by means of several upper and tree-like lower bounds. Moreover, [IS14] introduces DPLL algorithms, which can “branch” on arbitrary linear forms over , as well as parity decision trees, and showed a correspondence between parity decision trees and tree-like Res() refutations. In both [RT08] and [IS14] the dag-like lower bound question for resolution over linear equations remained open.

As it happens, resolution over linear equations, holds a special place in the theory of proof complexity: it can be viewed as a natural “minimal” subsystem of important propositional proof systems, as we now explain. Resolution operates with clauses, which are De Morgan formulas (, unbounded fan-in and ) of a particular kind, namely, of depth 1. Thus, from the perspective of the theory of proof complexity, resolution is a fairly weak version of the propositional-calculus, where the latter operates with arbitrary De Morgan formulas. Under a natural and general definition, propositional-calculus systems go under the name Frege systems: they can be (axiomatic) Hilbert-style systems or sequent-calculus style systems. The task of proving lower bounds for general Frege systems is notoriously hard: no nontrivial lower bounds are known to date. Basically, the strongest fragment of Frege systems, for which lower bounds are known are  systems, which are Frege proofs operating with constant-depth formulas. For example, both and do not admit sub-exponential proofs in  [Ajt88, PBI93, KPW95, BS02]. However, if we extend the De Morgan language with counting connectives such as unbounded fan-in mod (-Frege) or threshold gates (-Frege), then we step again into the darkness: proving super-polynomial lower bounds for these systems is a long-standing open problem on what can be characterized as the “frontiers” of proof complexity. In this sense, resolution over linear equations over prime fields and over the integers is interesting as a first step towards -Frege lower and -Frege lower bounds, respectively. Works by Krajíček [Kra17], Garlik-Kołodziejczyk [GK17] and Krajíček-Oliveira [KO18] had suggested possible approaches to attack dag-like Res) lower bounds.

1.2 Our Results

In this paper we continue the study of the power of resolution over linear equations, while extending it to different rings , denoted Res), both finite and infinite. We prove a host of new lower bounds, separations and upper bounds for resolution over linear equations, including dag-like refutations. We focus mainly on finite fields , for different primes , and fields of characteristic , most importantly the rational numbers . Using our notation, R(lin) from [RT08] is simply Res) and Res() from [IS14] is Res).

The refutation system Res) is defined as follows (see [RT08]). The proof-lines of Res) are linear clauses, that is, disjunctions of linear equations. More formally, they are disjunctions of the form:

where is some number (the width of the clause), and . The resolution rule is the following:

from and derive

where , and some linear clauses. A Res) refutation of an unsatisfiable over 0-1 set of linear clauses is a sequence of proof-lines, where each proof-line is either , for , a boolean axiom for a some variable , or was derived from previous proof-lines by the above resolution rule, or by the weakening rule that allows to extend clauses with arbitrary disjuncts, or a simplification rule allowing to discard false constant linear forms (e.g., ) from a linear clause. The last proof-line in a refutation is the empty clause (standing for the truth value false).

We are interested in the following questions:

  • For a given ring , what kind of counting can be efficiently performed in Res) and tree-like Res)?

  • Can dag-like Res) be separated from tree-like Res)?

  • Can tree-like systems for different rings be separated?

In order to be able to do some non-trivial counting in tree-like versions of resolution over linear equations we define a semantic version of the system as follows: Tree-like Res) with semantic weakening.

The system ) is obtained from Res) by replacing the weakening and the simplification rules, as well as the boolean axioms, with the semantic weakening rule (the symbol will denote in this work semantic implication with respect to 0-1 assignments): () . Let be the characteristic of the ring . In case , deciding whether an -linear clause is a tautology (that is, holds for every 0-1 assignment to its variables) is at least as hard as deciding whether a 3-DNF is a tautology (because over characteristic linear equations can express conjunction of three conjuncts). For this reason ) proofs cannot be checked in polynomial time and thus ) is not a Cook-Reckhow proof system unless (namely, the correctness of proofs in the system cannot necessarily be checked in polynomial-time, as required by a Cook-Reckhow propositional proof system [CR79]).

