Proof Complexity of Substructural Logics

In this paper, we investigate the proof complexity of a wide range of substructural systems. For any proof system $\mathbf{P}$ at least as strong as Full Lambek calculus, $\mathbf{FL}$, and polynomially simulated by the extended Frege system for some infinite branching super-intuitionistic logic, we present an exponential lower bound on the proof lengths. More precisely, we will provide a sequence of $\mathbf{P}$-provable formulas $\{A_n\}_{n=1}^{\infty}$ such that the length of the shortest $\mathbf{P}$-proof for $A_n$ is exponential in the length of $A_n$. The lower bound also extends to the number of proof-lines (proof-lengths) in any Frege system (extended Frege system) for a logic between $\mathsf{FL}$ and any infinite branching super-intuitionistic logic. We will also prove a similar result for the proof systems and logics extending Visser's basic propositional calculus $\mathbf{BPC}$ and its logic $\mathsf{BPC}$, respectively. Finally, in the classical substructural setting, we will establish an exponential lower bound on the number of proof-lines in any proof system polynomially simulated by the cut-free version of $\mathbf{CFL_{ew}}$.


Introduction
Propositional proof complexity, as a new independent field, was established predominantly to address the fundamental unsolved problems in computational complexity. Starting steps in this systematic study were taken by Cook and Reckhow. In their seminal paper [6], they defined a propositional proof system, PPS, as a polynomial-time computable function whose range is the set of all classical propositional tautologies. Then, they defined a polynomially bounded proof system as a PPS having a short proof for any tautology, i.e., a proof whose length is polynomially bounded by the length of the tautology itself. They proved that the existence of a polynomially-bounded proof system for the classical logic is equivalent to NP " coNP. Accordingly, if for any PPS there are super-polynomial lower bounds on the lengths of proofs, as a result NP will be different from coNP and consequently, P will be different from NP. Since these are considered to be major open problems in computational complexity, providing super-polynomial lower bounds for all PPS's gained momentum in the field of proof complexity of classical proof systems. Thus far, exponential lower bounds on proof lengths have been established in many different propositional proof systems, including resolution [10], cutting planes [19], and bounded-depth Frege systems [4]. For more on the lengths of proofs, see [15].
Aside from the extensive study of some well-known classical proof systems, recently there have been some investigations into the complexity of proofs in non-classical logics on account of their various applications, their power in expressibility and their essential role in computer science. Therefore, it is important to fully understand the inherent complexity of proofs in non-classical logics, considering specially the impact that lower bounds on lengths of proofs will have on the performance of the proof search algorithms. Moreover, from the computational complexity perspective, the study of complexity of proofs in non-classical logics is associated with another major computational complexity problem, namely the NP vs. PSPACE problem. Various results have been acheived in this area, for instance exponential lower bounds for the intuitionistic and modal logics [12], and for modal and intuitionistic Frege and extended Frege systems [13]. A comprehensive overview of results concerning proof complexity of non-classical logics can be found in [5].
In the realm of non-classical logics, substructural ones are logics originally defined by the systems where some or all of the usual structural rules are absent. These logics include relevant logics, linear logic, fuzzy logics, and many-valued logics. However, the field is more ambitious than any limited investigation of possible effects of the structural rules. The purpose of the study of substructural logics is to uniformly investigate the non-classical logics that originated from different motivations. Complexity-theoretically, several substructural logics are PSPACE-complete, for instance the multiplicative-additive fragment of linear logic, MALL [17], and full Lambek calculus, FL [14]. Check also the PSPACE-hardness for a wide range of substructural logics and PSPACE-completeness for a class of extensions of FL in [11]. Some complexity results about the decision problem of some fragments of Visser's basic propositional logic, BPC, and formal propositional logic, FPL are also studied in [21].
In this paper, we will study the proof complexity of proof systems for substructural logics and basic logic, and hence a wide-range class of proof systems. More precisely, we will start with an arbitrary proof system P at least as strong as FL (or BPC) and polynomially simulated by an extended Frege system for some super-intuitionistic infinite branching logic L, denoted by L´EF. For such a P, we will provide a sequence of hard P-tautologies, namely a sequence of P-provable formulas tA n u 8 n"1 with length polynomial in n such that their shortest P-proofs are exponentially long in n. Our method is using a sequence of intuitionistic tautologies for which we know there exists an exponential lower bound on the length of their proofs in any L´EF, where L is infinite branching. Since these formulas are not necessarily provable in P, the essential step is their modification so that they become provable in FL (or BPC) and hence in P, while they remain hard for L´EF. Finally, since L´EF is shown to be polynomially as strong as P, the length of any P-proofs of the P-tautologies must be exponential in n. Furthermore, using the same FL-tautologies, one can infer an exponential lower bound also for proof systems polynomially simulated by CFLé w , where the superscript "´" means the sequent calculus does not have the cut rule.

Preliminaries
In this section we provide some background and also some new notions needed in the future sections. Throughout the paper we mainly work with substructural logics and we follow [8] as the canonical source for the study of the theory of such logics. Nevertheless, to make the paper as self-contained as possible, we include all necessary background information.

Substructural logics
Consider the propositional language t^, _,˚, J, K, 1, 0, {, z, Ñu. The logical connective˚is called fusion and the connectives { and z are called left and right residuals, respectively. Throughout the paper, small Roman letters, p, q, . . ., are reserved for propositional variables, Greek small letters φ, ψ, . . ., and Roman capital letters A, B, . . ., are meta-variables for formulas and Greek capital letters Γ, Σ, . . ., are meta-variables for (possibly empty) finite sequences of formulas, separated by commas (unless specified otherwise).
Consider the following set of rules over sequents of the form Γ ñ ∆. The meta-variable Γ is called the antecedent of the sequent and ∆ its succedent. All the rules are presented in the form of schemes. Therefore, an instance of a rule is obtained by substituting formulas for lower case letters and finite (possibly empty) sequences of formulas for upper case letters.
