aa r X i v : . [ m a t h . L O ] J un Interpretability in PRA
Marta B´ılkov´a, Dick de Jongh and Joost J. Joosten2009
Abstract
In this paper from 2009 we study IL (PRA), the interpretability logicof PRA. As PRA is neither an essentially reflexive theory nor finitelyaxiomatizable, the two known arithmetical completeness results do notapply to PRA: IL (PRA) is not IL M or IL P . IL (PRA) does of coursecontain all the principles known to be part of IL (All), the interpretabilitylogic of the principles common to all reasonable arithmetical theories. Inthis paper, we take two arithmetical properties of PRA and see whattheir consequences in the modal logic IL (PRA) are. These propertiesare reflected in the so-called Beklemishev Principle B , and Zambella’sPrinciple Z , neither of which is a part of IL (All). Both principles and theirinterrelation are submitted to a modal study. In particular, we prove aframe condition for B . Moreover, we prove that Z follows from a restrictedform of B . Finally, we give an overview of the known relationships of IL (PRA) to important other interpetability principles. The notion of a relativized interpretation occurs in many places in mathematicsand in mathematical logic. If a theory T interprets a theory S , we shall write T ✄ S , which then, roughly, means that there is a translation · t from symbolsin the language of S to formulas in the language of T such that any theorem of S becomes a theorem of T under the canonical extension of this translation toformulas. In the notion of interpretation that we are interested in, the logicalstructure of formulas has to be preserved under the translation. Thus, forexample, ( ϕ ∨ ψ ) t = ϕ t ∨ ψ t and in particular ⊥ t = ( ∨ ∅ ) t = ∨ ∅ = ⊥ . We referthe reader to [17], [5] and [15] for precise definitions and examples.In this paper, we shall not go much into the technical details of interpre-tations. Rather, we are interested in the structural behavior of this notion ofinterpretability. In particular, we are interested in the structural behavior of We thank Lev Beklemishev for his help and suggestions. Evan Goris did a thoroughproofread of an early draft and suggested a simplification of the notion of B-simulation. Wethank Albert Visser for fruitful discussions and challenges. We also thank Franco Montagnafor his many contributions to the subject. Two unknown referees improved our paper con-siderably with their remarks and suggestions. Supported by grants GA CR 401/06/0387 andIAA900090703. T . An easyexample of such a structural property is the transitivity of interpretations:( T + α ✄ T + β ) ∧ ( T + β ✄ T + γ ) → ( T + α ✄ T + γ ) . We can use so-called interpretability logics to capture, in a sense, the completestructural behavior of interpretability between sentential extensions of a certainbase theory. We shall soon say a bit more on this. For now it is important tonote that for a large collection of theories, the interpretability logic is known.We call a theory reflexive if it proves the consistency of any of its finitesub-theories (as sets of axioms). We call a theory essentially reflexive if anyfinite sentential extension of it is reflexive. It is easy to see that any theorywith full induction, like Peano Arithmetic, is essentially reflexive. The inter-pretability logic of essentially reflexive theories was determined independentlyby Berarducci and Shavrukov ([4], [13]). We shall encounter this logic belowunder the name of IL M . The principle ( A ✄ B ) → ( A ∧ ✷ C ✄ B ∧ ✷ C ) whichis the particular feature of this system. It is called Montagna’s principle sinceit arose during the original discussions between Franco Montagna and AlbertVisser about the modal principles underlying interpetability logic. It was knownto Lindstr¨om and ˇSvejdar in arithmetic disguise before.It turns out that theories which are finitely axiomatizable and which containa sufficient amount of arithmetic, have a different interpretability logic which iscalled IL P . In [17], the first proof was given.For no theory that is neither finitely axiomatizable nor essentially reflexive,the interpretability logic is known. PRA is one such theory. In this paper, weshall make some first attempts to work out the interpretability logic of PRA.As such, this paper also fits into a larger project. As pointed out above, dif-ferent arithmetical theories have different interpretability logics. A question thatis open since a long time concerns the logic of the core principles that pertain to all reasonable arithmetical theories - IL (All). As PRA is certainly a ‘reasonablearithmetical theory’, this core logic should also be a part of IL (PRA). In thispaper we shall not focus too much on the principles in the core logic. Rathershall we consider the interpretability behavior of PRA that is typical for thistheory.One such principle that is characteristic for PRA is Beklemishev’s principlethat shall be studied closely in this paper. This principle exploits the fact thatany theory which is an extension of PRA by Σ sentences is reflexive. Wegive a characterization of this principle in terms of the modal semantics forinterpretability logics.A topic that is closely related to interpretability logics, is that of Π -conservativitylogics. A theory S is Π conservative over a theory T in the same language ofarithmetic, we shall write S ✄ Π T whenever S proves any Π theorem that isproven by T . In symbols: T ⊢ π = ⇒ S ⊢ π for any π ∈ Π . It is easy to see thatfor any Σ sentence σ , the following is a valid principle S ✄ Π T → S + σ ✄ Π T + σ .This principle is the basis for Montagna’s principle for interpretability logic, and2eklemishev’s principle which is studied in this paper is a restriction of Mon-tagna’s principle.When T and S are both reflexive theories we have that S ✄ T ↔ S ✄ Π T .This equivalence