A Arithmetization of the Solovay functions

Positive provability logic for uniform reflection principles

Abstract

We deal with the fragment of modal logic consisting of implications of formulas built up from the variables and the constant ‘true’ by conjunction and diamonds only. The weaker language allows one to interpret the diamonds as the uniform reflection schemata in arithmetic, possibly of unrestricted logical complexity. We formulate an arithmetically complete calculus with modalities labeled by natural numbers and , where corresponds to the full uniform reflection schema, whereas corresponds to its restriction to arithmetical -formulas. This calculus is shown to be complete w.r.t. a suitable class of finite Kripke models and to be decidable in polynomial time.

keywords:
provability logic, reflection principle, positive modal logic
Msc:
03F45, 03B45
1 \fntext

[aff]Also affiliated at: Moscow M.V. Lomonosov State University and National Research University Higher School of Economics.

1 Introduction

Several applications of provability logic in proof theory made use of a polymodal logic due to Giorgi Japaridze (17); (9). This system, although decidable, is not very easy to handle. In particular, it is not Kripke complete (9). It is complete w.r.t. the more general topological semantics, however this could only be established recently by rather complicated techniques (2).

A weaker system, called Reflection Calculus and denoted , was introduced in (8). It is much simpler than yet expressive enough to regain its main proof-theoretic applications. It has been outlined in (8) that allows to define a natural system of ordinal notations up to and serves as a convenient basis for a proof-theoretic analysis of Peano Arithmetic in the style of (4); (5). This includes a consistency proof for based on transfinite induction up to , a characterization of its -consequences in terms of iterated reflection principles, and a combinatorial independence result.

From the point of view of modal logic, can be seen as a fragment of polymodal logic consisting of implications of the form , where and are formulas built-up from and propositional variables using just and the diamond modalities. We call such formulas and strictly positive and will often omit the word ‘strictly.’2

A somewhat different but equivalent axiomatization of (as an equational calculus) has been earlier found by Evgeny Dashkov in his paper (11) which initiated the study of strictly positive fragments of provability logics. Dashkov proved two important further facts about which sharply contrast with the corresponding properties of . Firstly, is complete with respect to a natural class of finite Kripke frames. Secondly, is decidable in polynomial time, whereas most of the standard modal logics (including and ) are PSpace-complete.

Another advantage of going to a strictly positive language is explored in the present paper. Strictly positive modal formulas allow for more general arithmetical interpretations than those of the standard modal logic language. In particular, propositional formulas can now be interpreted as arithmetical theories rather than individual sentences. (Notice that the ‘negation’ of a theory would not be well-defined.)

Similarly, the diamonds need no longer be interpreted as individual consistency assertions but as the more general reflection schemata not necessarily having finite axiomatizations. Thus, for example, the full uniform reflection schema can be considered as a modality in the context of positive provability logic (see (18); (5) for general information on reflection principles). Such interpretations are not only natural but can be useful for further development of the approach to proof-theoretic analysis via provability algebras. Thus, positive provability logic allows to speak about certain notions not nicely representable in the context of the standard modal logic.

The main contribution of this paper is a Solovay-style arithmetical completeness result for an extension of by a new modality corresponding to the unrestricted uniform reflection principle. This is the primary example of a modality not representable in the full modal logic language. The system obtained is shown to be decidable and to enjoy a suitable complete Kripke semantics along with the finite model property.

Whereas the modal logic part of our theorem is a simple extension of Dashkov’s results, the arithmetical part is more substantial. We introduce a new modification of the Solovay construction using some previous ideas from (17); (16); (7). Since the arithmetical complexity of the unform reflection schema is unbounded, a single Solovay-style function is not enough for our purpose. Instead, we deal with infinitely many Solovay functions, of increasing arithmetical complexity, uniformly and simultaneously.3

The paper is organized as follows. Firstly, we introduce positive modal language and the systems leading to the arithmetically complete reflection calculus . Secondly, we present the details of its arithmetical interpretation and somewhat tediously prove the corresponding soundness theorem. Thirdly, we study the Kripke semantics of positive provability logics and obtain completeness results, along with a suitable version of the finite model property. Fourthly, we obtain polynomial complexity bounds for the derivability problem in by adapting the techniques of Dashkov. Finally, we prove the main result of this paper, the arithmetical completeness theorem for .

