 # Induction rules in bounded arithmetic

We study variants of Buss's theories of bounded arithmetic axiomatized by induction schemes disallowing the use of parameters, and closely related induction inference rules. We put particular emphasis on Π̂^b_i induction schemes, which were so far neglected in the literature. We present inclusions and conservation results between the systems (including a witnessing theorem for T^i_2 and S^i_2 of a new form), results on numbers of instances of the axioms or rules, connections to reflection principles for quantified propositional calculi, and separations between the systems.

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## 1 Introduction

Commonly studied theories of arithmetic, weak and strong alike, are typically axiomatized by variants of induction or other axiom schemes (comprehension, collection, …) restricted to suitable classes of formulas, where these formulas may freely use parameters: arbitrary numbers or other objects manipulated by the theory that enter the induction formula by means of free variables, unrelated to the induction variable. This generally makes the theories robust in their formal properties, and intuitive to work with. Nevertheless, induction schemes without parameters proved fruitful to study in the context of strong subtheories of Peano arithmetic (-induction), revealing a landscape of strange, and yet familiar systems: see e.g. Kaye, Paris, and Dimitracopoulos , Adamowicz and Bigorajska , Bigorajska , Beklemishev [3, 4], and Cordón-Franco and Lara-Martín .

On the one hand, the parameter-free induction schemes and are close to the original schemes with parameters , as the theories are conservative over each other with respect to large classes of sentences (though the correspondence is a bit off, as is on the same level as and ). On the other hand, there are substantial differences: as already alluded to, the  schemes without parameters become genuinely distinct from (and weaker than) the matching  schemes, whereas ; neither nor  are finitely axiomatizable, in contrast to .

The parameter-free schemes and  are intimately connected to induction rules and : here, instead of theories generated just by axioms on top of the usual rules of first-order logic, we consider a form of induction as an additional (Hilbert-style) rule of inference. It turns out is the weakest theory all of whose extensions are closed under , and likewise for . An important role in the analysis of and  is played by reflection principles for fragments of arithmetic [3, 4]: while is equivalent to a certain uniform (global) reflection principle, the theories and  can be characterized using relativized local reflection principles. There are also intricate connections relating the nesting of applications of rules and the number of instances of axioms. As an alternative to reflection principles, parameter-free induction schemes can be analysed using local induction .

In contrast to all these results, much less is known about parameter-free induction axioms and induction rules in the context of bounded arithmetic: the early work of Kaye  introduced the parameter-free subtheories of , while the only investigation of parameter-free Buss’s theories was done by Bloch , who studied proof-theoretically  parameter-free induction rules111Warning: the proof of Theorem 27, which effectively claims that , is incorrect. in a sequent formalism, and Cordón-Franco, Fernandéz-Margarit, and Lara-Martín , whose main results concern conservativity of the theories and  over the parameter-free and induction-rule versions of - and -, and conservativity of over its rule version. They rely on model-theoretic methods exploiting variants of existentially closed models.

The purpose of this paper is to study parameter-free versions of Buss’s theories in a more systematic way, filling in various gaps in our knowledge to obtain a more complete picture. Some highlights are as follows. We will investigate  schemes and rules, which were so far entirely ignored in the literature, alongside their  counterparts; in particular, we will prove conservation results of and  over -. We try to get as complete a description of the relationships among the systems in question as possible; to this end, we also include tentative separation results (conditional or relativized). While bounded arithmetic is too weak to prove the consistency of interesting first-order theories, it has a well-known connection to propositional proof systems; in accordance with this, we will present characterizations of our systems in terms of variants of reflection principles for fragments of the quantified propositional sequent calculus. We also include some results on the nesting of rules, namely conditions ensuring that closure under the induction rules collapses to unnested closure, and conservation results of  instances of parameter-free induction axioms over  applications of induction rules.

The paper is organized as follows. After some preliminary background in Section 2, we introduce in Section 3 the main axioms and rules that we are interested in, and we prove some of their elementary properties—primarily reductions between the rules (þ3.5), but also a result on a collapse of - to unnested applications (þ3.7). We discuss various variants of the axioms and rules in Section 4, and we show them mostly equivalent to our main systems (þ4.2).

The most substantial technical part of the paper comes in Section 5, which is devoted to conservation results. We recall the conservation of and  over - (þ5.1) from [6, 18], and we set out to prove an analogous conservation result over - (þ5.9). A key part of the proof is a new witnessing theorem for consequences (and consequences) of and , which may be of independent interest (þLABEL:thm:conservp,prop:conservs). We obtain conservation results over -, summarized in þ5.14, and a result on collapse of nesting of - (þ5.10). We also prove more direct conservation results of over for arbitrary theories 5.20).