was exploited by H´ajek and Montagna who were the first toshow that the Π -conservativity logic of PA is IL M as well [9]. The observationabout the equivalence is more generally important when looking at the repercus-sions of Π -conservativity principles on interpretability logics. In this paper weshall consider Zambella’s principle for Π -conservativity logics and look at itsrepercussions for the interpretability logic of PRA. We shall show that Zambelladoes not add new information in the sense that its modal-logical consequencesare already implied by Beklemishev’s principle.It is remarkable that the notion of interpretability is, in a sense, less stablethan that of Π -conservativity. H´ajek and Montagna show that their resultsextends to all reasonable theories containing IΣ . This was strengthened byBeklemishev and Visser in [3]: all theories extending the parameter-free in-duction schema IΠ − have the same Π -conservativity logic ( IL M ) whereas inthis range the interpretability logics expose a diverse and wild behavior. Notethough that PRA does not prove IΠ − , and, in fact, the Π -conservativity logicof PRA remains unknown.A number of the results in this paper was first proved in [10]. Let us first fix some arithmetical notation. We use modal symbols ✷ , ✸ , ✄ bothin modal and arithmetical statements, here we fix their arithmetical meaning.We write, for an arithmetical sentence α , ✷ T α for formalized provability inT, ✷ T ,n α for formalized provability of α in T using only non-logical axiomswith G¨odel numbers ≤ n and formulas of logical complexity ≤ n . Dually, ✸ T α = ¬ ✷ T ¬ α means formalized consistency of α over T (i.e. nonexistence ofa proof of a contradiction from α ), while ✸ T ,n α means ¬ ✷ T ,n ¬ α . For theoriesT , S we use T ✄ S to denote formalized interpretability of S in T. For arithmeticalsentences α, β , α ✄ T β means T + α ✄ T + β . Similarly for theories T , S, ✄ Π denotes formalized Π -conservativity of T over S and for arithmetical sentences α, β , α ✄ Π β means T + α ✄ Π T + β . PRA ? In the literature there are many definitions of PRA. Probably the best knowndefinition uses a language that contains a function symbol for every primitive re-cursive function. The axioms contain the defining equations of these functions.Moreover, there are induction axioms for each ∆ -formula in this enriched lan-guage. Since PRA proves superexponentiation this is, in the case under study , equivalent to therestriction of axioms to those ≤ n ω . Here, (EA) ω is the theory that isobtained by starting with EA (= I∆ + exp ) and iterating ‘ ω many times’ Π -reflection. In symbols: (EA) = EA, and (EA) n +1 = RFN (EA) n (Π ).In this paper, we shall use the definition:PRA := (EA) ω . Under this definition, the following lemma is immediate.
Lemma 2.1.
Any r.e. extension of
PRA by Σ sentences is reflexive. All theories that are mentioned here are supposed to be consistent and havea poly-time recognizable axiomatization. Orey and H´ajek have given severalequivalent conditions on theories which express that the one interprets the other.In this subsection we shall briefly mention the one we shall need and refer tothe literature for proofs.
Lemma 2.2.
Whenever T is reflexive we have that T ✄ S ⇔ ∀ x T ⊢ ¬ ✷ S,x ⊥ Moreover in the presence of the totality of exponentiation this equivalence canbe formalized. ⊢ T ✄ S ↔ ∀ x ✷ T ¬ ✷ S,x ⊥ In [10] an overview is given of all the implications, corresponding require-ments and necessary arguments regarding Orey-H´ajek. In the above Lemma the ⇐ does not need the requirement of reflexivity and can actually be formalizedin S . For the other direction reflexivity is needed, and for its formalization,the totality of exp as well.Note that, using the above characterization, the prima facie Σ notion ofinterpretability becomes Π . Similarly as formalized provability can be captured by modal provability logic,we can use modal logic to reason about formalized interpretability. Modal logicproved to be an extremely useful tool to reason about such formalized phenom-ena since it can visualize their behaviour using a simple language and an intuitiveframe semantics. Perhaps the most significant point where modal logic showsits skills are completeness proofs - arithmatical completeness proofs are basedon modal completeness proofs obtained by rather standard method of modeltheory of modal logics. For more on material contained in this section we referto [17, 10, 8]. 4e will work with modal propositional language containing two modalities -a unary ✷ modality for provability and a binary ✄ modality for interpretability.Modal interpretability formulas are defined as follows: A ::= p | ⊥ | ( A ∧ A ) | ( A → A ) | ( ✷ A ) | ( A ✄ A )We will use standard abbreviations ✸ , ∨ , ¬ , ⊤ , ↔ , and we write A ≡ B in-stead of ( A ✄ B ) ∧ ( B ✄ A ). We shall often omit brackets writing formulas. We saythat ¬ , ✷ , and ✸ bind equally strong, they bind stronger then equally strongbinding ∨ and ∧ which in turn bind stronger then ✄ . The weakest bindingconnectives are → and ↔ .An arithmetical interpretation of modal formulas is given by arithmeticalrealizations : for an arithmetical theory T, an arithmetical T-realization is a map ∗ sending propositional variables p to arithmetical sentences p ∗ . It is extendedto interpretability modal formulas as follows: first ∗ commutes with all booleanconnectives. Moreover ( ✷ A ) ∗ = ✷ T A ∗ and ( A ✄ B ) ∗ = A ∗ ✄ T B ∗ , i.e. ∗ translates modal operators to formalized provability and interpretability over Trespectively.An interpretability principle of an arithmetical theory T is a modal formula A such that ∀∗ T ⊢ A ∗ . The interpretability logic of a theory T, denoted IL (T),is then the set of all the interpretability principles of T. The logic IL is in a sense the