2 Reflection calculus and its basic properties

Consider a modal language with propositional variables ,…, a constant and connectives and , for each ordinal (understood as diamond modalities). Strictly positive formulas (or simply formulas) are built up by the grammar:

Sequents are expressions of the form where are strictly positive formulas. The system is given by the following axioms and rules:

  1. if and then (syllogism);

  2. if and then ;

  3. if then ; ;

  4. for ;

  5. for .

The systems and are obtained from by adding respectively one or two of the following principles:

  1.  (monotonicity)

  2. .  (persistence)

Dashkov (11) showed that , restricted to the language without modality, axiomatizes the strictly positive fragment of the polymodal logic (6), whereas axiomatizes the strictly positive fragment of .

Notice that Axioms 4 are redundant in the presence of Axiom 6: if then and .

If is a logic, we write for the statement that the sequent is provable in . As a simple example, consider the sequent

It is provable in as follows: We have , hence . Similarly, , therefore , by the conjunction rules. In contrast, , as we shall see below by a simple Kripke model argument.

Formulas and are called -equivalent (written ) if and .

We also consider the fragments of various logics obtained by restricting the language to a subset of modalities. Such a subset is called a signature. We denote by the set of all strictly positive formulas in . Similarly, for a logic in we denote by the restriction of the axioms and rules of to the language .

For a positive formula , let denote .

Lemma 2.1
  1. If then ;

  2. If where is or , then .

Proof. In each case, this is proved by an easy induction on the length of the derivation of .

Let denote the result of replacing in all occurrences of a variable by . If a logic contains Axioms 1, 2 and the first part of 3, then satisfies the following positive replacement lemma.

Lemma 2.2

Suppose , then , for any .

Proof. Induction on the build-up of .

A positive logic is called normal if it contains the rules 1, 2, and the first part of 3, and is closed under the following substitution rule: if then . It is clear that , and , as well as their restricted versions, are normal.

3 Arithmetical interpretation

We define the intended arithmetical interpretation of the positive modal language. The idea is that propositional variables (and positive formulas) now denote possibly infinite theories rather than individual sentences. To avoid possible problems with the representation of theories in the language of , we deal with primitive recursive numerations of theories rather than with the theories as sets of sentences.

All theories in this paper will be formulated in the language of Peano Arithmetic and contain the axioms of . It is convenient to assume that the language of contains the symbols for all primitive recursive programs. A primitive recursive numeration of a theory is a bounded arithmetical formula defining the set of Gödel numbers of the axioms of in the standard model of arithmetic. Given such a , we have a standard arithmetical -formula expressing the provability of in (see (12)). We often write for . The expression denotes the numeral ( times). If contains a parameter , then denotes a formula (with a parameter ) expressing the provability of the sentence in .

Given two numerations and , we write if

We write if that is, if the theory numerated by contains the one numerated by . We will only consider the numerations such that where is some standard numeration of .

With any finite extension of of the form we will associate its standard numeration that will be denoted . For obvious reasons we have: iff iff . (The statement implies by the soundness of , the converse is formalizable in .)

Given a numeration of , the consistency of is expressed by . A theory is called -consistent if together with the set of all true -sentences is consistent. The -consistency of is expressed by the formula

\mathrm{Con}_{n}(σ):

where is the standard -truthdefinition for -formulas (see (14)) and denotes the primitive recursive formula expressing that is a Gödel number of a -sentence.

Concerning the truthdefinitions we assume that for each -sentence . Moreover, this very fact can be formalized in uniformly in :

(1)

as the sequence of formulas is primitive recursive in and the corresponding proofs are constructed inductively.

The formula is often called the global -reflection principle for and is denoted (see (19); (3)). We note that the formula is -provably equivalent to .

The uniform reflection principle for is the schema

\mathrm{Con}_{\omega}(σ):

It is well-known that is -provably equivalent to the schema

for each arithmetical formula , which is usually denoted .