Initial sequents: Structural rules: Weakening rules: previous rules except for: the commutative implication rules, the structural rules out of the set S, and the initial sequents for K and J. Define FL K over the language L˚Y tKu as FL with the initial sequent for K. Figure  2.1, which is adapted from [8], shows the relationship between these sequent calculi. Moreover, define the system weak Lambek, denoted by WL, over the language t1, K,^, _,˚, zu similar to FL K , excluding the following rules: pL{q, pR{q, and pLzq. Some other useful calculi are introduced in Table 1.
For a sequent calculus S and a set of sequents Γ by the notation S`Γ we mean the sequent calculus obtained from adding the elements of Γ as initial sequents to S. By the notation φ ô ψ we mean both φ ñ ψ and ψ ñ φ.
The formula φ n is defined inductively. φ 1 is φ and by φ n`1 , we mean φ˚φ n .
Multi-conclusion. In the absence of the exchange rules, there are many possible ways to define the multi-conclusion rules for fusion and implications and the systems are in some respects more difficult than the commutative case. In this paper, we only consider the commutative case and hence we will use the language t^, _,˚, Ñ, 0, 1u, assuming only one implication. The interpretation of any sequent Γ ñ ∆ is defined as IpΓ ñ ∆q "˚Γ Ñ p˚ ∆q, where φ is an abbreviation for φ Ñ 0. Let S Ď te, i, o, cu such that e P S. By CFL S , we mean the system consisting of the multi-conclusion version of the previous rules except for: the structural rules out of the set S, the non-commutative implication rules, and the initial sequent for K. By CFL S´, we mean CFL S without the cut rule. For a sequent calculus S, proofs and provability of formulas are defined in the usual way, and by its logic, S, we mean the set of provable formulas in it, i.e., all formulas φ such that pñ φq is provable in S. Remark 2.1. Note that if e P S, it is easy to show that in the system FL S the two connectives ψ{φ and φzψ are provably equivalent and we can denote them by the usual connective φ Ñ ψ. Moreover, it is also possible to axiomatize the system FL S over the language L˚´t{, zu Y tÑu, using all the rules in FL S , replacing the non-commutative implication rules with the commutative ones. Similarly, in the sequent calculus FL ecw , the formulas φ˚ψ and φ^ψ become equivalent and 0 and 1 will be equivalent to K and J, respectively. Hence, it is possible to axiomatize FL ecw over the language L " t^, _, Ñ, J, Ku, using all the initial sequents and rules for the corresponding connectives. This is nothing but the usual system LJ, for the intuitionistic logic, IPC. A similar type of argument also applies on CFL S when e P S and for CFL ecw " LK, where LK is the sequent calculus for the classical logic, CPC. Finally, it is worth mentioning that the logic CFL e is essentially equivalent to the multiplicative additive linear logic, MALL, introduced by Girard [9] and the logic FL e is known as its intuitionistic version, called IMALL. CFL ew is sometimes called the monoidal logic and CFL ec is essentially equivalent to the relevant logic R without the distributive law. For more details, see [8].
The sequent calculi FL S and CFL S enjoy cut elimination. This fact has been shown independently by several authors. For instance, see [9], [16], and [18].
Definition 2.2. We say a formula φ is provable from a set of formulas Γ in the logic FL and we write it as Γ $ FL φ when the sequent ñ φ is provable in the sequent calculus FL by adding all ñ γ for γ P Γ as initial sequents, i.e., tñ γu γPΓ $ FL ñ φ. When Γ is the empty set we sometimes write FL $ φ for $ FL φ.
We will use a similar convention that for S a logic or a proof system or a sequent calculus, $ S φ and S $ φ are used interchangeably. If the sequent φ 1 , . . . , φ n ñ ψ is provable in the sequent calculus FL, then we have tφ 1 , . . . , φ n u $ FL ψ. However, the converse, which is the deduction theorem, does not hold. In fact, unlike the classical and intuitionistic logics, most other substructural logics, including FL, do not have a deduction theorem. We will see in Theorem 2.5 that only a restricted version of the deduction theorem (called parametrized local deduction theorem) holds for $ FL . However, note that by definition for a formula φ we have $ FL φ if and only if ñ φ is provable in the sequent calculus FL.
So far, we have defined some basic substructural logics with their sequent calculi. Now, it is a good point to introduce a substructural logic in a general sense. From now on, when no confusion occurs, we will write the fusion φ˚ψ as φψ. For a set of formulas Γ Y tφu, define Γ $ L φ as Γ Y L $ FL φ. We have $ L φ is equivalent to φ P L.
When L is the logic FL, then $ FL defined above will be the same as the one defined in Definition 2.2. Therefore, there will be no ambiguity. As a corollary of Theorem [8, 2.16], it is shown that the above definition can be replaced by the following: a substructural logic over FL is a set of formulas closed under both substitution and $ FL . It is easy to see that for any subset S of te, i, o, cu, the logic FL S is a substructural logic. We can see that if $ FL S Γ ñ φ, then Γ $ FL S φ. This can be easily shown since we can simulate each rule in te, i, o, cu by the corresponding axiom below and using the cut rule: peq : pφ˚ψqzpψ˚φq , pcq : φzpφ˚φq , piq : φz1 , poq : 0zφ Moreover, note that for all the sequent calculi in Table 1, the sequent calculus FL is present and hence all their corresponding logics are closed under the conditions in Definition 2.3. Therefore, they are substructural logics.
It can be easily shown ([8, Lemma 2.13.]) that if a sequent Γ, α, β, Σ ñ φ is provable in FL, then the following sequents are also provable in FL: The following theorem states the parametrized local deduction theorem for FL.
Theorem 2.5. [8, Theorem 2.14.] Let L be a substructural logic and Φ Y Ψ Y tφu be a set of formulas. Then, for some n, where each γ i pψ i q is an iterated conjugate of a formula ψ i P Ψ.
Remark 2.6. Note that the definition of $ L in Definition 2.3 depends on the sequent calculus FL and not the mere logic FL. The reason is that $ FL , which is defined in Definition 2.2, uses the sequent calculus FL. It is possible to use Theorem 2.5 to provide the following proof system-independent definition of for some n and m and some A i P Γ and B i P Γ Y tLu.