core interpretability logic - it is a (proper) part ofthe interpretability logic of any reasonable arithmetical theory: IL ⊂ IL (T). Itcaptures the basic structural behaviour of interpretability. IL is defined as the smallest set of formulas containing all propositionaltautologies, all instantiations of the following schemata, and is closed under theNecessitation and Modus Ponens rules: L1 ✷ ( A → B ) → ( ✷ A → ✷ B ) L2 ✷ A → ✷✷ A L3 ✷ ( ✷ A → A ) → ✷ A J1 ✷ ( A → B ) → A ✄ B J2 ( A ✄ B ) ∧ ( B ✄ C ) → A ✄ C J3 ( A ✄ C ) ∧ ( B ✄ C ) → A ∨ B ✄ C J4 A ✄ B → ( ✸ A → ✸ B ) J5 ✸ A ✄ A Note that the part of IL not containing the ✄ modality is the well-knownG¨odel-L¨ob provability logic GL , axiomatized by the first three schemata. It iseasy to show that ✷ can be defined in terms of ✄ modality: ⊢ IL ✷ A ↔ ¬ A ✄ ⊥ .More interpretability logics are obtained extending IL by new interpretabil-ity principles. Some of such principles are listed below:5 A ✄ B → A ✄ B ∧ ✷ ¬ A W ∗ A ✄ B → B ∧ ✷ C ✄ B ∧ ✷ C ∧ ✷ ¬ A M A ✄ B → ✸ A ∧ ✷ C ✄ B ∧ ✷ C M A ✄ B → A ∧ ✷ C ✄ B ∧ ✷ C P A ✄ B → ✷ ( A ✄ B ) R A ✄ B → ¬ ( A ✄ ¬ C ) ✄ B ∧ ✷ C R ∗ A ✄ B → ¬ ( A ✄ ¬ C ) ✄ B ∧ ✷ C ∧ ✷ ¬ A All of these principles are in IL (All) except the principles M and P whichwere mentioned above already. For an overview, see [17] and [8]. For the lastword on IL (All) see [11].For X a set of principles we denote IL X the logic extending IL with schematafrom X .There are some results considering arithmetical completeness of interpretabil-ity logics: it was shown in [4],[13] that the interpretability logic of an essentiallyreflexive theory (as e.g. PA) is IL M . For finitely axiomatizable theories con-taining supexp the interpretability logic is known to be IL P ([16]).An important consequence of IL M that expresses the Π -conservativity ofinterpretability more directly is ( A ✄ ✸ B ) → ✷ ( A → ✸ B ). Modal frame semantics of interpretability logics is based on GL -frames ex-tended with a ternary accesibility relation interpreting the binary ✄ modality.The ternary relation is however given by a set of binary relations indexed bythe nodes: Definition 3.1. An IL -frame (a Veltman frame) is a triple h W, R, S i where W is a nonempty universe, R is a binary relation on W , and S is a set of binaryrelations on W , indexed by elements of W such that . R is transitive and conversely well-founded . yS x z ⇒ xRy & xRz . xRy ⇒ yS x y . xRyRz ⇒ yS x z . uS x vS x w ⇒ uS x w An IL -model is a quadruple h W, R, S, (cid:13) i where h W, R, S i is a IL -frame and (cid:13) is a subset of W × Prop , extending to boolean formulas as usualy and to modalformulas as follows: w (cid:13) ✷ A iff ∀ v ( wRv ⇒ v (cid:13) A ) w (cid:13) A ✄ B iff ∀ u ( wRu & u (cid:13) A ⇒ ∃ v ( uS w v (cid:13) B ))We adopt standard definitions of validity of a modal formula in a model andin a frame. Moreover, let X be a scheme of interpretability logic. We say that6 formula C in first or higher order logic is a frame condition for X if, for eachframe F , F | = C iff F | = X . Let us list some known frame conditions (to be read universally quantified): M xRyS x zRu ⇒ yRu M xRyRzS x uRv ⇒ yRv P xRyRzS x u ⇒ yRu ∧ zS y u W ( S w ; R ) is conversely well-founded R xRyRzS x uRv ⇒ zS y v We have the following completeness results: IL is sound and complete w.r.t.(finite) IL frames, IL P is complete w.r.t. (finite) IL P frames (all in [6]), IL W is complete w.r.t. (finite) IL W frames ([7], see also [8]), IL M is complete w.r.t.(finite) IL M frames (in [6], also in [4]), It is possible to write down a valid principle specific for the interpretability logicof PRA. This was first done by Beklemishev (see [17]). Beklemishev’s principle B exploits the fact that any finite Σ -extension of PRA is reflexive, together withthe fact that we have a good Orey-H´ajek characterization for reflexive theories.It turns out to be possible to define a class of modal formulae which are underany arithmetical realization provably Σ in PRA. These are called essentially Σ -formulas , we write ES . Let us start by defining this class and some relatedclasses.The idea behind this definition is as follows. It is clear that each modalformula that starts with a ✷ will become under any arithmetical realization aΣ formula. Likewise, taking Lemma 2 into account, we see that any formulaof the form A ✄ B where A is Σ , will be under any arithmetical realization ofcomplexity Π and hence, ¬ ( A ✄ B ) will again be Σ . Note that we are here onlyformulating sufficient conditions. It turns out to be rather tough to show theseclasses actually cover, up to provable equivalence, all formulae in the intendedcomplexity class.The class BS denotes the formulae that are boolean combinations of Σ formulae ad thus certainly ∆ . Likewise, ES and ES , stands for those modalformulae that are under any arithmetical realization always Σ or Σ respec-tively.In our definition, A will stand for the set of all modal interpretability for-mulae. BS ::= ✷ A | ¬ BS | BS ∧ BS | BS ∨ BS ES ::= ✷ A | ¬ ✷ A | ES ∧ ES | ES ∨ ES | ¬ ( ES ✄ A ) ES ::= ✷ A | ¬ ✷ A | ES ∧ ES | ES ∨ ES | A ✄ A ES ::= ✷ A | ¬ ES | ES ∧ ES | ES ∨ ES | A ✄ A n ≥ ES n := ES . We can now formulate Beklemishev’s principle B . B := A ✄ B → A ∧ ✷ C ✄ B ∧ ✷ C for A ∈ ES Note that B is just Montagna’s principle M restricted to ES -formulas. Lemma 4.1. IL B ⊢ B ′ , where B ′ : A ✄ B → A ∧ C ✄ B ∧ C with A ∈ ES and C a CNF ( a conjunction of disjunctions ) of boxed formulas.Proof. Easy. B By Lemma 2.1 we know that PRA + σ is reflexive for any Σ (PRA)-sentence σ .Thus, we get by Orey-H´ajek thatPRA ⊢ σ ✄ PRA ψ ↔ ∀ x ✷ PRA ( σ → ✸ PRA ,x ψ ) . (1)Consequently, for σ ∈ Σ (PRA), ¬ ( σ ✄ PRA ψ ) ∈ Σ (PRA) and we see that,indeed, ∀ A ∈ ES ∀ ∗ A ∗ ∈ Σ (PRA). This enables us to prove the arithmeticalsoundness of B . Theorem 5.1.