The uniform reflection principle is elementarily axiomatized, and we fix a standard function mapping any numeration to the numeration of (denoted ). Similarly, the formula

numerating the theory will be denoted .

The intended arithmetical interpretation maps positive modal formulas to primitive recursive numerations in such a way that corresponds to the standard numeration of , corresponds to the union of theories, corresponds to the standard numeration of , for each , and to the standard numeration of .

Definition 3.1

An arithmetical interpretation is a map from positive modal formulas to numerations satisfying the following conditions:

  • ;  ;

  • ;  .

It is clear that the value is completely determined by the interpretations of all the variables occurring in .

Proposition 3.2 (soundness)

Suppose , then , for all arithmetical interpretations .

Proof. Induction on the length of proof of in . The validity of the first two groups of rules of is obvious. We treat the modal axioms and rules.

If then clearly , for each . Since this fact is formalizable in , we also obtain . Also, the validity of the monotonicity axioms 6 is clear. Next we need the following lemma.

Lemma 3.3
  1. Let be numerated by and . If then ;

  2. Statement (i) holds provably in uniformly in , that is,

Proof. We only prove Statement (ii). We reason in as follows.

Assume and . Then . On the other hand, by the definition of

This yields

so we obtain , and hence by (1).

Corollary 3.4
  1. , for all ;

  2. .

Proof. Since the theories numerated by and are finite extensions of , for a proof of Statement (i) it is sufficient to show

(2)

Since is a -sentence, we can take in Lemma 3.3 and . This yields statement (2).

For a proof of (ii), we show an informal version of this statement by an argument formalizable in . We must prove that, for each ,

Using the monotonicity and Statement (i) we reason as follows:

This shows the claim.

Corollary 3.4 shows the soundness of the third group of rules of . As we mentioned above, the fourth group is actually derivable from the first three and the monotonicity, so we can skip it. We show the soundness of Axiom 5.

Lemma 3.5

If then .

Proof. We reason in as follows: If and , then by the formalized deduction theorem . Since , the formula belongs to . By we obtain whence . Since , from we infer . Hence , as required.

Corollary 3.6

.

Proof. Informally, we must prove, for each , that

We can assume and then use the previous lemma. This argument is formalizable in .

Corollary 3.7

.

Proof. We reason as follows:

This shows the soundness of the remaining Axiom 7 of and completes the proof of Proposition 3.2.

4 Kripke models for

Kripke frames and models are understood in this paper in the usual sense. A Kripke frame for the language consists of a non-empty set equipped with a family of binary relations . A Kripke frame is called finite if so is and all but finitely many relations are empty.

A Kripke model is a Kripke frame together with a valuation assigning a truth value to each propositional variable at every node of . As usual, we write to denote that a formula is true at a node of a model . This relation is inductively defined as follows:

  • , for each ;

  • ;  ;

  • .

We call a RJ-frame a Kripke frame satisfying the following conditions, for all and all :

  • implies , if ; (polytransitivity)

  • and implies , if . (condition J)

These conditions can be more succinctly written as and . An RC-frame is an RJ-frame that is monotone, that is, , for each . An RJ-model (RC-model), respectively, is a Kripke model based on an RJ-frame (RC-frame). We speak about RJ- and RC-frames and models whenever .

The persistence axiom does not correspond to a frame condition.4 We call a Kripke model (downwards) persistent if, for each variable ,

By a straightforward induction we obtain the following lemma.

Lemma 4.1

Let be a persistent Kripke model based on a polytransitive frame. Then, for each positive formula ,

We say that a sequent is true in a Kripke model , if

A logic is sound for a class of Kripke models (of the same signature), if every sequent provable in is true in any model from . It is easy to see that our logics are sound for their respective classes of models.

Lemma 4.2
  1. is sound for the class of all RJ-models;

  2. is sound for the class of all RC-models;

  3. is sound for the class of all persistent RC-models.

A proof of this lemma is routine.

Notice that the frame conditions for the logics and (that is, polytransitivity, condition J, and monotonicity) are closure conditions. Therefore, for any Kripke frame there is an RJ-frame (RC-frame) such that

  • , for each ;

  • For any other RJ-frame (RC-frame) with , for all , we have , for all .