The reason for studying ) is mainly the following: Let be an arbitrary set of tautological -linear clauses. Then, lower bounds for ) imply lower bounds for tree-like Res() with formulas in as axioms. For example, in case  is a field of characteristic 0, the possibility to do counting in tree-like Res() is quite limited. For instance, we show that requires an exponential-size in refutation (Corollary 5.3). On the other hand, such contradictions do admit short tree-like Res) refutations in the presence of the following generalized boolean axioms (which is a tautological linear clause):

(1)

where is the image of under 0-1 assignments. Similar to the way the Boolean axioms state that the possible value of a variable is either zero or one, the axiom states all the possible values that the linear form can take. If a lower bound holds for ) it also holds, in particular, for tree-like Res() with the axioms , and this makes ) a useful system, for which lower bounds against are sufficiently interesting.

Lower bounds and separations in characteristic zero.

First, we show that over any field , whenever is unsatisfiable (over 0-1 assignments), it requires dag-like Res) refutations proportional to the image of the linear form (under 0-1 assignments). Note that expresses the subset sum principle: iff there is a subset of the integral coefficients whose sum is precisely . Our result implies an exponential-lower bound for dag-like Res) refutations, for large enough coefficients, as follows: [Theorem 5.2; Dag-like lower bound] If is a field of characteristic zero, then Res) refutations of are of size .

This lower bound is proved by showing (see Lemma 5.2) that every (dag- or tree-like) refutation of a subset sum principle of the form can be transformed without much increase in size into a normal form refutation (in dag- or tree-like, resp.): a derivation of  , combined with a successive use of resolution with to derive the empty clause. This then provides the desired lower bound whenever is sufficiently large.

The idea behind the normal form transformation is as follows: given a refutation in which the only non-Boolean axiom is , we defer all resolution steps using this axiom. Namely, we mimic the same refutation had we not used resolution with . We show that in this case, each clause in the resulting refutation is essentially a weakening of the original clause, possibly weakened by (i.e., is a disjunction with) disjunct of the form , for some constant . This concludes the argument, since the last clause must be such a tautological weakening of the empty clause, but such a tautology ought to be a weakening of the subset sum principle itself (note that every proof-line in the transformation is a tautology (over 0-1 assignments), since the only axioms used throughout the derivation are the Boolean axioms).

Moreover, we prove an exponential-size lower bound on tree-like ) refutations of the pigeonhole principle for every field (including finite fields). This extends a previous result by Itsykson and Sokolov [IS14] for tree-like Res). Together with the polynomial upper bound for refutations in dag-like Res) for fields of characteristic zero demonstrated in [RT08], our results establish a separation between dag-like Res) and tree-like ) for characteristic zero fields.

[Theorem 5.4; Pigeonhole principle lower bounds] Let be any field. Then every tree-like ) refutation of has size .

[Theorem 3.2; Raz-Tzameret [RT08]; Short dag-like pigeonhole principle refutations] For every ring of characteristic zero there exists a Res) refutation of of polynomial size.

To prove this theorem, as well as some other lower bounds, we extend the Prover-Delayer game technique as originated in Pudlak-Impagliazzo [PI00] for resolution, and developed further by Itsykson-Sokolov [IS14] for Res), to general rings, including characteristic zero rings. Specifically, to prove Theorem 5.4 we need to prove that Delayer’s strategy from [IS14] is successful over any field. This argument is new, and uses a result of Alon-Füredi [AF93]

about the hyperplane coverings of the hypercube.

We prove another separation between dag-like Res) and tree-like ), as follows. We define the Image Avoidance principle to be:

where . In words, the Image Avoidance principle expresses the contradictory statement that for every , equals some element in .

[Theorem 3.1] For every ring  and every linear form , there are polynomial-size Res) refutations of . [Theorem 5.3] Let , where , and let be a field of characteristic zero. Then, the following hold:

  1. Any tree-like ) refutation of is of size at least .

  2. Any tree-like Res) derivation of the clause is of size at least .

Together with the above mentioned normal form lemma (Lemma 5.2) that we establish for (both dag- and tree-like) refutations of  , we get the following:

[Corollary 5.3] Let and be as in the previous theorem. Then the shortest tree-like Res) refutation of is of size at least .