For any formulas B and C we have that ∀ A ∈ ES ∀ ∗ PRA ⊢ ( A ✄ B → A ∧ ✷ C ✄ B ∧ ✷ C ) ∗ .Proof. For some A ∈ ES and arbitrary B and C , we consider some realization ∗ and let α := A ∗ , β := B ∗ and γ := C ∗ . We reason in PRA and assume α ✄ PRA β . As α is Σ (PRA), we get by (1) that ∀ x ✷ PRA ( α → ✸ PRA ,x β ) . (2)We now consider n large enough (dependent on γ ) such that ✷ PRA ( ✷ PRA γ → ✷ PRA ,n ✷ PRA γ ) . (3)From general observations we have that, for large enough n , ✷ PRA ,n ( δ → ¬ ǫ ) ∧ ✷ PRA ,n δ → ✷ PRA ,n ¬ ǫ, whence ✸ PRA ,n ǫ ∧ ✷ PRA ,n δ → ✸ PRA ,n ( δ ∧ ǫ ) (4)Combining (2), (3), and using (4), we see that for any n , ✷ ( α ∧ ✷ γ → ✸ PRA ,n ( β ∧ ✷ γ )). Clearly, α ∧ ✷ γ is still a Σ (PRA)-sentence. Again by (1)we get α ∧ ✷ γ ✄ β ∧ ✷ γ .Let M ES n be the schema A ✄ B → A ∧ ✷ C ✄ B ∧ ✷ C with A ∈ ES n . Theorem5.1 can be generalized using results of [1] to the theory IΣ Rn , which is Robinson’sarithmetic Q plus the Σ n induction rule , for n = 1 , , Theorem 5.2. IL (IΣ Rn ) ⊢ M ES n + for n = 1 , , . Actually, this observation is not necessary as we use the direction in the Orey-H´ajekCharacterization that does not rely on the reflexivity. A frame condition for B Let us first fix some notation. If C is a finite set, we write xR C as short for VV c ∈C xRc . Similar conventions hold for the other relations. The A -criticalcone of x , C Ax is in this section defined as C Ax := { y | xRy ∧ ∀ z ( yS x z → z (cid:13) A ) } .By x ↑ we denote the set of worlds that lie above x w.r.t. the R relation.That is, x ↑ := { y | xRy } . With yS x ↑ we denote the set of those z for which yS x z .We will consider frames both as modal models without a valuation and asstructures for first- (or sometimes second) order logic. We say that a model M is based on a frame F if F is precisely M with the (cid:13) relation left out.In this subsection we give the frame condition of Beklemishev’s principle.Our frame condition holds on the class of finite frames. At first sight, thecondition might seem a bit awkward. On second sight it is just the framecondition of M with some simulation built in. First we approximate the class ES by stages. Definition 6.1. ES := BS ES n + := ES n2 | ES n + ∧ ES n + | ES n + ∨ ES n + | ¬ ( ES n2 ✄ A )It is clear that ES = ∪ i ES i2 . We now define some first order formulas S i ( b, u )that say that two nodes b and u in a frame look alike. The larger i is, the morethe two points look alike. We use the letter S as to hint at a simulation. Definition 6.2. S ( b, u ) := b ↑ = u ↑S n +1 ( b, u ) := S n ( b, u ) ∧∀ c ( bRc → ∃ c ′ ( uRc ′ ∧ S n ( c, c ′ ) ∧ c ′ S u ↑ ⊆ cS b ↑ ))By induction on n we easily see that ∀ n F | = S n ( b, b ) for all frames F andall b ∈ F . For i ≥ S i ( b, u ) is in general not symmetric. However itis not hard to see that the S i are transitive and reflexive. Lemma 6.3.