Super-basic logics
In [22], Visser introduced basic propositional logic, BPC, and formal propositional logic, FPL, to interpret implication as formal provability. In [20], Ruitenberg reintroduced BPC via philosophical reasons and produced its predicate version, BQC. In the following, we present the sequent calculus introduced in [2] for the logic BPC, denoted by BPC. It was shown that this proof system is complete with respect to transitive persistent Kripke models. Since formulas pA Ñ pA Ñ Bqq Ñ pA Ñ Bq and pA Ñ pB Ñ Cqq Ñ pB Ñ pA Ñ Cqq are not always true in transitive models (the former formula corresponds to the contraction rule and the latter to the exchange rule), one may view BPC as a substructural logic. In this logic modus ponens is weakened and hence BPC is weaker than the intuitionistic logic. BPC is also connected with the modal logic K4 via Gödel's translation T , as shown in [22].
The language of BPC is L " t^, _, J, K, Ñu and negation is defined as the abbreviation for φ " φ Ñ K. In this subsection capital Greek letters denote (possibly empty) multisets of L-formulas. By Γ, φ or φ, Γ, we mean the multiset ΓYtφu. Sequents of BPC are of the same form of the sequents of LK and they are interpreted in the same way, i.e., IpΓ Ñ ∆q " Ź Γ Ñ Ž ∆. The initial sequent and rules of BPC are as follows: Note that since we are assuming multisets of formulas, in this proof system the exchange rules are built in. Moreover, the left and right weakening and contraction rules are admissible in this proof system and it enjoys the cut elimination (see [2] Lemma 2.2, Lemma 2.12, Lemma 2.14, and Theorem 2.17, respectively). An extension of BPC augmented by the axiom J Ñ K ñ K is given in [3], denoted by EBPC. It is shown that this proof system is complete with respect to transitive persistent Kripke models that are serial [3]. Logic of the sequent calculus BPC is defined as the set of all formulas φ such that BPC $ pñ φq, and is denoted by BPC. In a similar way, we can define the logic of the sequent calculus EBPC which we denote by EBPC. It is shown that BPC Ř EBPC Ř IPC [3].
Definition 2.7. We say a formula φ is provable from a set of formulas Γ in the logic BPC and we write it as Γ $ BPC φ when the sequent ñ φ is provable in the sequent calculus BPC by adding all ñ γ for γ P Γ as initial sequents.
Remark 2.8. Note that although the modus ponens rule is neither present nor admissible in the sequent calculus BPC, its logic BPC admits the modus ponens rule. I.e., if φ P BPC and φ Ñ ψ P BPC, then ψ P BPC. The reason is that if φ Ñ ψ P BPC then BPC $ pñ φ Ñ ψq. By cut elimination, there exists a cut-free proof of pñ φ Ñ ψq in BPC. Then by induction on the structure of this cut-free proof we can show that BPC $ φ ñ ψ. Finally, since φ P BPC, we have BPC $ pñ φq, and then using the cut rule we get BPC $ pñ ψq which means ψ P BPC. The same property also holds for the logic EBPC. The proof is an easy consequence of the completeness of EBPC with respect to serial transitive persistent Kripke models. Definition 2.9. Let L be a set of L-formulas. L is a super-basic logic (over BPC) if it is closed under substitution and satisfies the following conditions: piq L includes all formulas in BPC, Note that $ L φ is equivalent to φ P L. One direction is obvious; if φ P L then $ L φ. For the other direction, we will prove a stronger result that if Γ $ L φ then Ź Γ Ñ φ P L. This can be proved using induction on the structure of the proof. For this matter, we transform every rule of BPC into a BPC-provable formula. To complete the proof of the other direction, since Ź Γ " J for Γ " H, we have J Ñ φ P L, which by modus ponens implies φ P L.
As an example, using Remark 2.8, both BPC and EBPC are super-basic logics. Moreover, super-intuitionistic logics (changing the first condition by including all formulas in IPC) are also super-basic, since BPC Ă IPC and they are closed under modus ponens.
For a logic L and a set of formulas Γ, by L`Γ we mean the smallest logic containing L and all the substitutions of formulas in Γ. We can define Jankov's logic, KC, as follows: it is the smallest logic containing IPC and the weak excluded middle formula, i.e., KC " IPC` p _ p. The condition on the Kripke models for this logic is being directed. The axioms BD n are defined in the following way: The logic of bounded depth BD n is then defined as IPC`BD n . Define logic T k as We say a superintuitionistic logic L has finite branching if there exists a number k such that L has branching less than or equal to k, otherwise we call it infinite branching. We will not use the following theorem by Jeřábek in our future discussions. However, it is worth mentioning since it presents a nice characterization of super-intuitionistic infinite branching logics.

Frege and extended Frege systems
The purpose of this section is to introduce Frege and extended Frege systems for substructural and super-basic logics. For that matter, we will recall or generalize some basic concepts in proof complexity. For more background the reader may consult [15].
Definition 3.1. Let L be a set of finite strings over a finite alphabet. A (propositional) proof system for L is a polynomial-time function P with the range L. Any string π such that Ppπq " φ is a P-proof of the string φ, sometimes written as P $ π φ. We denote proof systems by bold-face capital Roman letters.
By length of a formula φ, or a proof π, we mean the number of symbols it contains and we denote it by |φ| and |π|, respectively. We usually consider proof systems for a logic L. The usual Hilbert-style systems with finitely many axiom schemes and Gentzen's sequent calculi are instances of propositional proof systems, because they are complete and in polynomial time one can decide whether a finite string is a proof in the system or not.
Definition 3.2. Let P and Q be two proof systems with the languages L P and L Q , respectively. Let tr be a polynomial-time translation function from the strings in the language L P to the strings in the language L Q . We will denote it by tr : L P Ñ L Q . We say that the proof system Q simulates the proof system P (or P is simulated by Q, or Q is at least as strong as P) with respect to tr, if there is a function f such that Qpf pπqq " trpPpπqq and we denote it by P ď tr Q. We say that the proof system Q polynomially simulates (p-simulates) the proof system P (or P is polynomially simulated by Q) with respect to tr, if the function f is also polynomially bounded in length, i.e., there exists a polynomial qpnq such that |f pπq| ď qp|π|q. We denote this reduction by P ď tr p Q. In the simpler case that L P Ď L Q and the translation function is the inclusion function, we say Q simulates (p-simulates) P and denote it by P ď Q (P ď p Q). If L P " L Q and the translation function is the identity function, we say that the proof system P and Q are polynomially equivalent when they p-simulate each other. Finally, in a similar manner, for two logics L and M and a translation function tr : L L Ñ L M , by L Ď tr M, we mean that for any φ P L L , if φ P L then trpφq P M.