We discuss connections to propositional proof systems in Section 6, the main result being a characterization of - and - in terms of reflection principles for quantified propositional calculi (þ6.5). Section 7 is devoted to separations between our systems: we present some conditional separations in Section 7.1, and unconditional relativized separations in Section 7.2. We conclude the paper with a few remarks in Section 8.

## 2 Notation and preliminaries

We assume the reader is familiar with the basics of bounded arithmetic. We will work in the framework of Buss’s one-sorted theories and , as presented e.g. in Buss , Hájek and Pudlák [20, Ch. V], or Krajíček . It would not be too difficult to adapt our results to the setting of two-sorted theories as in Cook and Nguyen , but we find the one-sorted setting simpler to use for the present purpose.

In order not to get bogged down in trivial technicalities, we will employ a robust base theory in a rich language in place of Buss’s : let denote the basic first-order theory for , in a language with function symbols for all  functions so that is a universal theory. We are not very particular about its exact definition; for example, we may axiomatize it as the theory of Johannsen and Pollett  expanded with function symbols for all -definable functions of the theory, or as the equivalent theory of Clote and Takeuti . Note that is -isomorphic to the theory (or rather, ) of Cook and Nguyen . Unless stated otherwise, we will assume all first-order theories to be formulated in  and to extend .

If is a (possibly empty) set of sentences, and a sentence, we write if is provable in the theory . We may omit outermost universal quantifiers when writing down or , as is the customary fashion. We may also write for a set of sentences , meaning for all . We stress that is only closed under the standard deduction rules of first-order logic (i.e., it includes logically valid sentences, and it is closed under modus ponens); it is not supposed to be closed under the rule even if we define as in .

Let and  denote the classes of strict and  formulas in : that is, is the class of sharply bounded formulas, and for , a  formula ( formula) consists of  alternating (possibly empty) blocks of bounded quantifiers followed by a  formula, where the first block is existential (universal, resp.). Equivalently, we could further restrict the blocks to a single quantifier apiece. Note that every  formula is equivalent to an atomic formula in . The class of all bounded formulas is denoted .

We will combine notations such as and  with symbolic prefixes denoting unbounded quantifiers: for example, denotes the class of formulas (in most contexts, sentences) consisting of a block of universal quantifiers, followed by a block of existential quantifiers, followed by a  formula.

Let be a class of sentences, and a theory. The -fragment of  is the theory axiomatized by . If is another theory, is -conservative over  if the -fragment of  is included in .

Let denote the least class of formulas that includes bounded formulas, and is closed under existential and bounded universal quantifiers; denotes the dual class. A model-theoretic characterization of these classes is that  formulas are preserved downwards in cuts, and  formulas upwards.

###### Theorem 2.1 (Parikh)

þ Let be a -axiomatized extension of , and . If , there exists a term  such that .

We will occasionally use that -sentences true in the standard model of arithmetic  are provable in .

Another fundamental tool for studying systems of bounded arithmetic is Buss’s witnessing theorem. We are actually not interested in witnessing per se, but in the following consequence:

###### Theorem 2.2 (Buss)

þ For any , is a -conservative extension of .

We will in fact use it in an ostensibly stronger form:

###### Corollary 2.3

þ For any and , is -conservative over .

Proof:   Assume that , where , and . Then proves . By Parikh’s theorem, we may bound the  quantifier by a term in , which makes the statement (equivalent to) a  sentence. Thus, it is provable in  by þ2.2, and this implies .

Our basic objects of study will be rules rather than just axiom schemes. Here, a rule  is a set of pairs , where is a sentence, and is a finite set of sentences; each is called an instance of , and will be written more conspicuously as , or

 (1) φ1φ2…φnφ0.

The instance above is -ary. We will identify axiom schemes with -ary rules. Again, we will often omit outermost universal quantifiers from the sentences  when writing down rules like (1).

If is a theory, and a rule, then denotes the least theory  (i.e., deductively closed set of sentences) which includes , and which is closed under , meaning that for any instance of , if , then .

A rule is weakly reducible to a rule  if for all theories , and and  are weakly equivalent if they are weakly reducible to each other. Note that is weakly reducible to  iff for any instance of , .