Let F be a model. For all n we have the following. If F | = S n ( b, u ) ,then b (cid:13) A ⇒ u (cid:13) A for all A ∈ ES n2 .Proof. We proceed by induction on n . If n =0, A ∈ ES can be written as WW i ( ✷ A i ∧ VV j ✸ A ij ). Clearly, if b ↑ = u ↑ then b (cid:13) A ⇒ u (cid:13) A .Now consider A ∈ ES n + and b and u such that F | = S n +1 ( b, u ). We can write A = __ i ( A i ∧ ^^ j =0 ¬ ( A ij ✄ B ij )) , with A ij in ES n2 . If b (cid:13) A , then for some i , b (cid:13) A i ∧ VV j =0 ¬ ( A ij ✄ B ij ). As S n +1 ( b, u ) → S n ( b, u ), and by the induction hypothesis we see that u (cid:13) A i .So, we only need to see that u (cid:13) ¬ ( A ij ✄ B ij ) for j =0. As b (cid:13) ¬ ( A ij ✄ B ij ),for some c ∈ C B ij b we have c (cid:13) A ij . By S n +1 ( b, u ) we find a c ′ such that uRc ′ ,9nd c ′ S u ↑ ⊆ cS b ↑ (thus cS b c ′ ). This guarantees that c ′ ∈C B ij u . Moreover we knowthat S n ( c, c ′ ), thus by the induction hypothesis, as c (cid:13) A ij , we get that c ′ (cid:13) A ij .Consequently u (cid:13) ¬ ( A ij ✄ B ij ). Lemma 6.4.
Let F be a finite frame. For all i , and any b ∈ F , there is avaluation V bi on F and a formula A bi ∈ ES i2 such that F | = S i ( b, u ) ⇔ u (cid:13) A bi .Proof. The proof proceeds by induction on i . First consider the basis case, thatis, i =0. Let b ↑ be given by the finite set { x j } j ∈ J . We define y (cid:13) p j ⇔ y = x j y (cid:13) r ⇔ bRy. Let A b be ✷ r ∧ VV j ✸ p j . It is now obvious that u (cid:13) A ⇔ u ↑ = b ↑ .For the inductive step, we fix some b and reason as follows. First, let V bi and A bi be given by the induction hypothesis such that u (cid:13) A bi ⇔ F | = S i ( b, u ).We do not specify the variables in A i but we suppose they do not coincide withany of the ones mentioned below. Let b ↑ = { x j } j ∈ J . The induction hypothesisgives us sentences A ji (no sharing of variables) and valuations V ji such that F, u (cid:13) A ji ⇔ F | = S i ( x j , u ).Let { q j } j ∈ J be a set of fresh variables. V bi +1 will be V bi and V ji on the oldvariables. For the { q j } j ∈ J we define V bi +1 to act as follows: y (cid:13) q j ⇔ y x j S b ↑ . Moreover we define A bi +1 := A bi ∧ ^^ j ¬ ( A ji ✄ q j ) . Now we will see that under the new valuation V bi +1 ,( i ) u (cid:13) A bi +1 ⇒ F | = S i +1 ( b, u ),( ii ) F | = S i +1 ( b, u ) ⇒ u (cid:13) A bi +1 .For ( i ) we reason as follows. Suppose u (cid:13) A bi +1 . Then also u (cid:13) A bi and thus F | = S i ( b, u ). It remains to show that F | = ∀ c ( bRc → ∃ c ′ ( uRc ′ ∧ S i ( c, c ′ ) ∧ cS b c ′ ∧ c ′ S u ↑ ⊆ cS b ↑ )) . To this purpose we consider and fix some x j in b ↑ . As u (cid:13) A bi +1 , we getthat u (cid:13) ¬ ( A ji ✄ q j ). Thus, for some c ′ ∈ C q j u , c ′ (cid:13) A ji . Clearly c ′ (cid:13) ¬ q j whence x j S b c ′ . Also ∀ t ( c ′ S u y ⇒ y (cid:13) ¬ q j ) which, by the definition of V bi +1 translatesto c ′ S u ↑ ⊆ x j S b ↑ . Clearly also uRc ′ . By c ′ (cid:13) A ji and the induction hypothesiswe get that S i ( x j , c ′ ). Indeed we see that F | = S i +1 ( b, u ).For ( ii ) we reason as follows. As F | = S i +1 ( b, u ), also F | = S i ( b, u ) and bythe induction hypothesis, u (cid:13) A bi . It remains to show that u (cid:13) ¬ ( A ji ✄ q j ) for10ny j . So, let us fix some j . Then, by the second part of the S i +1 requirementwe find a c ′ such that uRc ′ ∧ S i ( x j , c ′ ) ∧ x j S b c ′ ∧ c ′ S u ↑ ⊆ x j S b ↑ . Now, uRc ′ ∧ x j S b c ′ ∧ c ′ S u ↑ ⊆ x j S b ↑ gives us that c ′ ∈ C q j u . By S i ( x j , c ′ ) and theinduction hypothesis we get that c ′ (cid:13) A ji . Thus indeed u (cid:13) ¬ ( A ji ✄ q j ).Note that in the proof of this lemma, we have only used conjunctions toconstruct the formulas A bi . Definition 6.5.
For every i we define the frame condition C i to be ∀ a, b ( aRb → ∃ u ( bS a u ∧ S i ( b, u ) ∧ ∀ d, e ( uS a dRe → bRe ))) . Lemma 6.6.