Note that if we take L " CPC to be the range of both proof systems P and Q, and let the translation function tr to be the identity function, we reach Cook and Reckhow's original definition of p-simulation in [6].
In the following we present a translation function t that enables us to carry out results in systems with the language L to systems with the language L˚. This translation function is nothing but bringing back the structural rules: Definition 3.3. Define the function t : L˚Ñ L as follows: For Γ, a finite sequence of formulas γ 1 , γ 2 , . . . , γ n , by Γ t we mean the sequence of formulas γ t 1 , γ t 2 . . . , γ t n . It is easy to see that |φ t | " |φ|.
The following lemma, which will be used in the future sections, is an example of how the translation t works. It expresses the relation between sequents provable in the sequent calculus WL and the translated version of the sequents in the system BPC.
Lemma 3.4. Let Γ be a sequence of formulas and A be a formula. Then Proof. It can be shown by an easy induction on the structure of the proof. Note that as mentioned earlier, the left contraction rule and both right and left weakening rules are derivable in BPC and exchange rules are built in. As an example, suppose the last rule in the proof of Γ ñ A is pR˚q: Remark 3.5. For any substructural logic L and any super-intuitionistic logic M, it is easy to see that L Ď t M implies the stronger form: In the following we will define Frege and extended Frege systems for substructural and super-basic logics.
Definition 3.6. An inference system P is defined by a set of rules of the form φ 1 . . . φ m φ where φ i and φ are formulas. A P-proof , π, of a formula φ from a set of formulas X is defined as a sequence of formulas φ 1 , . . . , φ n " φ, where φ i P X or φ i is obtained by substituting some φ j 's, j ă i, in a rule of the system P. Each φ i is called a step or a line in the proof π. The number of lines of a proof π is denoted by λpπq and it is clear that it is less than or equal to the length of the proof (the number of symbols in the proof). The set of all provable formulas in P is called its logic. If there is a P-proof for φ from assumptions φ 1 , . . . , φ n , we write φ 1 , . . . , φ n $ P φ. Specially, for every rule of the above form we have φ 1 , . . . , φ m $ P φ. Finally, the number of lines of the proof π is defined as the number of formulas in the proof π. Definition 3.7. In a sequent calculus a line in a proof is a sequent of the form Γ ñ ∆. We denote the number of proof-lines in a proof π in a sequent calculus by λpπq, as in an inference systems. It is obvious that the number of proof-lines of a sequent is less than or equal to the length of the proof.
There are two measures for the complexity of proofs in proof systems. The first one is the length of the proof and the other is the number of proof steps (also called proof-lines). This only makes sense for proof systems in which the proofs consist of lines containing formulas or sequents. Hilbert-style proof systems, Gentzen's sequent calculi, and Frege systems are examples of such proof systems.
Definition 3.8. Let L be a substructural logic or a super-basic logic. P is called a Frege system for L, if it satisfies the following conditions: p1q P is an inference system, p2q P has finitely many rules, Moreover, a Frege system P is called standard if p3 1 q P is strongly sound: if φ 1 , . . . , φ n $ P φ, then φ 1 , . . . , φ n $ L φ.
We will use the convention that all Frege systems are standard. Note that for the substructural logic L, the relation $ L has the property mentioned in Theorem 2.5. Hilbert-style proof systems for basic substructural logics and for BPC are examples of Frege systems. See Hilbert-style systems HFL S for S a subset of te, i, o, cu in [8, Section 2.5], and a Hilber-style system for BPC due to Došen in [7]. The usual Hilbert-style systems for classical and intuitionistic logics, HK and HJ respectively, are also examples of Frege systems; see [8, Sections 1.3.1 and 1.3.3] for the definitions of these systems. It is easy to see that these systems satisfy the conditions of a standard Frege system. Remark 3.9. Note that, by an easy induction, it can be shown that for a system P to satisfy the condition p3 1 q in Definition 3.8, it is enough to show that each rule of P is standard, i.e., for any rule of P of the form we have φ 1 , . . . , φ n $ L φ. In our discussions in the future sections, we use this condition.
Definition 3.10. An extended Frege system for a substructural logic (or a super-basic logic) L, denoted by L´EF, is a Frege system for L together with the extension axiom which allows formulas of the form p " φ :" ppzφ^φzpq (or p " φ :" pp Ñ φ^φ Ñ pq) to be added to a derivation with the following conditions: p is a new variable not occurring in φ, in any lines before p " φ, or in any hypotheses to the derivation. It can however appear in later lines, but not in the last line.
It is easy to check that the definition of equivalence introduced in Definition 3.10 is closed under substitution, i.e., if A " B then for any formula φpp,qq we have φpA,qq " φpB,qq.
Lemma 3.11. For any two Frege system P and Q for a logic L,there exists a number c such that for any formula φ and any proof π, there exists a proof π 1 such that P $ π φ implies Q $ π 1 φ and λpπ 1 q ď cλpπq. In the case that P and Q are extended Frege systems, they are polynomially equivalent.
Proof. The proof is easy and originally shown in [6]. The reason is that any instance of a rule in P can be replaced by its proof in Q, which has a fixed number of lines. Take c as the largest number of proof-lines of these proofs. Since there are finite many rules in P, finding c is possible. Therefore, λpπ 1 q ď cλpπq. A similar argument also works for the length of the proofs.
As a result of Lemma 3.11, since we are concerned with the number of proof-lines and lengths of proofs, we can talk about "the" Frege (extended Frege) system for L and denote it by L´F (L´EF).
