A Modal View on Resource-Bounded Propositional Logics

Classical propositional logic plays a prominent role in industrial applications, and yet the complexity of this logic is presumed to be non-feasible. Tractable systems such as depth-bounded boolean logics approximate classical logic and can be seen as a model for resource-bounded agents whose reasoning style is nonetheless classical. In this paper we first study a hierarchy of tractable logics that is not defined by depth. Then we extend it into a modal logic where modalities make explicit the assumptions discharged in propositional proofs, thereby expressing blueprints for proofs. A natural deduction system is provided that permits to reason about and manage such proof blueprints.


Introduction
Classical propositional logic plays a prominent role in many industrial applications [27,34]. Its satisfiability and validity problems, though, are NPcomplete resp. coNP-complete [8] and so under the widely accepted conjecture that P = NP, classical reasoning would lie beyond what is considered computationally feasible. Resource-bounded logics for propositional languages [10,11,13,15,18,19,39,40] address these concerns and offer more realistic models of reasoning agents. 1 These logics are obtained by setting bounds on certain inference rules of the proof system , thereby trading deductive power (completeness of ) for a lower complexity. This is a useful abstraction for modelling physical bounds on the memory and time available to agents -and even perhaps accurate for human reasoners. More importantly, propositional resource-bounded logics such as DBBL are tractable [10]. In the present paper, we focus on the informational approach to classical  : if B can be inferred under the assumption that A and also under the assumption that ¬A, then infer B. Both assumptions are discharged (denoted with braces) by this application of RB . (Right) A rule for the introduction of implication used commonly in natural deduction for classical logic [37] UBBL, that measures proof complexity by the number of RB -formulas, rather than their depth as in [11]. This measure, moreover, enables a modal logic for reasoning about proof blueprints. In this logic, a formula [±A 1 ] . . . [±A k ]B represents that a classical proof of B exists whose RB -instances involve at most the pairs {A 1 , ¬A 1 }, . . . {A k , ¬A k }. This modal logic can equip agents with tools for merging and optimizing blueprints. Related work Different hierarchies of classically sound but incomplete systems, called resource-bounded logics, have been shown to approximate classical logic. Under a 2-valued semantics, Schaerf and Cadoli [39] present a chain of logics |= S that validate modus ponens for increasing sets S of atoms; its valuations can be paraconsistent for p / ∈ S, i.e. V (p) = 1 = V (¬p). 4 In the same vein, Finger and Wassermann [19] describe a general system that validates each rule ρ within a set of applicable formulas S ρ only; these sets S ρ (and so the logic) can expand during proof search, thus imposing proof heuristics based on rule precedence. In an algebraic setting, Dalal [15] builds upon boolean constraint propagation to bound the size of the cut formula. This is generalized in the lattice approach of Finger and Gabbay [18] built on an infinite layer of truth-values between 0 and 1; the cut rule is not eliminable and sequent calculi bound its use by the number of atoms or formulas. Using a 3-valued non-deterministic semantics, D'Agostino et al. [10,11,13] define the DBBL hierarchy by bounds on the nesting of RB , its only branching rule. The same bounded logics are found in Stålmarck's method [40], which further extends them with rules for truth-value equivalences A ≡ B, among other novel features (see [11,40] for a comparison). For resource-bounded logics in first-order languages, an overview can be found in [23].
Epistemic logics have also been designed to avoid the logical omniscience problem [46]. These proposals distinguish explicit from implicit knowledge either by weakening axiom K [45] or by modelling reasoning steps dynamically or as the result of the passing of time or a number of derivation steps [1,2,5,30,42]. Hawke et al. [28] discusses the plausibility of modal versions of classical inferences for resource-bounded agents, in partial disagreement with D'Agostino [10] or the present approach. A related approach is that of logics of awareness, see [25,43] for recent work along this line. The above described [39] also considers modalities S for knowledge in the paraconsistent logics |= S . In comparison, our dynamic modalities validate the axioms for knowledge from Schaerf and Cadoli [39] plus some interaction axioms; their modular character also suits better applications in distributed reasoning. Along this line, epistemic justification logics [3,36] describe modalities (proof terms) that encode full proofs of the formulas under their scope, similarly to proof blueprints. These modal logics mostly build upon classical propositional logic, unlike the present work or Klassen [31], which describes a multi-agent belief logic with separate update modalities [A], [¬A] and Kleene's semantics (see footnote 6).
Finally, modal logics of distributed knowledge address what agents do know as a group [26,44]. Recent extensions of DBBL with multi-agent [7] and dynamic epistemic [33] modalities also address logical omniscience. Distributed reasoning in formal argumentation has been considered with DBBL in place of classical logic [14]. For first-order languages, a natural deduction system for distributed reasoning is considered in [22]. Structure of the paper Section 1 starts by recalling depth-bounded boolean logics DBBL. Section 2 studies a measure of proof complexity based on the number of RB -formulas used. Section 3 presents the lattice of valuations. This and Section 4 on models for 0-depth logic pave the way for degree-bounded logics UBBL. Section 5 studies applications of RB as semantic updates. Section 6 introduces the UBBL hierarchy and Section 7 presents a modal logic with update modalities. Section 8 describes a complete deductive system for this logic. Section 9 addresses the complexity of its validity problem and discusses applications in distributed reasoning. The paper concludes with directions for future research. Appendix A recalls fundamental facts from lattice theory and Appendix B contains proofs of some results. Figure 2. The information order ≤ on truth-values ⊥ ≤ 1 and ⊥ ≤ 0 according to the information they carry: from less to more information Figure 3. Truth-tables f * for * ∈ {∧, ∨, ¬, →}. See [11] for other connectives and for an intuitive explanation of the truth-tables in terms of information modules