The lower bounds in Theorem 5.3 and Corollary 5.3 are novel applications of the Prover-Delayer game argument, together with the notion of immunity from Alekhnovich and Razborov [AR01], as we now explain briefly. Let be a linear form as in Theorem 5.3. We consider two instances of the Prover-Delayer game: for and for . A position in the games is determined by a set of linear non-equalities of the form , which we think of as the set of non-equalities learned up to this point by Prover. For each of the two games we define Delayer’s strategy in such a way that for an end-game position, there is a satisfiable subset such that for some , and Delayer earns at least coins. Because is of characteristic zero, it follows that and thus the -immunity of ([AR01]) implies . To conclude, we use the standard argument that shows that if Delayer always earns coins, then the shortest proof is of size at least .

Table 1 sums up our knowledge up to this point with respect to characteristic 0 fields.

c—[1pt]c—c—c—c—c & & & & &

[1pt]

t-l Res) & & & & &

t-l ) & poly & poly & & & poly

Res) & poly & & poly & poly [RT08] & poly

Table 1: Lower bounds for fields of characteristic . The notation t-l Res) stands for tree-like Res). The rightmost column describes bounds on derivations, in contract to refutations.

Lower bounds and separations in finite fields.

Apart from the lower bounds for the pigeonhole principle that hold both for infinite and finite fields as discussed above, we prove a separation between tree-like Res) (resp. tree-like )) and tree-like Res) (resp. tree-like )) for every pair of distinct primes and every . The separating instances are mod Tseitin formulas , which are reformulations of the standard Tseitin graph formulas for counting mod . Furthermore, we establish an exponential lower bound for tree-like ) on random -CNFs.222We thank Dmitry Itsykson for telling us about the lower bound for random -CNF for the case of tree-like Res), that was proved by Garlik and Kołodziejczyk using size-width relations (unpublished note). Our result extends Garlik and Kołodziejczyk’s result to all finite fields. Similar to their result, we use a size-width argument and simulation by the polynomial calculus to establish the lower bound.

We now explain the general lower bounds argument, followed by the precise results. The lower bounds for tree-like Res) for finite fields are obtained via a variant of the size-width relation for tree-like Res) together with a translation to polynomial calculus over the field , denoted [CEI96], such that Res) proofs of width are translated to proofs of degree (the width of a clause is defined to be the total number of disjuncts in a clause). This establishes the lower bounds for the size of tree-like Res) proofs via lower bounds on degrees.

We show that

where is what we call the principal width, which counts the number of linear equations in clauses after the identification of those defining parallel hyperplanes, and denotes the minimal size of a tree-like Res) refutation of .

Specifically, over finite fields the following upper and lower bounds provide exponential separations:

[Theorem 6; Size-width relation] Assume is an unsatisfiable CNF formula. The following relation between principal width and size holds for tree-like ): . If is a finite ring, then the same relation holds for the (standard) width of a clause .

This extends to every field a result by Garlik-Kołodziejczyk [GK17, Theorem 14] who showed a size-width relation for a system denoted tree-like , which is a system extending tree-like Res) by allowing arbitrary constant-depth De Morgan formulas as inputs to (XOR gates) (though note that our result does not deal with arbitrary constant-depth formulas).

[Theorem 6] Let be a field and be a Res) refutation of an unsatisfiable CNF formula . Then, there exists a refutation of (the arithmetization of) of degree .

[Corollary 6; Tseitin mod lower bounds] For any fixed prime there exists a constant such that the following holds. If , is a -regular directed graph satisfying certain expansion properties, and is a finite field such that , then every tree-like ) refutation of the Tseitin mod formula has size .

[Corollary 6; Random -CNF formulas lower bounds] Let be a randomly generated -CNF with clause-variable ratio , and where is such that , and let  be a finite field. Then, every tree-like ) refutation of has size

with probability

.

The tree-like Res) upper bounds for mod Tseitin formulas in the case stem from the following proposition: [Proposition 3.1; Upper bounds on unsatisfiable linear systems] Let  be a ring and assume that the linear system , where is a matrix over , has no solutions (over ). Let be a CNF formula encoding the linear system . Then, there exists tree-like Res) refutations of of size polynomial in the sum of sizes of encodings of all coefficients in .