We may stratify this definition by counting the nesting depth of applications of the rules. Let denote the closure of  under unnested applications of -instances, i.e., the theory axiomatized by

 T∪{φ:Γ/φ∈R,T⊢Γ},

and we define , by induction on . Notice that . We say that is reducible to , written , if for every theory , and and  are equivalent, written , if . As above, we have that iff for each instance of . See also þ3.6.

We remark that just like sets of axioms are represented uniquely up to equivalence by theories, rules can be represented up to weak equivalence by finitary consequence relations, extending the standard first-order consequence relation of .

Aside from bounded arithmetic, we will also assume (especially in Section 6) familiarity with basic propositional proof complexity, and in particular with the quantified propositional sequent calculus  (see [31, 16]). The classes and  of quantified propositional formulas are defined as usual: consists of quantifier-free formulas; and  include , and are closed under and ; is closed under existential quantifiers, and under universal quantifiers; negations of  formulas are , and vice versa.

Following , we define for  as restricted so that all cut-formulas are . When the sequent to be proved consists of  formulas, this is equivalent to the original definition as in . Note that up to polynomial simulation, we could allow cut-formulas in  as well; on the other hand, we could restrict cut-formulas to prenex  formulas only . Let denote the tree-like version of . For , we define as extended Frege, optionally considered as a proof system for prenex formulas (the system introduced as  in ).

If is a quantified propositional proof system, and , then denotes the -reflection principle for . If , we take this to mean the  reading of the principle: “for every proof of a quantifier-free formula , and every evaluation of subformulas of  that respects the connectives, the value assigned to  is ” ( in the notation of [16, §X.2.3]). (This can make a difference, as does not necessarily prove that any given quantifier-free formula can be evaluated.) Note that for all proof systems we are going to consider, this form of is -provably equivalent to consistency.

## 3 Main systems

We are ready to introduce the main axioms and rules that will be the topic of this paper. In the rest of this section, we will show their basic properties, most importantly reductions (inclusions) among the rules.

###### Definition 3.1

þ Let or , where . The induction and polynomial induction axiom schemes are defined as usual:

 (Γ-IND) φ(0,y)∧∀x(φ(x,y)→φ(x+1,y))→∀xφ(x,y), (Γ-PIND) φ(0,y)∧∀x(φ(⌊x/2⌋,y)→φ(x,y))→∀xφ(x,y),

where . The corresponding induction rules are

 (Γ-INDR) φ(0,y)φ(x,y)→φ(x+1,y)φ(x,y), (Γ-PINDR) φ(0,y)φ(⌊x/2⌋,y)→φ(x,y)φ(x,y).

The variable  is a parameter of these axioms and rules (we could equivalently allow a tuple of parameters, as this can be encoded by a single parameter using a pairing function). The corresponding parameter-free schemes, denoted by superscript , are obtained by omitting , i.e., has no free variables besides .

The familiar theories and  are defined as and , respectively.

###### Remark 3.2

The cases of our schemes and rules are idiosyncratic in various ways: first, ; second, is closed under neither bounded existential nor bounded universal quantifiers, which is going to break some constructions; and third, - and their parameter-free and rule variants are already derivable in the base theory  (that is, in our language, , whereas is essentially ).

The standard theories with parameters and  are axiomatizable by bounded formulas (i.e.,  sentences), since the  axiom as stated above is equivalent to

 ∀z(φ(0,y)∧∀x

and similarly for . The proof of this equivalence uses as a parameter, hence it is not obvious that this should hold for the parameter-free schemes as well. Nevertheless, the - schemes do have, for , bounded axiomatizations (specifically, by  sentences), similarly to the case with parameters: if , then

 (2) ∀x(φ(0)∧∀y

is provable by induction on the formula , as

 ⊢∀y

and similarly for . This argument does not seem to work for -, though.

A crucial property is that induction rules are equivalent to their parameter-free versions. The case of was already proved in , but we include it for completeness anyway.

###### Lemma 3.3

þ If or  for , then .

Proof:   Let be a  pairing function nondecreasing in  such that , provably in . If , let denote the number whose binary representation consists of the th through th binary digits of , where the most significant digit has index ; i.e., .

An instance of - for a formula  can be reduced to - for the formula , where : we have either , or , , and .

Since and , we may assume in the remaining cases.

For -, let , and put . Then

 φ(0,y) ⊢ψ(0), φ(0,y),φ(x,y)→φ(x+1,y) ⊢ψ(z)→ψ(z+1), ⊢ψ(⟨x,y⟩)→φ(x,y).

