1 Definition of proof systems
The main example of a logical propositional calculus to keep in mind is a Frege system. It is any calculus operating with propositional formulas over a complete basis of logical connectives (i.e. all Boolean functions can be defined in the language), having a finite number of sound axiom schemes and inference rules that are implicationally complete. The latter term means that if a formula is a logical consequence of formulas then it can be derived from them in the calculus. An example of a complete language is the DeMorgan language with constants and connectives and . We shall denote by TAUT the set of tautologies in this language and we shall tacitly assume that , with formulas being encoded by binary strings in some natural way.
There is a number of such systems described in logic text-books and they are often called Hilbert-style, referring to Hilbert’s work in proof theory [14, 15, 16]. The form of calculi is based on Frege’s , hence the name Cook and Reckhow  chose for this class of propositional calculi.
The calculi are sound (every provable formula is a tautology) and complete (every tautology is provable). In addition, the key property singled out by  is that to recognize whether a string of symbols is a valid proof in the calculus or not is computationally feasible: it can be done by a p-time algorithm. This leads to the following fundamental definition.
Definition 1.1 (Cook-Reckhow )
A propositional proof system is any p-time computable function
Any such that is called an -proof of .
Cook and Reckhow [6, Def.1.3] actually define more generally a proof system for any by the condition , and consider proof systems for the set of tautologies in any fixed language.
A Frege system can be represented by a function which takes a string and maps it to the last formula of , if is a sequence of formulas that forms a valid -proof, or to constant if is not an -proof. The soundness of implies that and its completeness implies the opposite inclusion .
It is easy to see that a number of other classes of propositional calculi considered in mathematical logic literature fit the definition in the same sense as Frege systems do. These calculi include resolution, sequent calculus or natural deduction. Less usual examples of propositional proof systems can be constructed as follows. Take a consistent first-order theory axiomatized by a finite number of axioms and axiom schemes that is sound and contains some simple base theory (in order to guarantee both the correctness and the completeness) and interpret it as a proof system: a proof of formula is a proof in the theory of the formalized statement . Yet another examples are logic calculi that are set-up to prove the unsatisfiability of formulas: these can be interpreted as proof systems by accepting a refutation of as a proof of .
In addition, the general form of the definition allows us to interpret various calculations in algebra as propositional proofs. Here it is more natural to speak about refutation systems. If we have a CNF formula that is a conjunction of clauses , we can represent each by a constraint of an algebraic form and use a suitable algebraic calculus to derive the unsolvability of the formula. For example, a clause
together with the requirement that we look for solution can be represented by polynomial equations
the first equation states that the clause contains a true literal while the last three equations force solutions over any integral domain. In this case we can use a calculus deriving elements of the ideal generated by the equations representing similarly all clauses of the formula, trying to derive as a member of the ideal and thus demonstrating the unsolvability of the equations and hence the unsatisfiability of the formula.
Another approach is to represent the clause as integer linear inequalities
and use some integer linear programing algorithm to derive the unsolvability of the system of inequalities representing the whole CNF formula. It is a great advantage of Definition1.1 that it puts all these quite different formal system under one umbrella.
Proof systems can be also defined equivalently in a relational form. A relational propositional proof system is a binary relation that we interpret as the provability relation is a proof of . It is required that it is p-time decidable and that for any formula it holds:
This is closer in form to logical calculi (and can be represented by the function version as Frege systems were before) but it is equally general: a functional proof system is represented by the relation .
A proof system is p-bounded iff there exists such that for all , ,
In the relational form this would read
and combining this with the soundness we get
The right-hand side expression has the well-known general form in which any set can be defined. Hence we get as a simple but important corollary to the definition the following statement (the second equivalence uses Cook’s theorem: the -completeness of SAT, cf.Cook ).
Theorem 1.2 (Cook - Reckhow )
A p-bounded proof system exists iff iff .
This theorem determines
Problem 1.3 (Main problem of proof complexity)
Is there a p-bounded proof system for TAUT?
By Theorem 1.2 showing that no p-bounded proof system exists would imply, in particular, that because is closed under complementation. On the other hand, defining a p-bounded proof system would allow to witness various -properties by short witnesses (-proofs);  mentions the property that two graphs are not isomorphic.
One may consider variants of the definition of proof systems when the provability relation is not necessarily decidable by a p-time algorithm but only by more general algorithm; for example, using some randomness. My view is that this changes the basic problems of proof complexity substantially. While it may link propositional proof systems with various other proof systems considered in different parts of complexity theory, it is not clear that it will shed light on proof complexity proper. This may change if some of these other parts of complexity theory advance significantly on their own fundamental open problems.
The Cook-Reckhow definition is handy for establishing Theorem 1.2 and the connection to complexity theory but the reader may wonder if it does not deviate from logical form of calculi too much. In fact, it can be shown that every proof system can be p-simulated (in the sense of the next section) by a Frege system whose set of axioms is not given just by a finite number of axiom schemes but is possibly infinite but easy to recognize (in p-time, in particular) sparse subset of TAUT. Doing this precisely is rather technical and we refer the reader to [24, 20, 23].