Table 2 shows the results for Res) over finite fields.

l—[1pt]c—c—c—c—c & & & & random -CNF &

[1pt]

t-l Res) &

?
& poly & & &

t-l Res() & poly [IS14] & poly [IS14] & & & [IS14]

t-l ) & poly & poly &

?
&

?
&

Table 2: Lower bounds over finite fields. Here is -regular graph and is the clause density (number of clauses divided by the number of variables), stands for a linear system over that has no 0-1 solutions in the first and the third rows, and in the second row the linear system is over . The notation stands for in the first and the third rows and for in the second raw. t-l Res) stands for tree-like Res), and are primes (in the second raw we assume ). Circled “?” denotes an open problem. The results marked with [IS14] were proved in the corresponding paper. All other results are from the current work.

Nondeterministic linear decision trees. There is well-known size preserving (up to a constant factor) correspondence between tree-like resolution refutations for unsatisfiable formulas and decision trees, which solve the following problem: given an assignment for the variables of , determine which clause is falsified by querying values of the variables under the assignment . In Itsykson-Sokolov [IS14] this correspondence was generalized to tree-like refutations and parity decision trees. In this paper we extend the correspondence to a correspondence between tree-like Res() (and )) derivations to nondeterministic linear decision trees (NLDT).

NLDTs for an unsatisfiable set of linear clauses are binary rooted trees, where every edge is labeled with a non-equality for a linear form and every leaf is labeled with a linear clause , which is violated by the non-equalities on the path from the root to the leaf.

As a way to prove lower bounds on Res) tree-like for we show a correspondence between tree-like Res) refutations and what we call nondeterministic linear decision trees (NLDT). This correspondence generalizes the one between tree-like Res() and parity decision trees described in [IS14]. Just like parity decision trees arise as protocols of execution of the DPLL() procedure on some unsatisfiable formula , NLDTs may be considered as protocols of execution of a DPLL-like procedure on . [Theorem 4] If is an unsatisfiable CNF formula, then every tree-like Res) or tree-like ) refutation can be transformed to an NLDT for of the same size up to a constant factor, and vise versa.

?

Such a procedure, denoted and referred to as a DPLL() procedure, has two inputs: a formula in CNF and a system of linear non-equalities (informally, stands for the “partial assignment” accumulated until the current step). Initially runs on with the empty system . At each step chooses some non-equality and makes recursive calls and until either (i) we can conclude that is unsatisfiable (in which case we have to backtrack as our “current assignment” is not good), or (ii) there is only one satisfying assignment to in which case we will backtrack if this assignment satisfies , or (iii) contradicts a clause in (that is, there is a clause in such that all the satisfying assignments to falsifies this clause). We prove that NLDTs are related to tree-like Res) as follows:

2 Preliminaries

2.1 Notation

Denote by the set . We use to denote variables, both propositional and algebraic. Let be a linear form (equivalently, an affine function) over a ring , that is, a function of the form with . We sometimes refer to a linear form as a hyperplane, since a linear form determines a hyperplane. We denote by the image of under 0-1 assignments to its variables; , where .

For a set of clauses or linear clauses, denotes the set of variables occurring in and let Vars denote the set of all variables.

Let be a matrix over a ring. We introduce the notation for a system of linear non-equalities, where a non-equality means (note the difference between , which stands for , for all rows in , and , which stands for , for some row in ).

If is a linear form over  and is a matrix over , denote by the sum of sizes of encodings of coefficients in and by the sum of sizes of encodings of elements in .

If is a linear clause (i.e., a disjunction of linear equations), denote by the set of non-equalities . Conversely, if is a set of non-equalities, denote .

If is a set of linear clauses over a ring and is a linear clause over , denote by and semantic entailment over 0-1 and -valued assignments respectively.

Let be a linear form not containing the variable . If is a linear clause, denote by the linear clause, which is obtained from by substituting for everywhere in . If is a set of clauses, denote . We define a linear substitution to be a sequence such that linear forms does not depend on . For a clause or a set of clauses we define .

2.2 Propositional Proof Systems

A clause is an expression of the form , where is a literal, where a literal is a propositional variable or its negation . A formula is in Conjunctive Normal Form (CNF) if it is a conjunction of clauses. A CNF can thus be defined simply as a set of clauses. The choice of a reasonable binary encoding of sets of clauses allows us to define the language of unsatisfiable propositional formulas in CNF. We sometimes interpret an element in UNSAT as a formula and sometimes as a set of clauses. Dually, a formula is in Disjunctive Normal Form (DNF) if it is a disjunction of conjunctions of literals and TAUT is the language of tautological propositional formulas in DNF. There is a bijection between TAUT and UNSAT, which preserves the size of the formula, given by negation.