For -, we may use in a similar fashion. In order to verify

 φ(0,y),φ(⌊x/2⌋,y)→φ(x,y)⊢ψ(⌊z/2⌋)→ψ(z),

assume , and let . Put , . If , we have , and holds by assumption. Otherwise put , , , and . We have , , and , hence by the induction hypothesis, which implies by assumption.

For -, let be a formula of the form with . Fix a suitable sequence encoding with being the th element of the sequence coded by , and a term such that every sequence  of length at most , each of whose entries is bounded by for some , satisfies . Let be the formula

 ∃w≤b(z)∀i,j≤|z|(⟨i,j⟩<|z|→(w)⟨i,j⟩≤t(z[j,i+j),z[0,j))∧θ(z[j,i+j),z[0,j),(w)⟨i,j⟩)).

Again, the least obvious property to check is that assuming the premises of - for , we can derive . Let , , and assume that is a sequence of length  witnessing . We will construct a sequence  witnessing . If , then , thus and , and we may take . If , put , . Either , in which case and holds, or , , and . We have as witnessed by , hence . Either way, we can extend to  so that is a witness for , and then witnesses .

###### Corollary 3.4

þ is the weakest theory all of whose extensions are closed under -.

Proof:   On the one hand, it is clear that any extension of - derives , hence - by þ3.3. On the other hand, assume that all extensions of  are closed under -. Let be any instance of - as in þ3.1 (here, and  are sentences). Then is an instance of -, thus by assumption. The deduction theorem then gives .

The next result presents all reductions between our core rules that we know about; they are summarized in Fig. 3.1. We will argue in Section 7 that no other reductions are likely waiting to be discovered.

###### Theorem 3.5

þ Let , and be or .

1. .

2. .

3. , and .

4. , , and .

6. , and .

Proof:   (i) is an immediate consequence of þ3.3.

(ii) is well known: for  follows from for , and  for follows from  for , where is an additional parameter.

(iii): We may assume . Consider an instance of - for a formula , where , and let be the formula

 φ(a˙−x,y)∧z≤t(a,y)→θ(a,y,z).

Then

 ⊢ψ(0,y,a,z), φ(x,y)→φ(x+1,y) ⊢ψ(x,y,a,z)→ψ(x+1,y,a,z), ψ(x,y,x,z) ⊢φ(0,y)→φ(x,y),

showing that - reduces to -.

In order to show , assume further that is parameter-free. Then proves as it is closed under by (i), hence proves - by the deduction theorem.

The cases of  and  are similar, using in place of , as in (ii).

(iv): We may assume , as .  for a  formula  follows from  for the  formula , and likewise for  or .  for a  formula follows from  for the formula with an additional parameter , and this also applies to . The result for  follows from the result for  as in the proof of (iii).

(v): Let , and let be the formula

 φ(0,y)∧¬φ(a,y)→∃u≤a,v≤⌈a/2|x|⌉(u+v≤a∧φ(u,y)∧¬φ(u+v,y)).

Then it is easy to check that proves

 ψ(0,y,a), ψ(⌊x/2⌋,y,a)→ψ(x,y,a), ψ(a,y,a)→(φ(0,y)∧∀u

thus derives the induction axiom for .

(vi): Let , and let be the formula

 ∀x′≤z(φ(x′,y)∧x+x′≤z→φ(x+x′,y)).

Then

 ⊢ψ(0,y,z), φ(x,y)→φ(x+1,y) ⊢ψ(1,y,z), ⊢ψ(x0,y,z)∧ψ(x1,y,z)→ψ(x0+x1,y,z), ψ(x,y,x) ⊢φ(0,y)→φ(x,y),

whence . The result for - follows as in (iii).

###### Remark 3.6

þ Recall that we defined by counting the nesting depth of applications of , which is in general necessary in order to make a deductively closed first-order theory. However, observe that unnested applications of  for formulas , …, may be reduced to a single application of the same rule for the formula . It follows that if is closed under  (such as or ), then coincides with the set of formulas provable using  instances of -; the same applies to .

Surprisingly, a simple argument shows that the closure of  under - collapses to unnested applications of the rule (thus a single application is enough to prove any given consequence) under very mild assumptions on the complexity of the theory . In particular, note that all traditional subsystems of  such as are axiomatized by sentences.

###### Theorem 3.7

þ If is -axiomatized, and , then

 T+^Πbi-(P)INDR=[T,^Πbi-(P)INDR].

Proof:   In view of þ3.6, it is enough to show that includes all formulas provable using two instances of : this implies , i.e., is closed under