Preliminaries: DBBL Depth-Bounded Boolean Logics
Let us start with a reminder of 0-depth logic and the inductive definition of k+1-depth logics that together define the DBBL logical hierarchy [10].
We also use the logical constant (falsum), denoting an absurd proposition.
An informational semantics is proposed in [10,12] with a set of truthvalues {1, 0, ⊥} reading: 1 = known as true, 0 = known as false and ⊥ = unknown. A partial order (or poset, Definition A.1) of information ({1, 0, ⊥}, ≤) is depicted in Figure 2. The truth-tables for boolean connectives in Figure 3 coincide with the classical truth-tables over {1, 0} and expand Kleene's 3valued semantics [32] making it non-deterministic, e.g. ⊥ ∨ ⊥ ∈ {1, ⊥} = f ∨ (⊥, ⊥). Under this reading, A and B can be unknown (A, B → ⊥) but their disjunction known A ∨ B → 1 (or the conjunction known to be false). 6 5 Let us note that a valuation (Definition 1.2) might assign different truth-values to classically equivalent formulas such as {A → B, ¬A ∨ B}. This is not the case if L is defined from minimally functionally complete sets, e.g. {¬, ∨} defining A → B := ¬A ∨ B and so on. 6 The truth-tables in Kleene's 3-valued logic are as in Figure 3 except for: ⊥ ∨ ⊥ = ⊥ ∧ ⊥ = ⊥ → ⊥ = ⊥. Quine [38] motivates the informational reading of ∨ and ∧ with the examples this is a mouse or a chipmunk and this is a mouse and a chipmunk. The constant function V ⊥ : L → {⊥} satisfies the truth-tables, and so is a (null information) valuation. The set of designated values is just {1}, from which the 0-depth logic (L, |= 0 ) is defined as usual.
exists, is a tree (N, R) of nodes labelled 7 with formulas N ⊆ L and such that: (i) A is the root node and leaf nodes are in Γ; (ii) if A 0 ∈ N has as R-successors A 1 , A 2 (possibly with A 1 = A 2 ), then is an instance of an intelim rule (Fig. 4). The logic of k-depth |= k strengthens |= 0 with at most k nested applications of bivalence. Under the informational semantics, each application of bivalence expands the information possessed by the agent with the virtual (or temporary) possession of some formula A, and in parallel of its negation ¬A, and extracts their shared consequences. 8 Figure 4. The natural deduction system 0 consists of rules I * , E * (called intelim rules) for the introduction and elimination of connectives and falsum * ∈ {∧, ∨, →, ¬, }. D'Agostino [10] presents these rules in a language with signed formulas {T A, F A}, where they enjoy the separation property [4] so any rule for * only mentions this connective * Figure 5. (Left) A schematic application of the RB rule. A is called the RB -formula. Number labels are also used to track which assumptions are discharged at each RB application. (Right) Given a proof Π1 of B from A and a proof Π2 of B from ¬A, an application of RB results in a proof of B from Γ ∪ Δ. It combines Π1, Π2 and a root node labelled B Definition 1.6. (|= k consequence) Define |= 0 as in Definition 1.3, and inductively: Each consequence relation |= k is monotonic and satisfies a weaker version of cut (see Proposition 6.6). For the corresponding system k , one simply needs to add the rule of bivalence RB ( Figure 5) and limit its nesting in proofs. An important property of the DBBL hierarchy is that to decide whether Γ B, the RB -formulas can be searched among sets of bounded size such as the sets of subformulas sub(Γ ∪ {B}) or atoms at(Γ ∪ {B}).
is closed under subformulas, and (iii) the size of f(Δ) is polynomially bounded w.r.t. Δ. Any such function f defines a consequence |= f k and calculus f k as follows. Let |= f 0 = |= 0 resp. f 0 = 0 and: The length of a formula A, denoted |A|, is the number of symbol occurrences in A. For a finite set Γ ⊆ L, we define |Γ| = Σ A∈Γ |A|.