2 Simulations among proof systems
When studying the problem whether some proof system is p-bounded it is useful to be able to compare two proof systems with respect to their efficiency. The following two notions333P-simulations are also defined in Cook . are aimed at that.
Definition 2.1 (Cook-Reckhow )
Let be two proof systems. A simulation of by is any function
such that for all , , for some independent constant and all , and such that
Simulation is p-simulation if it is p-time computable.
Proof system (p-)simulates ( and in symbols, respectively) iff there is a (p-)simulation of by .
In other words, the statement that says that if we replace by we can speed-up proofs at most polynomially, while the statement that says that we can even efficiently translate -proofs into -proofs. Both these relations are quasi-orderings (we get partial orderings after factoring by the equivalence relations of mutual simulations).
There are other options how to define a quasi-ordering of proof systems. In particular, if we did not insist in Definition 1.1 that all proof systems prove tautologies in the same language (we have defined TAUT using the DeMorgan language only) but allowed tautologies in different languages then a (p-)simulation should allow to translate also formulas and not just proofs. By insisting that the target set is TAUT we forced that such a translation of formulas is incorporated into the definition of particular proof systems that may operate with formulas in other languages or even with polynomials or other objects. In fact, considering instead of propositional proof systems proof systems for any -complete set we ought to allow p-reductions between such sets and TAUT.
However, for positive results (as is Theorem 2.2 bellow) p-simulations allow to formulate the strongest possible statements while strongest negative results (obtained by proving a super-polynomial lower bound for -proofs of formulas for which there are polynomial size -proofs) talk about super-polynomial speed-ups and hence about the non-existence of simulations. Thus the two types of simulations serve their purpose very well.
Cook and Reckhow  compared various logical proof systems in terms of p-simulations; the following statement summarizes their most memorable results in this respect.
Theorem 2.2 (Cook-Reckhow)
All Extended Frege systems in all languages p-simulate each other.
Frege systems and propositional parts of natural deduction and of sequent calculus mutually p-simulate each other.
Extended Frege system EF and Tseitin’s Extended resolution ER are p-equivalent and they are p-simulated by any Frege system with the substitution rule.
Extended Frege systems EF were defined in  in a direct analogy with Extended resolution ER of Tseitin . Any such system starts with a Frege system and allows, in addition, to abbreviate formulas by new atoms and use these in proofs. In particular, during an EF-proof we can take a new atom (an extension atom) not used so far and not occurring in the target formula to be proved, any formula not containing , and introduce the equivalence (represented in the language of the system) as a new extension axiom. Note that EF is not a Frege system as the introduction of extension axioms does not fit the schematic way Frege axioms are supposed to be defined. The first statement in the theorem is a weaker version of Reckhow’s theorem  which is stated for Frege systems. The version for Extended Frege system is much easier to prove (see [20, 23] for published proofs of the stronger version).
For the definition of natural deduction see , for sequent calculus see any of [11, 20, 23] (the sequent calculus part of the statement is just mentioned in  while natural deduction is treated in detail). The substitution rule allows to infer from a formula its arbitrary substitution instance in one inference. A Substitution Frege system SF is a Frege system augmented by this rule. It was proved later in  (indirectly) and in  (an explicit p-simulation) that EF actually p-simulates SF as well.
An illuminating description of EF is that it is essentially a Frege system that operates with circuits rather than with formulas; this has been made precise in . Perhaps even more useful is the statement that the minimum size of an EF-proof of formula is proportional to the minimum number of steps in a Frege proof of and , or to the minimum number of different formulas that need to occur as subformulas in any Frege proof of and , cf.  or [20, 23]. Hence moving from F to EF means that we are replacing the size as the measure of complexity of Frege proofs by the number of steps. This is interesting because from the point of view of mathematical logic the number of steps is a very natural complexity measure.
Extended Frege system is also important because of its relation to a particular theory PV introduced by Cook  at the same time (he used ER in his paper). This is discussed in S. Buss’s article in this volume. Theory PV (stands for Polynomially Verifiable) allows to formalize a number of standard computational complexity constructions and arguments. Understanding the power of proof system EF and, in particular, showing that it is not p-bounded, is considered in the field as the pivotal step towards solving the Main problem and proving that . In particular, it is also known that any super-polynomial lower bound for EF implies that is consistent with PV (cf. [23, Sec.12.4]).
We shall mention one problem formulated only later in  which is, however, natural and is implicit in the definition of simulations.
Problem 2.3 (Optimality problem)
Is there a proof system that (p-)simulates all other proof systems?