Definition 3.12. A proof in a Frege (extended Frege, Hilbert-style, Gentzenstyle) system is called tree-like if every step of the proof is used at most once as a hypothesis of a rule in the proof. It is called a general (or dag-like) proof, otherwise.
In this paper we will not use this distinction, because throughout the paper all the proofs are considered to be dag-like, which is the more general notion.

A descent into the substructural world
In this section, we will present a sequence of tautologies and then we show they are exponentially hard for any system L´EF for any substructural and super-basic logics. In order to do so, we first provide some sentences provable in the weak system WL. This uniformly provides two sequence of formulas provable in FL K and BPC. In the case of FL K , since the system FL K is conservative over FL and the formulas we are interested in do not contain K, we will automatically have a proof in FL.
To provide tautologies in WL, we pursue the following strategy: First, using the representations tK, 1u for true and false, we encode every binary evaluation of an LK-formula by a suitable WL-proof. Then, using this encoding, we map a certain fragment of LK into the system WL, without any essential change into the original sequent. Finally, applying this map on a certain hard intuitionistic tautology provides the intended hard WLtautology that we are looking for. Let v be a Boolean valuation assigning truth values tt, f u to the propositional variables. For a formula A in the language L, by vpAq we mean the Boolean valuation of A by v, defined in the usual way. The substitution σ v for a formula A is defined in the following way: if an atom is assigned "t" in the valuation v, substitute 1 for this atom in A and if an atom is assigned "f " in v then substitute K for this atom in A. We write A σv for the formula obtained from this substitution. if vpAq " f, then WL $ A σv ô K.
Proof. The proof is simple and uses induction on the structure of the formula A. If it is an atom, then the claim is clear by the definition of A σv . If A " B^C then if vpAq " t we have V pBq " vpCq " t. Therefore, by induction hypothesis we have WL $ B σv ô 1 and WL $ C σv ô 1 Using the following proof-trees in WL we obtain WL $ B σv^C σv ô 1, which is WL $ A σv ô 1. If A " B^C and vpAq " f , then one of the following happens vpBq " t, vpCq " f or vpBq " f, vpCq " t or vpBq " vpCq " f We investigate the first case, the other cases are similar. If vpBq " t and vpCq " f , by induction hypothesis we get WL $ B σv ô 1 and WL $ C σv ô K Therefore, the following are provable in WL where the right sequent is an instance of the axiom for K. Hence, we get WL $ A σv ô K. Finally, if A " B _ C, based on whether vpAq " t or vpAq " f we proceed as before. All the cases are simple, therefore here we only investigate the case where vpAq " vpB _ Cq " t and vpBq " f and vpCq " t, as an example. Using the induction hypothesis for B and C, consider the following prooftrees in WL: The following theorem is our main tool in proving the lower bound and it provides a method to convert classical tautologies to tautologies in WL.
Proof. The theorem states that due to the commutativity of conjunction in classical logic, any order on the elements of I, i.e. the sequence i 1 , . . . , i k , can be used and˚k j"1 pp i j^1 q ñ Appq is provable in WL. However, the order must be fixed throughout the proof. Since Ź i j PI p i j Ñ Appq is a classical tautology, it will be true under any assignment of truth values to the propositional variables, especially the valuation v assigning truth to every p i , for i P I, and falsity to the rest. It is easy to see that under this valuation we have vp Ź i j PI p i j q " t and since we also have vp Ź i j PI p i j Ñ Appqq " t (because the formula is a classical tautology), we get as a result vpAq " t. Therefore, using Lemma 4.2 we obtain WL $ A σv ô 1 and since WL $ñ 1, using the cut rule we get On the other hand if we show WL $ k j"1 pp i j^1 q, A σv ñ A p:q then using the cut rule on the sequents in p‹q and p:q we get We will prove p:q by induction on the structure of the formula A. If A is equal to p i j , for some j where i j P I, then since vpp i j q " t, we have A σv " p σv i j " 1. Therefore, the following proof-tree represents a proof in WL: . pL˚q k j"1 pp i j^1 q, 1 ñ p i j where the first vertical dots means using the rules p1wq and pL^2q consecutively. Note that based on the rule p1wq, we can add 1 in any position on the left hand-side of the sequents. Using this fact together with the rule pL^2q we obtain all formulas in the appropriate order. The second vertical dots represents applications of the rule pL˚q consecutively until one reaches the conclusion. Therefore, we have proved The case where A " p i j where i j R I is easier. Since for such j we have vpp i j q " f , using Lemma 4.2 we get WL $ A σv ô K. Using the initial sequent for K we have WL $˚k j"1 pp i j^1 q, K ñ A and using the cut rule we get p:q.
If Appq " Bppq^Cppq, and the induction hypothesis holds for Bppq and Cppq, i.e., then first using the rule pL^1q for the left sequent and rule pL^2q for the right sequent, and then using the rule pR^q we get If Appq " Bppq _ Cppq then first using the rule pR_ 1 q for the left sequent in p;q and rule pR_ 2 q for the right sequent in p;q, and then using pL_q we get

A brief digression into hard tautologies
The formulas we are going to introduce as our hard tautologies for the system FL´EF and BPC´EF are inspired by the hard formulas for IPC´F introduced by Hrubeš [12] and their negation-free version introduced by Jeřábek [13]. In this subsection, we briefly explain these formulas and what combinatorial facts they represent.
Let us first define formulas Clique n,k and Color n,m which will be used in Hrubeš's formulas. By an undirected simple graph on rns we mean the set of strings of length`n 2˘. We say a graph has a clique when there exists a complete subgraph, which is a subgraph with all possible edges among its vertices. Define Clique n,k to be the set of undirected simple graphs on rns that have a clique of size at least k, and define Color n,m to be the set of garphs on rns that are m-colorable, and they are defined by the following two sets.
The set of clauses denoted by Clique k n pp,qq uses`n 2˘a toms p ij , ti, ju P`n 2˘, one for each potential edge in a graph on rns, and k.n atoms q ui intended to describe a mapping from rks to rns. It consists of the following clauses: ‚ q ui _ q uj , all u P rks and i ‰ j P rns, ‚ q ui _ q vi , all u ‰ v P rks and i P rns, ‚ q ui _ q vj _ p ij , all u ‰ v P rks and ti, ju P`n 2˘.