Measures of Proof Complexity
A quick look at Definition 1.6 shows that k-depth proofs contain a branch with k nested applications of RB while other branches may differ from this 9 Proofs based on analytic cuts (that is, f = sub for RB ) can be exponentially longer than those based on unrestricted cut, as witnessed by pigeon-hole formulas [6]. Analytic cuts, on the other hand, enjoy the weak subformula property [13] and reduce the search space in implementations by controling the choice of RB -formulas. In Section 7, we return to the general case and describe a modal logic based on unrestricted cuts.
number. Another measure of proof complexity would then simply be: Thus it also holds that Γ∪{±A 1 , . . . , Or, after renaming the formulas, A hierarchy of degree-bounded logics ||= k k<ω satisfying Γ ||= k B iff h(Γ, B) ≤ k, will be defined (Definition 6.1) using updates on valuations.

Preliminaries: a Complete Lattice of Valuations
The lattice structure of valuations obtains by lifting the order ≤ from truthvalues ( Figure 2). A similar construction for partial valuations is found in [9].
For a poset (X, ≤) and a set Y ⊆ X, the meet Y is the greatest lower bound of Y (Definition A.2). The join Y is the least upper bound of Y . We also use the notation x y = {x, y} and x y = {x, y}.     To keep the terminology simple, we henceforth rename + as and call any element of V + a valuation. Note that although V does not satisfy the truth-tables and so V / ∈ V, its choice as a top element is not arbitrary.

A bit of Model Theory for 0-depth Logic
One can first observe that valuations and 0-depth theories are closely connected through the corresponding partial orders and ⊆.
An interesting fact about 0-depth logic |= 0 is that among all the valuations satisfying a set of formulas Γ, one can construct their minimum. This minimum captures what all these valuations agree upon.
Whenever Γ is 0-depth consistent, V Γ will contain some valuation in V. All 0-depth inconsistent sets Γ share the same set of valuations V Γ = {V }. In both cases, the infimum V Γ (Proposition 3.4) happens to be in V Γ and so it is a minimum: V Γ = V Γ . A stepwise version of a construction used in the completeness proof [10,Proposition 4.7] gives us a direct proof of this fact.  From Propositions 3.4-4.5 it follows that model-checking V Γ suffices for checking any |= 0 -consequence A of Γ, offering an insight into the low complexity of its validity problem, namely O(n 2 ) for n = |Γ∪{A}|. It also follows that a 1-1 correspondence exists between theories and minimum valuations.   Proof. We use that Γ is a theory plus Proposition 1.5 and Corollary 4.8 to obtain: The connectives ∧, ∨ need not be distributive over each other, as witnessed by valuations V satisfying one of the following: In a similar vein, the lattice of valuations (V + , ) is semi-distributive (see Lemma A.8) but not distributive as shown in Example 5 below.

Updates and the RB-Rule
Intuitively, updating a valuation V with a formula A should result in a valuation that refines V , satisfies A and is minimal with these properties.