A formula is in -CNF (resp. -DNF) if it is in CNF (resp. DNF) and every clause (resp. conjunct) has at most literals. -UNSAT (resp. -TAUT) is the language of unsatisfiable (resp. tautological) formulas in -CNF (resp. -DNF).

[Cook-Reckhow propositional proof system [CR79]] A propositional proof system is a polynomial time computable onto function . -proofs of are elements in . Definition 2.2 can be generalized to arbitrary languages: proof system for a language is polynomial time computable onto function . In particular, a refutation system is a proof system for UNSAT. Post-composition with negation turns a propositional proof system into a refutation system and vise versa.

Denote by , and alternatively by , the size of the binary encoding of a proof in a proof system . For and a refutation system denote by (we sometimes omit the subscript when it is clear from the context) the minimal size of a -refutation of .

The resolution system (which we denote also by Res) is a refutation system, based on the following rule, allowing to derive new clauses from given ones:

(Resolution rule). A resolution derivation of a clause from a set of clauses is a sequence of clauses such that for every either or is obtained from previous clauses by applying the resolution rule. A resolution refutation of is a resolution derivation of the empty clause from , which stands for the truth value False.

(Resolution)

(Weakening) A resolution derivation is tree-like if every clause in it is used at most once as a premise of a rule. Accordingly, tree-like resolution is the resolution system allowing only tree-like refutations.

Let be a field. A polynomial calculus [CEI96] derivation of a polynomial from a set of polynomials is a sequence such that for every either , or is obtained from previous polynomials by applying one of the following rules: ()

() .

A polynomial calculus refutation of is a derivation of . The degree of a polynomial calculus derivation is the maximal total degree of a polynomial appearing in it. This defines the proof system for the language of unsatisfiable systems of polynomial equations over . It can be turned into a proof system for -UNSAT via arithmetization of clauses as follows: is represented as .

2.3 Hard Instances

2.3.1 Pigeonhole Principle

The pigeonhole principle states that there is no injective mapping from the set to the set for . Elements of the former and the latter sets are referred to as pigeons and holes, respectively. The CNF formula, denoted , encoding the negation of this principle is defined as follows. Let the set of propositional variables correspond to the mapping from to , that is, iff the pigeon is mapped to the hole. Then , where are axioms for pigeons and are axioms for holes.

Weaker (namely, easier to refute) versions of are obtained by augmenting it with the functionality axioms () or the surjectivity axioms ().

2.3.2 Mod Tseitin Formulas

We use the version given in [AR01] (which is different from the one in [BGIP01, RT08]). Let be a directed -regular graph. We assign to every edge a corresponding variable . Let . The Tseitin mod formulas are the CNF encoding of the following equations for all :

(2)

Note that we use the standard encoding of boolean functions as CNF formulas and the number of clauses, required to encode these equations is . is unsatisfiable if and only if . To see this, note that if we sum (2) over all nodes we obtain precisely which is different from ; but on the other hand, in this sum over all nodes each edge appears once with a positive sign as an outgoing edge from and with a negative sign as an incoming edge to , meaning the the total sum is 0, which is a contradiction.

In particular, are the classical Tseitin formulas [Tse68] and , where is the constant function (for all ), expresses the fact that the sum of total degrees (incoming outgoing) of the vertices is even.

The proof complexity of Tseitin tautologies depends on the properties of the graph . For example, if is just a union of (the complete graphs on vertices), then they are easy to prove. On the other hand, they are known to be hard for some proof systems if satisfies certain expansion properties.

Let be an undirected graph. For define . Consider the following measure of expansion for :

is -expander if is -regular and . There are explicit constructions of good expanders. For example:

[Lubotzky et. al [LPS88]] For any , there exists an explicit construction of -regular graph , called Ramanujan graph, which is -expander for any .

[Alekhnovich-Razborov [AR01]] For any fixed prime there exists a constant such that the following holds. If , is a -regular Ramanujan graph on vertices (augmented with arbitrary orientation of its edges) and , then for every function such that every refutation of has degree .

2.3.3 Random k-CNFs

A random -CNF is a formula with variables that is generated by picking randomly and independently clauses from the set of all clauses.

[Alekhnovich-Razborov [AR01]] Let and is such that . Then every refutation of has degree with probability for any field .