Let F be a finite frame. For all i , we have thatfor all A ∈ ES i2 , F | = A ✄ B → A ∧ ✷ C ✄ B ∧ ✷ C ,if and only if F | = C i .Proof. First suppose that F | = C i and that a (cid:13) A ✄ B for some A ∈ ES i2 and somevaluation on F . We will show that a (cid:13) A ∧ ✷ C ✄ B ∧ ✷ C for any C . Considertherefore some b with aRb and b (cid:13) A ∧ ✷ C . The C i condition provides us witha u such that bS a u ∧ S i ( b, u ) ∧ ∀ d, e ( uS a dRe → bRe ) ( ∗ )As F | = S i ( b, u ), we get by Lemma 6.3 that u (cid:13) A . Thus, as aRu and a (cid:13) A ✄ B ,we know that there is some d with uS a d and d (cid:13) B . If now dRe , by ( ∗ ), also bRe and hence e (cid:13) C . Thus, d (cid:13) B ∧ ✷ C . Clearly bS a d and thus a (cid:13) A ∧ ✷ C ✄ B ∧ ✷ C .For the opposite direction we reason as follows. Suppose that F = C i . Thus,we can find a, b with aRb ∧ ∀ u ( bS a u ∧ S i ( b, u ) → ∃ d, e ( uS a dRe ∧ ¬ bRe )) ( ∗∗ ) . By Lemma 6.4 we can find a valuation V bi and a sentence A bi ∈ ES i2 such that u (cid:13) A bi ⇔ F | = S i ( b, u ). Let q and s be fresh variables. Moreover, let D be thefollowing set. D := { d ∈ F | bS a dRe ∧ ¬ bRe for some e } . We define a valuation V that is an extension of V bi by stipulating that y (cid:13) q ↔ ( y ∈D ) ∨ ¬ ( bS a y ) ,y (cid:13) s ↔ bRy. We now see that( i ) a (cid:13) A bi ✄ q ,( ii ) a (cid:13) ¬ ( A bi ∧ ✷ s ✄ q ∧ ✷ s ). 11or ( i ) we reason as follows. Suppose that aRb ′ and b ′ (cid:13) A bi . If ¬ ( bS a b ′ ), b ′ (cid:13) q and we are done. So, we consider the case in which bS a b ′ . As S i ( b, b ′ ), ( ∗∗ ) nowyields us a d ∈D such that b ′ S a d . Clearly bS a d and thus, by definition, d (cid:13) q .To see ( ii ) we notice that b (cid:13) A bi ∧ ✷ s . But if bS a y and y (cid:13) q , by definition y ∈D and thus y (cid:13) ¬ ✷ s . Thus b ∈C q ∧ ✷ sa and a (cid:13) ¬ ( A i ∧ ✷ s ✄ q ∧ ✷ s ).The following theorem is now an immediate corollary of the above reasoning. Theorem 6.7.
A finite frame F validates all instances of Beklemishev’s prin-ciple if and only if ∀ i F | = C i . Definition 6.8.
Let B i be the principle A ✄ B → A ∧ ✷ C ✄ B ∧ ✷ C for A ∈ ES i2 . Corollary 6.9.
For a finite frame we have F | = B i ⇔ F | = C i . For the class of finite frames, we can get rid of the universal quantification inthe frame condition of Beklemishev’s principle. Remember that depth ( x ), thedepth of a point x , is the length of the longest chain of R -successors starting in x . Lemma 6.10. If S n ( x, x ′ ) , then depth ( x ) = depth ( x ′ ) .Proof. S n ( x, x ′ ) ⇒ S ( x, x ′ ) ⇒ x ↑ = x ′ ↑ . Lemma 6.11. If S n ( x, x ′ ) & depth ( x ) ≤ n , then S m ( x, x ′ ) for all m .Proof. The proof goes by induction on n . For n = 0, the result is clear. So, weconsider some x, x ′ with S n +1 ( x, x ′ ) & depth ( x ) ≤ n + 1. We are done if we canshow S m +1 ( x, x ′ ) for m ≥ n + 1.This, we prove by a subsidiary induction on m . The basis is trivial. For theinductive step, we assume S m ( x, x ′ ) for some m ≥ n + 1 and set out to prove S m +1 ( x, x ′ ), that is S m ( x, x ′ ) ∧ ∀ y ( xRy → ∃ y ′ ( yS x y ′ ∧ S m ( y, y ′ ) ∧ y ′ S x ′ ↑ ⊆ yS x ↑ ))The first conjunct is precisely the induction hypothesis. For the second conjunctwe reason as follows. As m ≥ n + 1, certainly S n +1 ( x, x ′ ). We consider y with xRy . By S n +1 ( x, x ′ ), we find a y ′ with yS x y ′ ∧ S n ( y, y ′ ) ∧ y ′ S x ′ ↑ ⊆ yS x ↑ . As xRy and depth ( x ) ≤ n +1, we see depth ( y ) ≤ n . Hence by the main induction,we get that S m ( y, y ′ ) and we are done. Definition 6.12. A B -simulation on a frame is a binary relation S for whichthe following holds.1. S ( x, x ′ ) → x ↑ = x ′ ↑ S ( x, x ′ ) & xRy → ∃ y ′ ( yS x y ′ ∧ S ( y, y ′ ) ∧ y ′ S x ′ ↑ ⊆ yS x ↑ )12f F is a finite frame that satisfies C i for all i , we can consider T i ∈ ω S i . Thiswill certainly be a B -simulation. Definition 6.13.
The frame condition C B is defined as follows. F | = C B if andonly if there is a B -simulation S on F such that for all x and y , xRy → ∃ y ′ ( yS x y ′ ∧ S ( y, y ′ ) ∧ ∀ d, e ( y ′ S x dRe → yRd )) . An immediate consequence of Lemma 6.11 is the following theorem.
Theorem 6.14.