Proof. Conditions (i) and (ii) in Definition 5.1 give that the set of
in the set of such refinements, and so it is the -minimum A-refinement of V .
Example 5. (±-updates simulate RB ) The following -chain below holds for any set Γ. The inequalities can all be proper, as in Γ = {±p → q}. For this Γ, each formula in the the lower line becomes true exactly at the valuation above it.
Each inequality can also become an identity if the update is made trivial (Corollary 5.3 or Fact 5.6 below); e.g. for the first and third inequalities, the sets Γ 1 = {±p → q, p ∨ ¬p} and resp. Γ 3 = {±p → q, p} give: .
seen above for Γ = {±p → q} shows that the lattice of valuations is not distributive. These two valuations are respectively: Any such valuation V [±]A enforces all (and only) the consequences of applying RB with the formula A. Since depth-bounded logics |= k admit different RB -formulas in each branch after an RB rule (for k > 1), depth and number of updates need not match beyond the k = 1 case shown below.
Example 7. (Degree and ±-updates) Recall the set from Table 1 (mid) , and let (j) denote the j-th premise from Γ. Three ±-updates (in any order, see Corollary 5.9 below) suffice for B: and so B → 1 by (4).
Proof. For the first claim, use commutativity and associativity (Lemma A.7) and Proposition 5.2 and reason as follows: For the second claim, we use Corollary 4.10(i) and expand (V A ) B as follows: The general property only holds if 0-depth logic is expanded with idempotence A ∨ A 0 A, an option discussed in [12].

Corollary 5.9. (Permutation of ±-updates) For any valuation
Proof. We split the identity and prove both directions and . For ( ), where each step follows from applications of: Definition 5.5; Proposition 5.2; Definition 5.5; ( ) Lemma A.8; Proposition 5.2; Lemma A.7 (commutativity); Lemma 5.8; and Definition 5.5 (twice). For ( ), switching A and B everywhere in the above reasoning gives , and so we are done.
Proof. (Base case 0) From definitions and Proposition 1.  B). And vice versa, any such pair of proofs can be merged so that all occurrences of A 1 , ¬A 1 are discharged by new RB -instances. This justifies the first equivalence below. We omit everywhere for some where each step follows from: the reasoning above; the inductive hypothesis; Lemma 5.4; def. of (V Γ ) A ; Definition 5.5 and meets; and resp. Definition 6.1.    is a U -model, whose -minimum V ∅ is more informative than f ⊥ . Functions in the striped area cannot be refined into C Lemma 6.9. For any V ∈ V, the following are equivalent: Hence, for a set V its join is the is k-depth consistent and k+1-depth refutable. If a valuation V satisfies it, k+1 updates with e.g. {±p 1 , . . . , ±p k+1 } suffice to render the 0-depth inconsistent formula p k+2 ∧ ¬p k+2 true, thus making

A Modal Logic for RB-Updates
The characterization of the RB rule via updates not only defines degreebounded propositional logics: it also provides a semantics for modalities that capture applications of RB and make the virtual assumptions explicit.
The information order extends to functions f over L U . Valuations now depend on updates (Definition 7.2), making them relative to a model (Definition 7.3). See Figure 7 for an illustration and Definition 7.6 for an explicit construction. Figure 3) and the update condition: We also call the function V : L U → {1} a valuation.
Since U -models are closed under A-updates, they contain V . We extend as before to V V for any valuation V , so that Remark 2. The meet of a set of L U valuations S corresponds to the restricted pointwise meet, just as in the propositional case: In order to show that non-trivial U -models exist, we build one with all valuations through fragments L Uk of the language L U = k<ω L Uk .
where A ∈ L and B i ∈ L U ≤k := i≤k L Ui .