The frame is unique up to isomorphism. We call it the RJ-closure (RC-closure) of .

Example 4.3

Consider a Kripke frame with . Relation consists of two pairs and , and the other relations are empty. Let be the RC-closure of . It is easy to see that , whereas, for each , is a total relation, and .

Further, we define iff , and iff . This makes a downwards persistent RC-model falsifying at . By Lemma 4.2 we conclude .

The completeness proofs in all these cases are also easy. As in Dashkov (11), we present an argument based on a (simplified) version of filtrated canonical model.

Let be a set of -formulas. Denote . A set is called adequate if is closed under subformulas, and

  • If and , then ;

  • For any variable , if then .

It is easy to see that any finite set of formulas can be extended to a finite adequate set.

Let be a set of -formulas and a logic. We shall take for one of , or , or their restricted versions in the language where . We write if there are formulas such that the sequent is provable in .

Fix an adequate set of formulas . An -theory in is a set such that and implies . Define a model as follows. The set of nodes is the set of all -theories in .5 We stipulate that iff and the following conditions hold for each formula :

  1. and implies ;

  2. and implies ;

  3. and implies .

We also let iff , for any -theory .

Lemma 4.4

Suppose contains with . Then is an RJ-model.

Proof. To check the polytransitivity assume and . We show by checking R1–R3. If and , then by the adequacy and hence . It follows that .

For R2 notice that . If and then by the adequacy and hence . This in turn implies . Condition R3 is obviously satisfied, as all three theories have the same formulas of the form for .

Second, we check condition (J). Assume and with . We show . R1: If and then . Since this implies . R2: If and then whence for the same reason. R3 is, again, obvious.

Lemma 4.5

For any , iff .

Proof. Induction on the build-up of . If is a variable, or has the form , the argument is obvious. Assume .

If then, for some such that , we have . By IH it follows that and hence .

Now assume . Let and let be the deductive closure of in . By the IH we have . We claim that which completes the argument.

Assume , then for some finite . Then . Hence, if then . Similarly, if then . Then . If then whence . Finally, if and then , hence .

Lemma 4.6
  1. If contains the monotonicity axiom and , then is an RC-frame;

  2. If contains the persistence axiom, then is persistent.

Proof. (i) Assume and . We show by checking the three conditions. If and then by the adequacy of . Hence, and therefore by the monotonicity axioms . Since we obtain . Similarly, if and then by the adequacy. Therefore, , whence by the monotonicity axioms . Since both and are in , it follows that which proves the second condition. The third condition is obviously satisfied.

(ii) Assume . If then ; by the adequacy and hence . It follows that and .

Taking and we obtain the completeness of , and w.r.t. their respective classes of models.

Theorem 1
  1. iff is true in all RJ-models;

  2. iff is true in all RC-models;

  3. iff is true in all persistent RC-models.

Proof. The three systems are sound by Lemma 4.2. The completeness is proved by observing that , for each of the three logics , is a model of the corresponding type. Assume . Then letting denote the -theory generated by we have , hence by Lemma 4.5 .

Next we discuss the finite model property of the three logics. For the answer is obvious, but for and we have a small complication due to the fact that modality is present in the language.

Corollary 4.7

iff is true in all finite RJ-models.

Proof. Assume , let be a finite adequate set of formulas containing both and . We have and . Moreover, is a finite RJ-model, where . By putting , for any , we expand to an RJ-model in falsifying .

A similar argument does not quite work for , as the expansion by empty relations leads, in general, outside the class of RC-models. However, for a finite signature we do have an analog of Theorem 1.

Corollary 4.8

Suppose is finite.

  1. iff is true in all finite RC-models;

  2. iff is true in all finite persistent RC-models.

Lemma 4.9

Let and . Any RC-model can be expanded to an RC-model.

Proof. Let be a given RC-model. Denote: and . If we can put . Otherwise, for any relation on , denote

Further, define , where ;

Notice that , for any . It follows that , for each . By the construction, is transitive and , hence condition (J) is satisfied for all . Moreover, the polytransitivity follows from the transitivity and the monotonicity properties. Therefore, is an RC-model.