Such a maximal proof system is called (p-)optimal after . We have (names for) three types of proof systems whose existence is considered by most researchers unlikely: p-bounded, p-optimal and optimal. Every p-bounded or p-optimal proof system is also optimal and this rules out three out of eight possibilities for the existence/non-existence of objects of these three types. At present we cannot rule out any of the remaining five scenarios:
A p-bounded, p-optimal proof system exists.
Having such an ideal proof system we do not need to consider any other: even searching for proofs in any other proof system can be reduced to searching for -proofs. (We ignore here that p-reductions themselves increase polynomially the time complexity of a proof search algorithm and may transform a combinatorially transparent one into a complex one, cf. the last paragraph of this section.)
A p-bounded proof system exists but no p-optimal does.
While p-size -proofs would exist for each tautology, finding them may be difficult and it may help to consider different proof systems for different (classes of) tautologies.
A p-optimal proof system exists but no p-bounded does.
Here we can restrict our attention to : it is also optimal and search for proofs in any proof system can be replaced by a search for -proofs.
An optimal proof system exists but no p-bounded or p-optimal does.
Proving lengths-of-proofs lower bounds (or upper bounds, for that matter) can be restricted to but proof search may benefit from considering different proof systems for different classes of tautologies.
None of these ideal objects exist.
This appears to be the most likely scenario.
At present we cannot rule out that a Frege system is one of . The Optimality problem is related to a surprising number of varied topics in proof theory (quantitative Gödel’s theorem), finite model theory, structural complexity, and some other (cf. [23, Chpt.21]).
3 Hard tautologies and the formula
In order to prove lengths-of-proofs lower bounds for a proof system we start with a suitable candidate tautology that we conjecture to be hard to prove (i.e. requiring long proofs) therein. A particular tautology for this purpose based on the pigeon-hole principle was proposed in . The formula, to be denoted , is built from atoms with and , for . Thinking of as representing the atomic statement that maps to , we can express that the map is defined at by the clause
the fact that can be the value of at most on by
and the fact that maps to at most one value by
Taking the conjunction of these clauses for all choices of and states that
is the graph of an injective map from into . No such map exists and hence the negation of the conjunction is a tautology. This leads to the following definition.
Definition 3.1 (Cook-Reckhow )
In fact, to reach a contradiction we do not need the assumption that it is the graph of a function, a multi-function suffices (if occupies more values it is harder to be injective). In other words, we do not need to include the clauses from (3) and  did not included them. Nowadays the definition of as formulated above is more customary and proving lower bounds for it yields stronger results than for the more economical version (the principle assumes more and hence it is logically weaker).
Cook and Reckhow  showed that it is possible to prove in Extended Frege systems by a proof of size polynomial in (note that the size of is also polynomial in ). In fact, they introduced EF in order to formalize smoothly the inductive argument: from an assignment violating we can define (using the extension rule) an assignment violating . Hence has also a proof in Frege systems with a polynomial number of steps (but having large size). Buss  improved the result (by a different construction formalizing counting) and proved that Frege systems actually also admit polynomial size proofs of .
On the other hand, in a breakthrough result, Haken  proved a first lower bound for resolution using and the same formula was proved to be hard for constant depth subsystems of any Frege system in the DeMorgan language by Ajtai  (Haken’s lower bound was exponential while Ajtai’s super-polynomial - its rate was later improved to exponential too by [25, 30]). The same formula (represented by polynomial equations similarly as in Section 1.1) served to Razborov  for his lower bound for polynomial calculus, an algebraic proof system manipulating polynomials.
There is an important variant of the PHP formula considered first by Paris, Wilkie and Woods  in the context of bounded arithmetic: allow to range over a much bigger set than ; for example, over or even . Similarly as the PHP principle is related to counting these weak PHP principles relate to approximate counting and  showed that they can be sometimes used in place of PHP proper and that, crucially, they are easier to prove. Their proof (formulated using bounded arithmetic) gives quasi-polynomial size proofs in constant depth Frege systems of formulas formalizing these weaker principles for .
Even if the formula itself cannot be used as a hard example for proof systems like F or EF, formulas formalizing a form of a weak PHP in a different way possibly can. It has been an insight of Wilkie (result reported in [20, Sec.7.3]) that the dual weak PHP for p-time functions is important in bounded arithmetic (this has been much extended by Jeřábek [17, 18, 19]). The principle says that no p-time function when restricted to any can can be onto . Now take an arbitrary , where is the restriction of to , and define propositional formula
expressing . The formula uses atoms for bits of and further atoms for bits of the computation of on and says, in a DNF form, that either is not a valid computation on input or the output of the computation differs from . Clearly
These formulas were defined in [2, 21] and lead to the theory of proof complexity generators proposing several candidate tautologies of the form above as possibly hard for strong (or all) proof systems. The reader may find an overview of the theory in [22, Chpts.29 and 30] (no need to read the first 28 chapters).
I thank Sam Buss, Bruce Kapron, Igor C. Oliveira and Jan Pich for comments on earlier versions of this paper.
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