P. Pardo
Definition 7.6. (Standard model S) Define S 0 = V and S n+1 as the set of functions V n+1 of the form: Assume as inductive hypothesis that the claims hold for S n . (Ind. case n + 1) (S n+1 is closed under meets.) Let S n+1 ⊆ S n+1 . We show that S n+1 can be built as in Definition 7.6 and so S n+1 ∈ S n+1 . To see this, let in Definition 7.
By the inductive hypothesis, V n and all updates V B n , V ¬B n are in S n . Then, For boolean L Un+1 -formulas, S n+1 (B * C) ∈ f * ( S n+1 (B), S n+1 (C)) is proved analogously to the propositional case (Proposition 3.4), and similarly for ¬B formulas. The function obtained this way is exactly S n+1 and so By the claim shown above, V A n+1 is in S n+1 and by definition it is clearly the -minimum Arefinement of V n+1 in S n+1 . (V n+1 ∈ S n+1 satisfies the truth-tables.) Since V n+1 extends some V n ∈ S n , it respects the truth-tables over all L U ≤nformulas (by the inductive hypothesis) and also over L Un+1 -formulas (by Definition 7.6). This concludes the inductive proof.
Let now V = V for some ⊆-chain V = V n n∈ω with V n ∈ S n . Since each finite fragment V n satisfies the truth-tables for L U ≤n -formulas, so does  It can also be pointed out that U enjoys a modal deduction theorem Γ U [±A]B iff Γ∪{±A} U B. Basic modal axioms are not valid, though 14 :

Completeness and Conservativeness of the U U U Logic
We define next the closure of a set Γ under all U -rules except for I [±]. This rule is added at the construction of minimum functions V Γ in Definition 8.2.  13 In the present setting, the RE rule is covered by these two cases A ≡0 B  where Proof. (⇒). Let Π be a proof of A from Γ, and let Π 1 , . . . , Π m be all the subtrees of Π with an instance of I[±] at the root node, say for a conclusion [±B i ]C i in Π i . Clearly, for each 1 ≤ i, j ≤ m either Π i Π j or Π i Π j or both, so assume an ordering Π 1 , . . . , Π m satisfying: all subtrees of Π i+1 also applying I[±] are among {Π 1 , . . . , Π i }. Let also A 1 , . . . , A n be a sequence with all formula labels (possibly repeated) in Π that respects the inverse tree order: if {B, C} ∴ A occurs in Π, then B and C are listed before A in the sequence. Let us rephrase this sequence A 1 , . . . , A n to mark each conclusion  Thus there are proofs of C i+1 from Γ(i + 2) ∪ {B i+1 } and from Γ(i + 2) ∪ {¬B i+1 }. By the construction of the sequence, any such instance C i+1 occurs before [±B i+1 ]C i+1 = A i+1 0 , and so in the worst case it is of the form

) Assume as inductive hypothesis that for each
. This is immediate, since V Γ is built by assigning 1 to every formula in Γ and closing the set of 1-values under U -rules, following Definitions 8.1-8.2.  In all subcases, V Γ (B ∨ C) = V Γ (C), and so V Γ satisfies the truth-table We only need to show that V Γ (B ∨ C) = 0, but this is immediate since otherwise V Γ (¬(B ∨ C)) = 1, and then we would have V Γ (¬B) = 1 and finally V Γ (B) = 0, contradicting the case assumption. C) holds for non-basic formulas B ∨ C ∈ L U ≤k+1 is proved as in Proposition [11, 2.26] but with intelim rules over L U . This inductive proof shows that V Γ respects the truth-tables for any L U -formula.
Proof. Let V ∈ S. Consider first the case V (A) = 1. Then, where each identities follows resp. from: Corollary 8.  Proof. S-valuations build from propositional ones in V (Definition 7.6), so one can rephrase classical consequence as: Δ |= A iff V (A) = 1 for all V ∈ C Δ . Then, we reason as follows: The ⇒ direction is immediate and ⇐ follows from Lemma 8.16. (Ind. case k+1.) Assume as inductive hypothesis that the claim holds for k. We omit everywhere that A 1 , . . . , A k ∈ f(Γ ∪ {B}) and that V ranges over S Γ unless specified otherwise:

Complexity of the Modal Logic U U U and Applications
Let us address next the computational complexity of the decision problems Val U and MC ∅ U of validity and resp. model checking for formulas in the minimum valuation V ∅ . See [35,41] for further details.
Proof. We show that the coNP -complete problem of classical validity We abusively call a modal path any choice sequence σ = A 1 , . . . , A m ∈ ±A 1 , . . . , ±A m . Finally, we refer to the set {A 1 , . . . , A m } also by σ.
0consq(σ, B) in line (2) is also based on [11] and returns (σ, B, v) (Correctness and termination.) Note that the algorithm takes any initial node of the form (σ, A). We show by induction on A that V σ ∅ (A) = v iff on input node (σ, A) the algorithm terminates and returns (σ, A, v).
(Base case A = p.) A quick inspection of 0consq(σ, p) (line 2) shows that: Thus, for all other cases, f * will coincide with k * , and so v = V σ ∅ (B 0 * B 1 ) will match the node (σ, B 0 * B 1 , v) (line 2.1). Let trace(σ, A) denote the trace of the algorithm for an input node (σ, A) and let us represent the two stacks in a state by α | β (or by different columns below): Thus, the property that the first unevaluated node in the second stack is preceded by its evaluated child nodes is preserved from B 0 and B 1 into B 0 * B 1 . The case for (σ, ¬B 0 ) is even simpler, as it only generates a left child node. (Ind. case, [±C]B.) Assume the claim for any modal path σ and a formula B. Let α | β and γ | δ denote states in trace(σ.C, B) resp. trace(σ.¬C, B) In view of Remark 2, an inspection of meet shows that if V σ.C . Thus, in all cases the algorithm terminates and returns a sound input (σ, B, V σ ∅ (B)). In particular, for inputs (∅, A) corresponding to the root of the tree T for A, we obtain that V ∅ (A) = v iff the algorithm returns (∅, A, v). (Space complexity) The space complexity of post-order traversal is the depth of the tree, or the depth d(A) of the formula A in our case, which is linear in |A|. Our algorithm keeps the same bound by discarding evaluated nodes as soon as they can be used to evaluate a more complex formula (line 2.1). 16 Hence, at any time the two stacks have size polynomial in the length of A, namely d(A) · |A|, as for any modal path and its formula |(σ, B)| ≤ |A|.  16 Moreover, the algorithm might not explore the full tree T : at the first sign of 0depth inconsistency or 0 consequence at a node (σ.Am, B), the subtree below this node is pruned.
Thus, the number of nodes in the tree is exponential in the length of A. This gives an exponential time upper bound.
In summary, MC ∅ U is in PSpace. We prove that Val U is also in PSpace by providing a polynomial reduction from Val U to MC ∅ U . This is simply given by the function A → (∅, A, 1). This is indeed a reduction since A ∈ Val U is equivalent to the truth of A in V ∅ , as the latter is -minimal among all valuations in S \ {V }.
In comparison with the general case, the validity problem for proof blueprints under some virtual space function f remains in coNP . The same argument shows that deciding whether a proof blueprint is a logical consequence of some propositional database Γ is also coNP -complete. For any given particular instance, moreover, deciding whether a blueprint A = [±A 1 ] . . . [±A k ]B is valid or a consequence from Γ has polynomial complexity, namely in O(|B| k+1 ) and resp. O(|Γ ∪ {B}| k+1 ). In practice, the actual coefficients will be lower as no choice of RB formulas is involved (compare with [11,Alg. 3

.1]).
Remark 3. Other decision problems are satisfiability and ⊥-satisfiability, denoted Sat U resp. ⊥Sat U . Each is defined as the set of formulas A for which there is some V = V with V (A) = 1 and resp. V (A) = ⊥. While Sat U can be shown to be NP -hard using the same function f as in Lemma 9.2, ⊥Sat U is in PSpace (by Theorem 9.4 and that A ∈ ⊥Sat U iff V ∅ (A) = ⊥.)

Applications in Distributed Reasoning
As argued in the Introduction, the form of deductive exchanges (messages) among a network of reasoning agents should satisfy certain desiderata, including: quick correctness tests, optimality preservation and succintness. For proof blueprints, generated directly from U or encoding some f k -proof, these properties are granted by: the discussion following Theorem 9.6 above, the proof normalization procedures for resource-bounded logics [13] and the fact that any proof blueprint for B ∈ L is linear in the length of B. Two further advantages of proof blueprints are described next. (cut) suppose the agent proves These operations facilitate the management of large collections of proof blueprints towards query solving from a propositional data base, or consistency checks upon it. In summary, an exchange format consisting of blueprints (and the actual premises used in a proof) fulfils the above desiderata and can speed up distributed proof and refutation methods.