For F a finite frame, we have F | = B ⇔ F | = C B . Note that the M -frame condition can be seen as a special case of the framecondition of B : we demand that S be the identity relation.It is not hard to see that the frame condition of M follows from C . Andindeed, IL B ⊢ M as ✸ A ∈ ES and A ✄ B → ✸ A ✄ B . Actually, we have that IL B ⊢ M . Zambella proved in [18] a fact concerning Π -consequences of theories with a Π axiomatization. As we shall see, his result has some repercussions on the studyof the interpretability logic of PRA. Lemma 7.1 (Zambella) . Let T and S be two theories axiomatized by Π -axioms. If T and S have the same Π -consequences then T + S has no more Π -consequences than T or S . In [18], Zambella gave a model-theoretic proof of this lemma. As was sketchedby G. Mints (see [3]), also a finitary proof based on Herbrand’s theorem can begiven. This proof can certainly be formalized in the presence of the superexpo-nentiation function, thus it yields a principle for the Π -conservativity logic ofΠ -axiomatized theories. We denote it here as Z ( EP c2 ). Z ( EP c2 ) ( A ≡ Π B ) → A ✄ Π A ∧ B for A and B in EP c2 .where the class EP c2 of modal formulas is defined as follows: EP c2 ::= ✷ A | ¬ ✷ A | EP c2 ∧ EP c2 | EP c2 ∨ EP c2 | A ✄ A . The class EP c is of course tailored so that any arithmetical realization will beprovably Π . Note that the superscript c is there to indicate that the ✄ modalityis to be interpreted as a formalization of the notion of Π conservativity. It isnot hard to see that the formalization of this notion is itself Π . Moreover, notethat this class coincides in extension with the earlier defined class ES .Since PRA is Π axiomatized and proves totality of the supexp function theprinciple Z ( EP ) c applies to PRA. 13ut there are repercussions for the interpretability logic of PRA as well. Weknow that for reflexive theories Π -conservativity coincides with interpretability.We also know that any Σ -extension of PRA is reflexive (Lemma 2.1). Alto-gether this means that a statement α ✄ β and α ✄ Π β are equivalent if α is inΣ and PRA + α is Π -axiomatized, i.e. α is in ∆ .We arrive at Zambella’s principle for interpretability logic: Z ( A ≡ B ) → A ✄ A ∧ B for A and B in BS For the Π -conservativity logic of PRA, the principle Z ( EP c2 ) is really informative(see [3]), it is the only principle known on top of the basic ones for the Π -conservativity logic of PRA. The principle Z for interpretability logic is veryinteresting as well but it does turn out to be derivable in IL B as we will nowproceed to show. (See however the final remark of this section.)Here modal logic again proves to be informative - to have such a proofis interesting since it is not at all clear to us how the two principles relatearithmetically.We shall give a purely syntactical proof of IL B ⊢ Z , B being a restrictionof B to BS formulas, see Definition 6.8. The proof in [10] of the same fact wasnot correct.Throughout the proof we consider a full disjunctive normal form of modalformulas: Definition 7.2. A full disjunctive normal form (a full DNF) over a finite setof formulas { C , . . . , C n } is a disjunction of conjunctions of the form ± C ∧ . . . ∧ ± C n where + C i means C i and − C i means ¬ C i , i.e., each C i occurs eitherpositively or negatively in each disjunct. Each propositional formula is clearly equivalent to a formula in full DNFover the set of propositional atoms occurring in it. Similarly each modal BS -formula, being a boolean combination of boxed formulas, is equivalent to aformula in full DNF over the set of its boxed subformulas, or even over anyfinite set of boxed formulas containing its boxed subformulas (or just its boxedsubforumulas maximal w.r.t. box-depth). Theorem 7.3. IL B ⊢ Z Proof.
Let
A, B ∈ BS and let { A , . . . , A m } be the set of boxed subformulas of both A and B . Assume w.l.o.g. that A and B are in full DNF over { A , . . . , A m } .Assume A ≡ B . We show that A ✄ A ∧ B . Since A comes in full DNF, thismeans to show, for each disjunct D of A , that D ✄ A ∧ B . In fact, we show thisfor any disjunct of A or B .A disjunct D of either A or B is fully determined by the set D ✷ of boxedformulas occurring positively in it. We shall write D ✷ also for the conjunctionof its members.We first show, if D is a member of A or B which has a maximal set D ✷ (nodisjunct E with E ✷ properly containing D ✷ occurs in A or B ) then D ✄ A ∧ B :14uppose such D is in A , the other case is symmetrical. Since D ✄ A wehave also D ✄ B . Then, noting that D ✷ is a conjunction of boxed formulas andapplying B , we obtain D ✄ B ∧ D ✷ .Now take any disjunct E of B for which E ✷ does not contain D ✷ . Then E contradicts D ✷ by its negative part. We distinguish two cases: if for all E in B the set E ✷ does not contain D ✷ , then B contradicts D ✷ . It follows from D ✄ B ∧ D ✷ that D ✄ ⊥ . Then clearly D ✄ A ∧ B .Otherwise B does contain E with E ✷ containing D ✷ . But since D has amaximal Box-set, E and D must be the same and D occurs in B as well. Thus D ✄ B ∧ D and, since ⊢ D → A , also D ✄ A ∧ B .We have shown that all maximal disjuncts interpret A ∧ B .We show by induction that the same is true for all other disjuncts of A and B .This suffices for the proof.Assume that, for all k ′ with m ≥ k ′ > k and all disjuncts D in either A or B with D ✷ of size k ′ , D ✄ A ∧ B (this has already been shown for k equalto the size of the