3 Resolution with Linear Equations over General Rings

In this section we define and outline some basic properties of systems that are extensions of resolution, where clauses are disjunctions of linear equations over a ring : . Disjunctions of this form are called linear clauses.

The rules of Res) are as follows (cf. [RT08]):

() (Resolution)

() (Simplification)

(Weakening) where are linear forms over and are linear clauses. The Boolean axioms are defined as follows:

A Res) derivation of a linear clause from a set of linear clauses is a sequence of linear clauses such that for every either or is a Boolean axiom or is obtained from previous clauses by applying one of the rules above. A Res) refutation of an unsatisfiable set of linear clauses is a Res) derivation of the empty clause (which stands for false) from . The size of a Res) derivation is the total size of all the clauses in the derivation, where the size of a clause is defined to be the total number of occurrences of variables in it plus the total size of all the coefficient occurring in the clause. The size of a coefficient when using integers (or integers embedded in characteristic zero rings) will be the standard size of the binary representation of integers.

In this definition we assume that is a non-trivial () ring such that there are polynomial-time algorithms for addition, multiplication and taking additive inverses.

Along with size, we will be dealing with two complexity measures of derivations: width and principal width.

A clause has width and principal width where identifies -linear forms and if they define parallel hyperplanes, that is, if or for some . For , the measure associated with a Res) derivation is . For , denote by the minimal value of over all Res) refutations .

Res) is sound and complete. It is also implicationally complete, that is if is a set of linear clauses and is a linear clause such that , then there exists a Res) derivation of from .

Proof:.

The soundness can be checked by inspecting that each rule of Res) is sound. Implicational completeness (and thus completeness) follows from Proposition 4. If is a set of linear clauses and , where , is a linear clause such that , then there exists tree-like Res() derivation of from of size .

Proof:.

The proof is by induction on . . Then . If then , otherwise if it is derivable in one step from . let and pick some . Denote, respectively, and the shortest dag-like and tree-like derivations of from where . ∎

We now define two systems of resolution with linear equations over a ring, where some of the rules are semantic: ) and Sem-Res(). ) is obtained from Res) by replacing the boolean axioms with , discarding simplification rule and replacing the weakening rule with the following semantic weakening rule:

() (Semantic weakening)

The system Sem-Res() has no axioms except for , and has only the following semantic resolution rule:

() (Semantic resolution)

It is easy to see that , where denotes that polynomially simulates .

In contrast to the case (see [IS14]), for rings  with both ) and Sem-Res() are not Cook-Reckhow proof systems, unless : The following decision problem is -complete: given a linear clause over a ring R with decide whether it is a tautology under 0-1 assignments.

Proof:.

Consider a 3-DNF and encode every conjunct as the equation , where . Then is tautological if and only if the disjunction of these linear equations is tautological (that is, for every 0-1 assignment to the variables at least one of the equations hold, when the equations are computed over a ring with characteristic zero or finite characteristic bigger than 3). ∎

We leave it as an open question to determine the complexity of verifying a correct application of the semantic weakening in case or in case and . In the case the negation of a clause is a system of linear equations and thus the existence of solutions for it can be checked in polynomial time. Therefore ) is a Cook-Reckhow propositional proof system. The definitions of Res), ) and Sem-Res() coincide with the definitions of syntactic , and from [IS14], respectively333There is, however, one minor difference in the formulation of syntactic and Res): the former does not have the boolean axioms, but has an extra rule (addition rule).. As showed in [IS14], Res), ) and Sem-Res() are polynomially equivalent.

We now show that if , then ) is polynomially bounded as a proof system for -UNSAT (that is, admits polynomial-size refutation for every instance): If , then dag-like ) and tree-like Sem-Res() are polynomially bounded (not necessarily Cook-Reckhow) propositionally proof systems for 3-UNSAT.

Proof:.

Let . Given define where . The linear clause is a tautology (under 0-1 assignments) and thus can be derived in ) in a single step as a weakening of or resolving with in tree-like Sem-Res().

In tree-like Sem-Res() the disjunct can be eliminated from by a single resolution with , thus the empty clause is derived by a sequence of resolutions of with .

Similarly, the disjuncts are eliminated from in ), but with a few more steps. Let be the empty clause and . Assume is derived and assume without loss of generality, that