1 Introduction
1.1 Background
The resolution refutation system is among the most prominent and wellstudied 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 nontrivial, 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 treelike 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 SATsolvers. The transcript of a run of DPLL on an unsatisfiable formula is a decision tree, which can be interpreted as a treelike resolution refutation. Thus, lower bounds on the size of treelike resolution refutations imply lower bounds on the runtime of DPLL algorithms.
In contrast to the apparent practical success of SATsolvers, a variety of hard instances that require exponentialsize 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 SATsolvers 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 treelike 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 treelike Res() refutations. In both [RT08] and [IS14] the daglike 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 fanin 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 propositionalcalculus, where the latter operates with arbitrary De Morgan formulas. Under a natural and general definition, propositionalcalculus systems go under the name Frege systems: they can be (axiomatic) Hilbertstyle systems or sequentcalculus 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 constantdepth formulas. For example, both and do not admit subexponential proofs in [Ajt88, PBI93, KPW95, BS02]. However, if we extend the De Morgan language with counting connectives such as unbounded fanin mod (Frege) or threshold gates (Frege), then we step again into the darkness: proving superpolynomial lower bounds for these systems is a longstanding 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], GarlikKołodziejczyk [GK17] and KrajíčekOliveira [KO18] had suggested possible approaches to attack daglike 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 daglike 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 prooflines 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 01 set of linear clauses is a sequence of prooflines, where each proofline is either , for , a boolean axiom for a some variable , or was derived from previous prooflines 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 proofline 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 treelike Res)?

Can daglike Res) be separated from treelike Res)?

Can treelike systems for different rings be separated?
In order to be able to do some nontrivial counting in treelike versions of resolution over linear equations we define a semantic version of the system as follows: Treelike 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 01 assignments): () . Let be the characteristic of the ring . In case , deciding whether an linear clause is a tautology (that is, holds for every 01 assignment to its variables) is at least as hard as deciding whether a 3DNF 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 CookReckhow proof system unless (namely, the correctness of proofs in the system cannot necessarily be checked in polynomialtime, as required by a CookReckhow 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 treelike Res() with formulas in as axioms. For example, in case is a field of characteristic 0, the possibility to do counting in treelike Res() is quite limited. For instance, we show that requires an exponentialsize in refutation (Corollary 5.3). On the other hand, such contradictions do admit short treelike Res) refutations in the presence of the following generalized boolean axioms (which is a tautological linear clause):
(1) 
where is the image of under 01 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 treelike 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 01 assignments), it requires daglike Res) refutations proportional to the image of the linear form (under 01 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 exponentiallower bound for daglike Res) refutations, for large enough coefficients, as follows: [Theorem 5.2; Daglike 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 treelike) 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 treelike, 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 nonBoolean 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 proofline in the transformation is a tautology (over 01 assignments), since the only axioms used throughout the derivation are the Boolean axioms).
Moreover, we prove an exponentialsize lower bound on treelike ) refutations of the pigeonhole principle for every field (including finite fields). This extends a previous result by Itsykson and Sokolov [IS14] for treelike Res). Together with the polynomial upper bound for refutations in daglike Res) for fields of characteristic zero demonstrated in [RT08], our results establish a separation between daglike Res) and treelike ) for characteristic zero fields.
[Theorem 5.4; Pigeonhole principle lower bounds] Let be any field. Then every treelike ) refutation of has size .
[Theorem 3.2; RazTzameret [RT08]; Short daglike 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 ProverDelayer game technique as originated in PudlakImpagliazzo [PI00] for resolution, and developed further by ItsyksonSokolov [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 AlonFüredi [AF93]
about the hyperplane coverings of the hypercube.
We prove another separation between daglike Res) and treelike ), 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 polynomialsize Res) refutations of . [Theorem 5.3] Let , where , and let be a field of characteristic zero. Then, the following hold:

Any treelike ) refutation of is of size at least .