The set of clauses Color m n pp,rq uses atomsp and n.m more atoms r ia where i P rns and a P rms, intended to describe an m-coloring of the graph. It consists of the following clauses: ‚ r ia _ r ib , all a ‰ b P rms and i P rns, ‚ r ia _ r ja _ p ij , all a P rms and ti, ju P`n 2˘.
Note that every occurrence of atoms p ij in Clique k n pp,qq is positive, or in other words it is monotone inp.
The exponential lower bound for intuitionistic logic is demonstrated in the following theorem due to P. Hrubeš. The main idea is that any short proof for the hard tautology provides a small monotone circuit to decide whether a given graph is a clique or colorable, which we know is a hard problem to decide [1].
We refer to the formulas Θ K n as Hrubeš's formulas. The superscript K in Θ K n stresses that the formulas contain negations. For our purposes, we need to use a negation-free version of Hrubeš's formulas. Define the negation-free Hrubeš formulas for k " t ? nu as follows: pr i,l _r 1 i,l q Ñ β k`1 n pq,r,r 1 qqs.
Notice that Color k n pp,sq " α k n pp,s, sq and Clique k n pp,rq " β k n p p,r, rq. The lower bound of Theorem 4.5 also applies to Θ n [13].
To make Hrubeš's formulas negation-free, Jeřábek introduced new propositional variables s 1 i,l and r 1 i,l to play the role of s i,l and r i,l , respectively. This trick provides some implication-free formulas α k n and β k n in the definition 4.6 to make the formulas Θ n more amenable to the technique that we provided in Section 4.

Weak hard tautologies
The following lemmas are easy observations. The first one states that fusion distributes over disjunction in substructural logics. The second one presents a property of the sequent calculus LK. Lemma 4.8. In the sequent calculus WL we have the following: where I Ď t1, 2,¨¨¨, nu and D I i " Proof. The proof is easy and uses induction on n. Note that in each disjunct in the right hand-side, D I i is either A i or B i , according to the subset I. However, the order of the subscripts must be increasing. For instance, for the case n " 2 we have Lemma 4.9. Suppose α 1 Ñ α 2 and β 1 Ñ β 2 have no propositional variables in common. If the formula α 1^α2 Ñ β 1 _ β 2 is provable in LK, then either α 1 Ñ α 2 or β 1 Ñ β 2 is provable in LK.
Proof. It is an easy corollary of Craig's interpolation theorem.
We are now ready to formulate hard tautologies in WL and prove the lower bound. By˚n´1 i"1˚n´1 j"1 A i,j , we mean that the indices first range over j and then over i, which will result in the lexicographic order, i.e., it has the following form For a set (sequence of formulas) Γ, by Γ we mean the number of elements of the set (the number of formulas the sequence contains). Proof. Let us denote the following formula by A: l"1 pps i,l^1 q_ps 1 i,l^1 qqzα k n pp,s,s 1 qs _ r n´1 i"1 k´1 l"1 ppr i,l^1 q_pr 1 i,l^1 qqzβ k`1 n pq,r,r 1 qs.
for any I Ď tpi, jq | i, j P t1,¨¨¨, n´1uu, such that Q i,j " For simplicity from now on, unless specified otherwise, we will delete the ranges of i, j and l, which are indicated in p:q.
It is easy to see how proving p:q will result in proving the theorem. The reason is the following. Since p:q is provable for any I Ď tpi, jq | i, j P t1,¨¨¨, n´1uu, using the left disjunction rule for 2 pn´1q 2´1 many times on p:q, we get Furthermore, Lemma 4.8 allows us to obtain and using the cut rule and the rule pRzq, we conclude On the other hand, Θ n is provable in LJ, and therefore also provable in LK.
Using the distributivity of conjunction over disjunction we have We consider the first case, the second one being similar. Therefore, suppose the first case holds. Using the cut rule, we have Note that by Theorem 4.3, we can choose any order on˚p i,jqPM pp i,j^1 q provided we do not change it throughout the proof. Since the order is arbitrary, for simplicity we do not explicitly write it down. Equivalently (using the fact that for any formulas A and B, we have WL $ A, B ñ A˚B and then using the cut rule), we have Since this sequent is provable for any U and V such that U Y V " tpi, lq | i ă n, l ă ku, using the left disjunction rule for 2 pn´1qpk´1q´1 many times we get Using Theorem 4.8 and the cut rule we have WL $ ril pps i,l^1 q _ ps 1 i,l^1 qqs , pMpp i,j^1 qq ñ α k n pp,s,s 1 q, and using the rule pRzq we get WL $Mpp i,j^1 q ñ ril pps i,l^1 q _ ps 1 i,l^1 qqs z α k n pp,s,s 1 q.
Now, using the rules pL1q and pL^2q consecutively for N -many times (each time we produce q i,j^1 for each element of N, in the same manner as in the proof of Theorem 4.3) and then using the rule pL˚q for N -many times and in the end using the rule pR_ 1 q we prove p:q.
Note that the formulas Θn depend on the variable k, as well. The reason for our notation is that we are only concerned with the case where k " t ? nu and we will prove the lower bound for this case.
Remark 4.11. It is worth noting that the system WL could have been defined in an alternative way by deleting { instead of z from the language, and having the same initial sequents and rules as FL and leaving the rules pR{q, pL{q, and pLzq out. Then, in a similar manner, the following formulas would be provable in this alternative calculus:  Proof. Clearly, WL is a subsystem of the sequent calculus FL K . Then, using the cut elimination theorem [8,Theorem 7.8] for FL K , and the fact that Θn do not contain K, we obtain the result.
To provide tautologies in BPC, we need the translation function, t, defined in Section 3.

The main theorem
In this section we will present the main result of the paper. We will prove that there exists an exponential lower bound on the lengths of proofs in proof systems for a wide range of logics. Furthermore, we will obtain an exponential lower bound on the number of proof-lines in a broad range of Frege systems. piq Let P be a proof system for a logic with the language L˚such that FL ď P ď t p L´EF. Then there is an exponential lower bound on the lengths of proofs in P.
piiq Let P be a proof system for a logic with the language L such that BPC ď P ď p L´EF. Then there is an exponential lower bound on the lengths of proofs in P.