To complete the argument we have to show that is an RC-model. To this end we prove that, for each and ,

  1. ;

  2. .

Both statements are verified by induction on . The basis of induction holds, since the original model was an RC-model. Assume the statements hold for and consider .

1. We have by the IH. Further, , since is transitive. For any , , since condition (J) holds in . Hence, , as required.

2. We have by the IH. Further, . Finally, for any , . Therefore, , as required.

Remark 4.10

The given proof also works for the more general analogs of , e.g., for logics with linearly ordered sets of modalities (see (1)).

Taking into account that expansions of persistent models are persistent, we obtain the following theorem for both and .

Theorem 2

Let be either or . The following statements are equivalent:

  1. ;

  2. , for some finite ;

  3. where .

Proof. Clearly, (iii) implies (i), and (i) implies (ii) since a finite derivation may only contain finitely many different modalities. We prove that (ii) implies (iii). Assume . By Corollary 4.8 there is a finite RC-model falsifying (which is persistent if ). Assume any finite be given. We may assume (otherwise clearly ). By Lemma 4.9, can be expanded to an RC-model falsifying the same sequent. Hence, .

Thus, even though we do not have the finite model property for and in the full language, these logics are conservatively approximated by their fragments with this property. Together with Corollary 4.8 this yields

Corollary 4.11

The systems and are decidable.

For the logics and a sharper result can be stated. As we have seen, the question whether a sequent is provable in such a logic is equivalent to the same question for the logic with . However, for any finite , the logic is modulo renaming of modalities the same logic as for (we identify with the set ). The systems are shown to be polytime decidable (11). Therefore, we obtain

Corollary 4.12

The systems and are polytime decidable.

The same result holds for , however we cannot directly refer to Dashkov’s theorem. This question is considered in the next section, where we also obtain somewhat sharper complexity estimates for the cases and . The material of that section, up to Theorem 3, is due to Dashkov (11).

5 Polytime decidability of

We have to develop some combinatorial techniques to deal with positive logics. It allows one to state the Kripke completeness results in a sharper form, from which the complexity bounds are easily read off.

Let be a Kripke model and . The submodel of generated by is obtained by restricting all the relations and the valuation of to the set of all nodes such that there is a path where . A model is called rooted if it has a distinguished element (called the root) such that . We notice that in polytransitive rooted frames every node is reachable from the root in one step.

Definition 5.1

We can associate with each positive formula a rooted treelike Kripke model in the signature called its canonical tree. It is essentially the parse tree of viewed as a Kripke model.

If is a variable or , then is a one-point model with the empty relations, and the only variable true at is .

If then is obtained from the disjoint union of the models and by identifying the roots. We declare any variable true at the root of iff it is true at the root of either or .

If then is obtained from by adding a new root (where all variables are false), from which the root of of is -accessible.

We write if is true at the root of . Then, one can easily verify the following properties:

  • Each on is an irreflexive forest-like binary relation;

  • .

Definition 5.2

A homomorphism of a Kripke model into a Kripke model (of the same signature ) is a function such that

  • , for each ;

  • If then , for each variable .

Let and be rooted Kripke models. A simulation of by is a homomorphism mapping the root of to the root of .

Lemma 5.3

If is strictly positive and is a homomorphism of into , then

Lemma 5.4

iff there is a homomorphism mapping the root of to .

Proof. Suppose is such a homomorphism. We have where is the root of . Since is strictly positive and , by Lemma 5.3, .

Suppose . We construct a homomorphism by induction on the complexity of . If is a variable or , the claim is obvious.

If then . By the IH, there are homomorphisms of the models and into mapping their respective roots to . The homomorphism of maps its root to and is defined as the union of and everywhere else on . We note that if then either or , by the definition of . In either case we have , therefore the variable condition at the root is met and we have a homomorphism of into .

If and , then there is a node such that and . By the IH, there is a homomorphism of into mapping its root to . We extend it by mapping the root of to . All the variables are false at the root of , so the variable condition is met.

Let () denote the RC-closure (respectively, RJ-closure) of , where .

Theorem 3
  1. iff