We shall study in this chapter the topic of hard tautologies: tautologies that are candidates for not having short proofs in a particular proof system. The closely related question is whether there is an optimal propositional proof system, that is, a proof system P such that no other system has more than a polynomial speed-up over P. We shall obtain a statement analogous to the NP-completeness results characterizing any propositional proof system as an extension of EF by a set of axioms of particular form. Recall the notions of a proof system and p-simulation from Section 4.1, the definitions of translations of arithmetic formulas into propositional ones in Section 9.2, and the relation between reflection principles (consistency statements) and p-simulations established in Section 9.3. We shall also use the notation previously used in Chapter 9.

Finitistic consistency statements and optimal proof systems

We shall denote by Taut (x) the formula Taut0 (x) from Section 9.3 defining the set of the (quantifierfree) tautologies, denoted TAUT itself.

Recall from Section 9.2 the definition of the translation

producing from a formula a sequence of propositional formulas (Definition 9.2.1, Lemma 9.2.2).

This chapter presents important definability results for fragments of bounded arithmetic.

A Turing machine M will be given by its set of states Q, the alphabet Σ, the number of working tapes, the transition function, and its clocks, that is, an explicit time bound. Most results of the form “Given machine M the theory T can prove …” could be actually proved in a bit stronger form: “For any k the theory T can prove that for any M running in time ≥ nk …” A natural formulation for such results is in terms of models of T and computations within such models, but in this chapter we shall omit these formulations.

An instantaneous description of a computation of machine M on input x consists of the current state, the positions of the heads, the content of all tapes, and the current time: That is, it is a sequence of symbols whose length is proportional to the time bound for n:= |x|.

A computation will be coded by the sequence of the consecutive instantaneous descriptions.

Now we shall consider several bounded formulas defining these elementary concepts. They are all in the language L+ and thus also (by Lemma 5.4.1) in L.

This chapter will present basic propositional calculus. By that I mean properties of propositional calculus established by direct combinatorial arguments as distinguished from high level arguments involving concepts (or motivations) from other parts of logic (bounded arithmetic) and complexity theory.

Examples of the former are various simulation results or the lower bound for resolution from Haken (1985). Examples of the latter are the simulation of the Frege system with substitution by the extended Frege system (Lemma 4.5.5 and Corollary 9.3.19), or the construction of the provably hardest tautologies from the fmitistic consistency statements (Section 14.2).

We shall define basic propositional proof systems: resolution R, extended resolution ER, Frege system F, extended Frege system EF, Frege system with the substitution rule SF, quantified propositional calculus G, and Gentzen's sequent calculus LK. We begin with the general concept of a propositional proof system.

Propositional proof systems

A property of the usual textbook calculus is that it can be checked in deterministic polynomial time whether a string of symbols is a proof in the system or not. This is generalized into the following basic definition of Cook and Reckhow (1979).

Definition 4.1.1. Let TA UT be the set of propositional tautologies in the language with propositional connectives: constants 0 (FALSE) and 1 (TRUE), ¬ (negation), ∨ (disjunction), and & (conjunction), and atoms p1, p2,…

A propositional proof system is a polynomial time function P whose range is the set TAUT.

This chapter is devoted primarily to proving several definability and witnessing theorems for the second order system and analogous to those in Chapters 6 and 7.

Our tool is the RSUV isomorphism (Theorem 5.5.13), or rather the definition of (Definition 5.5.3), together with the model-theoretic construction of Lemma 5.5.4.

The first section discusses and defines the second order computations. In the second section are proved some definability and witnessing theorems for the second order systems and further conservation results for first order theories (Corollaries 8.2.5-8.2.7). The proofs are sketched and the details of the RSUV isomorphism arguments are left to the reader.

Second order computations

Let A (a, βt(b)) be a second order bounded formula and (K, X) a model of. By Definition 5.5.3 we may think of K as of K = Log(M) for some M ⊨, with X being the subsets of K coded in M. Pick some a, b ∈ K of length n and some βt(b). Then

if and only if (see Theorem 5.5.13 for the notation)

where u codes βt(b) The length of u is thus.

In this chapter we briefly review the basic notions and facts from logic and complexity theory whose knowledge is assumed throughout the book. We shall always sketch important arguments, both from logic and from complexity theory, and so a determined reader can start with only a rough familiarity with the notions surveyed in the next two sections and pick the necessary material along the way.

For those readers who prefer to consult relevant textbooks we recommend the following books: The best introduction to logic are parts of Shoenfield (1967); for elements of structural complexity theory I recommend Balcalzár, Diáz, and Gabbarró (1988, 1990); for NP-completeness Garey and Johnson (1979); and for a Boolean complexity theory survey of lower bounds Boppana and Sipser (1990) or the comprehensive monograph Wegener (1987). A more advanced (but selfcontained) text on logic of first order arithmetic theories is Hájek and Pudlák (1993).

Logic

We shall deal with first order and second order theories of arithmetic. The second order theories are, in fact, just two-sorted first order theories: One sort are numbers; the other are finite sets. This phrase means that the underlying logic is always the first order predicate calculus; in particular, no set-theoretic assumptions are a part of the underlying logic.

From basic theorems we shall use Gödel completeness and incompleteness theorems, Tarski's undefinability of truth, and, in arithmetic, constructions of partial truth definitions.

In this chapter we shall discuss the complexity of Frege systems without any restrictions on the depth. There is some nontrivial information, in particular nontrivial upper bounds, but no nontrivial lower bounds are known at present (only bounds from Lemma 4.4.12).

Counting in Frege systems

Theorems 9.1.5 and 9.1.6 are useful sufficient conditions guaranteeing the existence of the polynomial size EF-proofs and of quasipolynomial size F-proofs, respectively. For example, U11 proves the pigeonhole principle PHP(R) and hence there are quasipolynomial size F-proofs of PHPn. A subtheory of corresponding to the polynomial size F-proofs, based on a version of inductive definitions, was considered by Arai (1991); see Section 9.6. Its axiomatization however, stresses a logical construction, whereas we would like a theory based on a more combinatorial principle.

The most important property of a Frege system relevant for the upper bounds is that it can count. We shall make this precise by showing that F simulates an extension of I△0(α) by counting functions, and that F p-simulates a propositional proof system cutting planes.

Definition 13.1.1.

(a) Let L0 be the language of the second order bounded arithmetic but without the symbol #.

Ten years ago I had the wonderful opportunity to attend a series of lectures given by Jeff Paris in Prague on his and Alec Wilkie's work on bounded arithmetic and its relations to complexity theory. Their work produced fundamental information about the strength and properties of these weak systems, and they developed a variety of basic methods and extracted inspiring problems.

At that time Pavel Pudlak studied sequential theories and proved interesting results about the finitistic consistency statements and interpretability (Pudlak 1985, 1986, 1987). A couple of years later Sam Buss's Ph.D. thesis (Buss 1986) came out with an elegant proof-theoretic characterization of the polynomial time computations. Then I learned about Cook (1975), predating the above developments and containing fundamental ideas about the relation of weak systems of arithmetic, propositional logic, and feasible computations. These ideas were developed already in the late 70s by some of his students but unfortunately remained, to a large extent, unavailable to a general audience. New connections and opportunities opened up with Miki Ajtai's entrance with powerful combinatorics applied earlier in Boolean complexity (Ajtai 1988).

The work of these people attracted other researchers and allowed, quite recently, further fundamental results.

It appears to me that with a growing interest in the field a text surveying some basic knowledge could be helpful. The following is an outline of the book.