maximal Box-set in A and in B which is certainly less then m ). Consider a disjunct D of A , the other case is again symmetrical. Assumew.l.o.g. that D ✷ has size k . We have to show D ✄ A ∧ B :Since D ✄ A and hence D ✄ B , we again have that D ✄ B ∧ D ✷ . Now D ✷ conflictswith all the disjuncts of B , Box-set of which is not a superset of D ✷ . Again, wedistinguish two cases: if there are no disjuncts of B with a Box-set which is asuperset of D ✷ then B conflicts with D ✷ and D ✄ ⊥ and thus D ✄ A ∧ B .Otherwise some disjuncts of B do have a Box-set which is a superset of D ✷ . Let E , . . . , E l be all such disjuncts of B . Then, since D ✄ B ∧ D ✷ and ⊢ B ∧ D ✷ → E ∨ . . . ∨ E l (where E ∨ . . . ∨ E l is the part of B not conflictingwith D ✷ ), we obtain D ✄ E ∨ . . . ∨ E l . Now it suffices to show that each E i interprets A ∧ B .Fix an E i and suppose E ✷ i have size k . But then E i = D and thus we have,as before, D ✄ ( B ∧ D ) ✄ ( B ∧ A ). If E ✷ i have size greater then k , the inductionhypothesis apply and we obtain that E i interprets A ∧ B .Actually it is possible to extend Zambella’s principle somewhat in such a waythat it is no longer clear whether the result is still derivable from B . First notethat the formulas in ES are just the propositional combinations of ✷ -formulas.Zambella’s principle for interpretability logic as studied in this paper reads A ≡ B → A ✄ A ∧ B where A and B should both be BS . However, to have access to the ideas behindZambella’s principle, it is sufficient that A and B be both provably of complexity∆ . We can thus look at those ES formulae who are provably equivalent to thenegation of some other ES formula and plug those formulae in. Reflecting this15hought in a formula yields ✷ (( A ↔ A ′ ) ∧ ( B ↔ B ′ )) → ( A ≡ B → A ✄ A ∧ B )where A , A ′ , B and B ′ are all from ES . It actually makes sense to call thisprinciple the Zambella principle for interpretability logic as it more preciselyreflects the arithmetical ingredients. We have chosen not to do so as to beconsistent with earlier papers.
PRA ) Let us see what we can conclude about IL (PRA) from the above. Certainly IL (PRA) includes IL (All) but it is more than that because B is not a principleof IL (All). The latter is clear from the fact that IL (All) ⊆ IL M ∩ IL P and Z isnot in IL P : consider the following model: wp qS w We have w (cid:13) ✸ p ≡ ✸ q and w p ✄ p ∧ q , thus Zambella fails. The model isclearly an IL P model.This shows, by derivability of Z from B , that indeed B is not a principle of IL (All).Also we know that IL (PRA) is not IL M since M is not in IL (PRA), as A.Visser discusses in [17]: the two logics cannot be the same because if IL M isa part of the interpretability logic of a theory then it is a part of the inter-pretability logic of any of its finite extensions as well. This cannot be the casefor PRA because not all of its finite extensions are reflexive. A more specificexample of a principle of IL M which is not in IL (PRA) can be given: A ✄ ✸ B → ✷ ( A ✄ ✸ B ) . That this formula is not in IL (PRA) can be shown using Shavrukov’s resultfrom [14] about complexity of the set { ψ | ψ ∈ Π & φ ✄ ψ } ; see [17] for the fullproof.We know that M is provable in IL B . The other principles surely containedin IL (PRA) are B , R and W ( R ∗ is the conjunction of R and W ). Let us showthey are mutually independent. Note that for nonderivability proofs soundnesssuffices. We would like to thank one of the referees for pointing out that our original extension ofZambella’s principle for interpetability logic could actually be even generalized to its currentform. W and R .The condition for W requires that the composition ( S w ; R ) is conversely well-founded, the condition for R is the following: xRyRzS x uRv ⇒ zS y v . W vs. B : It is easy to see that W B since the former is in IL (All) whilethe later is not in it. Since R is in IL (All) as well, W , R B . The followingframe wxy zS w is an IL B frame and it violates the frame condition for W : wRxRy and xS w yS w x and wRz . Now z is bi-similar to y and B is ensured. Thus B W .Moreover, the same frame, being an R frame, shows that B , R W :the only case to check is wRxRyS w xRy . Now the condition for R re-quires yS x y , but this is clearly the case since S x is reflexive over x . R vs. B : Again, since R ∈ IL (All), it cannot be that R ⊢ B . We have alreadydiscussed that neither R , W ⊢ B . The following frame xz ′ yz uvS x is an IL B -frame violating the frame condition of R : We have a basicsituation violating R , which is xRyRzS x uRv and ¬ zS y v . To ensure B for y we add an arrow yRv , to ensure B for z , we add a bi-similar world z ′ such that xRz ′ and z ′ has no successors at all.Moreover, since the frame is clearly a W frame as well, we have shownthat B , W R . R vs. W : already discussed in [8]. 17t is clear from our exposition that, though we have solved a number ofproblems concerning IL (PRA), many remain open, e.g. those connected withour incomplete knowledge of IL (All). Also, we lack a modal completeness theo-rem for IL B . Unfortunately, the complexity of the frame condition for B makesthis seem an intractable problem at the present time. In any case, the logic ofinterpetability is far from being a finished subject. References [1] L.D. Beklemishev. Induction rules, reflection principles, and provably re-cursive functions.
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