Proof. piq Since FL $ Θn by 4.10, and FL ď P, the formulas Θn are also provable in P. Take such a proof π, i.e., P $ π Θn. Since P ď t p L´EF, there exists a proof π 1 and a polynomial p, such that L´EF $ π 1 pΘnq t and |π 1 | " pp|π|q. We want to prove L´EF $ pΘnq t Ñ Θ n by a proof whose length is polynomial in n. First, since L is a super-intuitionistic logic, we have L´EF $ u^J Ø u, where u is any of the atoms present in the formulas Θ n . This proof has a fix number of proof-lines in L´EF. The claim then easily follows from the fact that the length of the formula Θ n is also polynomial in n. Therefore, Θ n is provable in L´EF with a proof polynomially long in n and π 1 . By 4.7, any L´EF-proof of Θ n has length at least 2 Ωpn 1{4 q . Therefore, the length of π must be exponential in n. piiq The proof for this part is similar to that of piq. Here, the formulas pΘnq t are provable in BPC and hence in P. Since L´EF polynomially simulates P, we obtain the exponential lower bound using the fact that L´EF $ pΘnq t Ñ Θ n .
The following theorem states an exponential lower bound on the number of proof-lines in a wide range of Frege systems. Proof. To prove piq, note that since FL $ Θn by 4.12, and FL Ď L, we have L $ Θn. Let π be a proof of Θn in L´EF. Using the assumption we will provide a proof π 1 of pΘnq t in M´EF such that λpπ 1 q ď cλpπq. Fix an extended Frege system Q for the logic M. Define the system P as the system consisting of all the rules in Q plus the rules: A t for any rule of L´EF of the form: We have to show that P is an extended Frege system for the logic M. To be more precise, we have to show that P is strongly sound and strongly complete with respect to M, because the other conditions (1 and 2 in Definition 3.8) are obvious. P is strongly complete wrt M, since it contains Q and Q is strongly complete wrt M. For strongly soundness, note that for any rule in L´EF of the form: Hence, all the new rules in P are standard with respect to M. To bound the number of proof-lines, let π " φ 1 , . . . , φ m be a proof for Θn in L´EF. Then, each φ i is either an extension axiom, or it is derived from tφ j 1 , . . . , φ j l u such that all j r 's are less than i. It is clear that π 1 " π t " φ t 1 , . . . , φ t m is a proof in P, since the translation t of the extension axiom of L´EF will be the extension axiom of M´EF and moreover, is an instance of a rule in P. Note that the number of proof-lines stay the same, i.e., λpπq " λpπ 1 q. Therefore, the formula pΘnq t has a proof in P whose number of lines is the same as the number of lines of the proof of Θn in L´EF. Since for any formula φ in the language L˚we have |φ| " |φ t |, therefore the length of π is the same as the length of π 1 . On the other hand, as we observed in the proof of Theorem 5.1, we can show that pΘnq t Ñ Θ n has a proof in P with polynomial number of lines. Gluing these proofs together, we will obtain a proof for Θ n in P. Since any proof for Θ n in P has exponential length (Theorem 4.7), any proof for Θn in L´EF will also have exponential length. Note that the above construction also works for the case of considering Frege systems. It is easy to see that the translation of every proof in L´F will be a proof in M´F, and the number of proof-lines stay the same. Therefore, the bound on the number of proof-lines follows.
For part piiq, using Corollary 4.13, pΘnq t is provable in BPC and hence in L. Fix an extended Frege system Q for the logic M. Add the rules of L´EF to Q. The resulting system, which we denote by P, is an extended Frege system for the logic M. The reason is similar to the argument in the part piq, using the facts that L Ď M and all the rules of L´EF are standard with respect to L. Let π be a proof for pΘnq t in L´EF, therefore, it will also be a proof in P with the same number of lines and same length. Again by gluing the short proof of P $ pΘnq t Ñ Θ n to π, we reach the result as in the proof for part piq. Corollary 5.3. Let S be any subset of te, c, i, ou, and L be FL S , BPC, EBPC, or any of the logics of the sequent calculi in Table 1. Then there is an exponential lower bound on the number of proof-lines in L´F and on the lengths of proofs in L´EF.

The lower bound for sequent calculi
So far, we have provided a lower bound for proof systems for logics as least as strong as FL and polynomially simulated by an extended Frege system for an infinite branching super-intuitionistic logic. It is very desirable to see if the lower bound also applies to proof systems for logics outside this range, for instance their classical counterparts. The result in this section is an attempt in this direction and we reach a positive answer for any proof system polynomially weaker than CFLé w , which is the system CFL ew without the cut rule. For that matter, we first transfer the lower bound from the previous section to the sequent-style proof system FL S for any S Ď te, c, i, ou. Then we use the observation that any cut-free proof of a single-conclusion sequent in the 0-free fragment of CFL ew is also an FL ew -proof. Theorem 6.1. Let Γ be a sequence of formulas γ 1 , . . . , γ m , A a formula and S any subset of te, c, i, ou. If FL S $ π Γ ñ A then there exists a Frege system P for FL S such that P $ π 1 m i"1 γ i zA such that λpπ 1 q " λpπq.