Conclusions
The informational 3-valued semantics and natural deduction system studied in [10,11,13] ∨ ¬p), and is a conservative extension of the RB -free fragment of classical logic. A natural deduction system U with introduction and elimination rules for [±A] and reduction rules for negation was proved to be sound and complete for |= U . We also identified derivable rules which permit a comparison with modal axioms and natural deduction systems for modal logics [21,29]. The computational complexity of its validity problem was also studied, setting a PSpace upper bound. Towards applications in distributed reasoning, we established the complexity class for validity over proof blueprints as coNP -complete, thus improving on the complexity of the (general) validity problem. Proof blueprints are thus a robust, computationally feasible message form for deductive exchanges in distributed reasoning. A number of syntactic operations on proof blueprints enable quick management techniques for merging and optimizing such proof blueprints.
As for future work, an interesting question is whether the natural deduction system U enjoys some form of the subformula property and proof normalization; for DBBL this has been shown to be the case for the weak subformula property [13]. Also left as an open question is a sharp characterization of the complexity of validity for U .
In relation to DBBL, one might also ask what modal logic of updates can capture the DBBL hierarchy, as |= U does for UBBL. To this end, instead of representing RB applications in a proof as a sequence [±A] . . . [±A ], the depth-bounded case might use conditional progams π built from: updates ±C, tests ?C, composition π; π and choice π ∪ π . We conjecture that a PDL-style language [20] with formulas [π]A built from these programs, the formula [±A; (?A; π) ∪ (?¬A; π )]B would express that B is a (k+1)-depth consequence if π and π are k-depth conditional programs.
Towards distributed reasoning applications, a further step would be to extend the modal logic |= U with modalities for public announcements of proof blueprints, with formulas Finally, a general open question is how arbitrary modal logics build upon resource-bounded logics, rather than classical logic. Of particular interest would be epistemic logics addressing the logical omniscience problem, but the question is equally pertinent to all modal logics starting from K, as studied in [17] for a tableaux system.
Acknowledgements. The author wishes to thank the anonymous referees for their many interesting comments and corrections. This research was funded by the Department of Philosophy "Piero Martinetti" of the University of Milan under the project "Departments of Excellence 2018-2022" awarded by the Ministry of Education, University and Research (MIUR).
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Basic concepts and results for lattices are listed next, taken from Davey and Priestley [16] or Grätzer [24].
Definition A.2. (Lower, upper bound; meet, join; minimum) Given a poset (X, ≤), a subset Y ⊆ X, we say that α ∈ X is Theorem A.7. (Join and meet [16, p. 39]) For any lattice (X, ≤), its join is associative, commutative , idempotent and satisfies the absorption law: x (y z) = (x y) z (assoc.)  [24, p. 14]) For any lattice (X, ≤) and arbitrary elements x, y, z ∈ X, it holds that: Proof. The pointwise meet of a set of 3-valued functions is clearly its greatest lower bound. Let us show then V ∈ V, i.e. that V respects the truth-tables. For negation ¬, we omit for all V ∈ V and reason as follows:     Let V ω = n<ω V n and finally define V Γ = V ω ∪{ A, ⊥ : A ∈ L\dom(V ω )}.

B. Appendix: Proofs
Since Γ is 0-depth consistent, an easy induction shows that no V n contains an inconsistent pair A, 1 , A, 0 and so each V n is a function V n : L → {1, 0}. Hence the same holds for V ω . This and the fact that V Γ is closed under intelim rules imply that V Γ satisfies the truth-tables (Fact 3.2) and so V Γ ∈ V. Vice versa, for any V ∈ V Γ , V Γ (A) = 1 implies V (A) = 1, again by  Proof. For (i) let V ∈ V Γ be arbitrary. Then,  This and Proposition 4.9(i) imply that V Γ V Δ = V Th(V Γ V Δ ) = V Cn 0 (Γ)∩Cn 0 (Δ) .
The inductive step V n → V n+1 clearly preserves classical consistency and membership in V. As a consequence, also V ω = n V n is classically consistent (V ω (A) = 1 whenever ¬A) and a valuation in V. The latter, together with V (p i ) ∈ {1, 0} for any p i ∈ Var, implies that V V ω ∈ C.