Any treelike 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 treelike) refutations of , we get the following:
[Corollary 5.3] Let and be as in the previous theorem. Then the shortest treelike Res) refutation of is of size at least .
The lower bounds in Theorem 5.3 and Corollary 5.3 are novel applications of the ProverDelayer 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 ProverDelayer game: for and for . A position in the games is determined by a set of linear nonequalities of the form , which we think of as the set of nonequalities 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 endgame 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.
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 treelike Res) (resp. treelike )) and treelike Res) (resp. treelike )) 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 treelike ) on random CNFs.^{2}^{2}2We thank Dmitry Itsykson for telling us about the lower bound for random CNF for the case of treelike Res), that was proved by Garlik and Kołodziejczyk using sizewidth relations (unpublished note). Our result extends Garlik and Kołodziejczyk’s result to all finite fields. Similar to their result, we use a sizewidth 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 treelike Res) for finite fields are obtained via a variant of the sizewidth relation for treelike 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 treelike 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 treelike Res) refutation of .
Specifically, over finite fields the following upper and lower bounds provide exponential separations:
[Theorem 6; Sizewidth relation] Assume is an unsatisfiable CNF formula. The following relation between principal width and size holds for treelike ): . 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 GarlikKołodziejczyk [GK17, Theorem 14] who showed a sizewidth relation for a system denoted treelike , which is a system extending treelike Res) by allowing arbitrary constantdepth De Morgan formulas as inputs to (XOR gates) (though note that our result does not deal with arbitrary constantdepth 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 treelike ) refutation of the Tseitin mod formula has size .
[Corollary 6; Random CNF formulas lower bounds] Let be a randomly generated CNF with clausevariable ratio , and where is such that , and let be a finite field. Then, every treelike ) refutation of has size
with probability
.The treelike 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 treelike 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.
Nondeterministic linear decision trees. There is wellknown size preserving (up to a constant factor) correspondence between treelike 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 ItsyksonSokolov [IS14] this correspondence was generalized to treelike refutations and parity decision trees. In this paper we extend the correspondence to a correspondence between treelike 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 nonequality for a linear form and every leaf is labeled with a linear clause , which is violated by the nonequalities on the path from the root to the leaf.
As a way to prove lower bounds on Res) treelike for we show a correspondence between treelike Res) refutations and what we call nondeterministic linear decision trees (NLDT). This correspondence generalizes the one between treelike 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 DPLLlike procedure on . [Theorem 4] If is an unsatisfiable CNF formula, then every treelike Res) or treelike ) 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 nonequalities (informally, stands for the “partial assignment” accumulated until the current step). Initially runs on with the empty system . At each step chooses some nonequality 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 treelike 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 01 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 nonequalities, where a nonequality 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 nonequalities . Conversely, if is a set of nonequalities, denote .
If is a set of linear clauses over a ring and is a linear clause over , denote by and semantic entailment over 01 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).
[CookReckhow 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. Postcomposition 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 treelike if every clause in it is used at most once as a premise of a rule. Accordingly, treelike resolution is the resolution system allowing only treelike 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 .
[AlekhnovichRazborov [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 kCNFs
A random CNF is a formula with variables that is generated by picking randomly and independently clauses from the set of all clauses.
[AlekhnovichRazborov [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 nontrivial () ring such that there are polynomialtime 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 treelike 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 daglike and treelike 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 SemRes(). ) 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 SemRes() 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 SemRes() are not CookReckhow 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 01 assignments.
Proof:.
Consider a 3DNF 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 01 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 CookReckhow propositional proof system. The definitions of Res), ) and SemRes() coincide with the definitions of syntactic , and from [IS14], respectively^{3}^{3}3There 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 SemRes() are polynomially equivalent.
We now show that if , then ) is polynomially bounded as a proof system for UNSAT (that is, admits polynomialsize refutation for every instance): If , then daglike ) and treelike SemRes() are polynomially bounded (not necessarily CookReckhow) propositionally proof systems for 3UNSAT.
Proof:.
Let . Given define where . The linear clause is a tautology (under 01 assignments) and thus can be derived in ) in a single step as a weakening of or resolving with in treelike SemRes().
In treelike SemRes() 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
Comments
There are no comments yet.