Proof. The proof is similar to the proof of Theorem 5.2. As noted in the discussion after Definition 2.3, since FL S $ Γ ñ A, we have Γ $ FL S A. Therefore, for any Frege system Q for the logic FL S , by strong completeness in Definition 3.8, we have Γ $ Q A. Fix such Q. The method is developing a Frege system P for FL S by transforming all the axioms and rules of the sequent calculus FL S to Frege rules in the new system. For the sake of completeness, we also add Q to the resulting system. Recall that for Γ " H, the formula˚Γ is defined as 1 and for any singleconclusion sequent T " pΓ ñ ∆q by IpT q, the interpretation of the sequent T , we meant˚Γz∆, if ∆ is non-empty, and˚Γz0 for ∆ " H. Now, define P as the system consisting of the rules of Q plus the following rules: for the axiom T in the sequent calculus FL S add IpT q and for any rule in the sequent calculus FL S of the form T m T add the following rule IpT m q IpT q where m " 1 or m " 2. We have to show that P is a Frege system for the logic FL S , i.e., P is strongly sound and strongly complete wrt the logic FL S . First, since Q is strongly complete wrt FL S , then so is P. Now for strongly soundness, we have to show that the new rules are standard wrt FL S . I.e., for any rule of the form in P we have to show IpT 1 q, . . . , IpT m q $ FL S IpT q. However, it is not hard to show that, since in the sequent calculus FL S the cut rule exists, we have Using the corresponding rule, T 1 , . . . , T m $ FL S T , the fact that T $ FL S ñ IpT q, and the cut rule we have ñ IpT 1 q, . . . , ñ IpT m q $ FL S ñ IpT q. Therefore, by definition, IpT 1 q, . . . , IpT m q $ FL S IpT q. Therefore, P is a Frege system for FL S . For the number of proof-lines, note that if π " T 1 , . . . , T n is a proof for T n " pΓ ñ Aq in FL S , then it is easy to see that IpT 1 q, . . . , IpT n q will be a proof for˚m i"1 γ i zA in P. Therefore, λpπ 1 q " λpπq. Corollary 6.2. For any S Ď te, i, o, cu we have FL S $ñ Θn and the number of lines of any proof of this sequent is exponential in n.
By a 0-free formula in CFL ew , we mean a formula only consisting of propositional variables, the constant 1, and the connectives t^, _, Ñ,˚u. Proof. Suppose pΓ ñq has a proof in CFLé w . Since the proof is cut-free and Γ is 0-free, by the subformula property of CFLé w , the whole proof is also 0-free. Therefore, there is no axiom in the proof with an empty succedent, because such an axiom must be in the form p0 ñq, which is not 0-free. Moreover, if the succedent of the conclusion of any rule is empty, then the succedent of at least one of its premises must be empty, as well. The reason is the following. First, note that the last rule is not an axiom, as stated. It cannot be a right rule either, because they always have at least one formula in the succedent of their conclusion. And for the left rules, the claim is evident by a simple case checking. The only non-trivial case to check is pL Ñq which also has such a premise: Υ ñ φ Π, ψ, Σ ñ (L Ñ) Π, φ Ñ ψ, Υ, Σ ñ Therefore, any sequent in the proof with an empty succedent has also a premise with an empty succedent. This is clearly a contradiction.
The following theorem, which is of independent interest, states that for positive formulas, a cut-free proof for a single-conclusion sequent in CFL ew is also a proof for the same sequent in FL ew . Theorem 6.4. Suppose Γ is a sequence of 0-free formulas and A is a 0-free formula. Then any proof π for Γ ñ A in CFLé w is also a proof in FL ew .
Proof. The sketch of the proof is the following: suppose π is a cut-free proof in CFL ew such that all the formulas in the proof are 0-free. Then, along the proof, the number of formulas in the succedent of the sequents does not decrease. The reason lies in the fact that neither the cut rule nor the contraction rules are present. Hence, in the special case that the sequent is also single-conclusion, the succedents of all the sequents in the whole proof will contain exactly one formula. Therefore, the proof is in FL ew . Let π be a proof for Γ ñ A in CFLé w . By induction on the structure of π we will show it is also a proof for the same sequent in FL ew . As stated in the proof of Lemma 6.3, every formula in the proof must be 0-free. If Γ ñ A is an instance of an axiom in CFLé w , then it is either ñ 1 or an instance of the axiom φ ñ φ, which are both also axioms in the sequent calculus FL e . For the induction step, note that the last rule in the proof cannot be p0wq. For all the other rules (except for the rule pL Ñq), it is easy to see that since the conclusion of the rule is single-conclusion, then every premise must also be single-conclusion. It remains to investigate the case where the last rule used in the proof is pL Ñq: There are two possibilities; either Λ is empty and ∆ is equal to A π 1 Υ ñ φ π 2 Π, ψ, Σ ñ A pL Ñq Π, φ Ñ ψ, Υ, Σ ñ A or ∆ is empty and Λ is equal to A π 1 Υ ñ φ, A π 2 Π, ψ, Σ ñ pL Ñq Π, φ Ñ ψ, Υ, Σ ñ A In the former since both premises are single-conclusion, by induction hypothesis, π 1 and π 2 are proofs in FL ew and by applying the rule pL Ñq we obtain a proof for Γ ñ A. On the other hand, the latter cannot happen since the right premise is of the form Π, ψ, Σ ñ and the antecedent of this sequent is 0-free. Therefore, Lemma 6.3 implies that it is not provable in CFLé w .
are provable in CFLé . Moreover, every CFLé -proof ofΘn contains at least 2 Ωpn 1{4 q proof-lines and hence has length exponential in terms of the length of Θn.
Proof. Since formulas Θn are provable in FL 4.10 and therefore in FL ew , they are provable in CFL ew . However, since in FL ew and CFL ew the exchange rules are present, as stated in the preliminaries the connectives z and { are substituted by Ñ. Therefore, the tautologies Θn will have the more recognizable formΘn. Using the cut elimination theorem for CFL ew , formu-lasΘn are also provable in CFLé w . By Theorem 6.4, sinceΘn are 0-free any cut-free proof for these formulas in CFLé w is also a proof in FL ew . However, Theorem 6.1 guaranties these proofs contain at least 2 Ωpn 1{4 q proof-lines and hence the lengths of these proofs are exponential in terms of the length of Θn.
Remark 6.6. So far, we do not have any method to extend the lower bound to the calculus CFL e , where the cut rule is present. Note that since there are no non-trivial lower bounds for the sequent calculus LK, we can not use a similar argument as that in the proof of Theorem 4.10.
Corollary 6.7. For any proof system P such that CFLé ď P ď p CFLé w , there is an exponential lower bound on the length of proofs in P. As a result, there are exponential lower bounds on the length of proofs in sequent calculi CFLé , CFLé i , and CFLé o .
Proof. It follows from Theorem 6.5.