Tractable depth-bounded approximations to FDE and its satellites

FDE , LP and K 3 are closely related to each other and admit of an intuitive informational interpretation. However, all these logics are co-NP complete, and so idealized models of how an agent can think. We address this issue by shifting to signed formulae, where the signs express imprecise values associated with two bipartitions of the corresponding set of standard values. We present proof systems whose operational rules are all linear and have only two structural branching rules that express a generalized Principle of Bivalence. Each of these systems leads to defining an infinite hierarchy of tractable approximations to the respective logic, in terms of the maximum number of allowed nested applications of the two branching rules. Further, each resulting hierarchy admits of an intuitive 5-valued non-deterministic semantics.


Introduction
Many interesting propositional logics are likely to be intractable.For instance, Classical Propositional Logic (CPL), First-Degree Entailment (FDE) [1], the Logic of Paradox (LP) [6,63] and Strong Kleene Logic (K 3 ) [50] are all co-NP complete [see 3,24,75].Thus, we cannot expect a real agent, no matter whether human or artificial, to be always able to recognize in practice that a certain conclusion follows from a given set of assumptions.This is a source of major difficulties in research areas that are in need of less idealized, yet theoretically principled, models of logical agents with bounded cognitive and computational resources.The 'depth-bounded approach' to CPL [e.g.[30][31][32] provides an account of how this logic can be approximated in practice by realistic agents in two moves: (i) by providing a semantic and proof-theoretic characterization of a tractable 0-depth approximation and (ii) by defining an infinite hierarchy of tractable k-depth approximations, which can be naturally related to a hierarchy of realistic, resource-bounded agents, and admits of an elegant proof-theoretic characterization.
A key idea underlying the 'depth-bounded approach' to CPL is that the meaning of a logical operator is specified solely in terms of the information that is actually possessed by an agent, i.e. information practically accessible to her and with which she can operate.This kind of information is called actual, and we use the verb 'to hold' as synonymous with 'to actually possess'.The semantics is ultimately based on intuitive, albeit non-deterministic, 3-valued tables that were first L-formulae; and Γ , Δ, Λ, . .., possibly with subscripts, to vary over sets of L-formulae. 2 Finally, we shall assume that all propositional languages share the same set of atomic variables and, so, we shall identify a language L with C(L).

DEFINITIONS 1
For every formula A, a subformula of A is defined inductively: (i) A is a subformula of A; (ii) for every binary operator •, if B • C is a subformula of A, then so are B and C; (iii) if ¬B is a subformula of A, so is B; (iv) nothing else is a subformula of A. In turn, a proper subformula of A is any subformula of A that is different from A. An immediate subformula of A is any proper subformula of A that is not a proper subformula of any proper subformula of A. NOTATION 2 We denote by sub the function that maps any given set Γ of formulae to the set of all its subformulae, and by at the function that maps any given Γ to the set of its atomic subformulae.Moreover, we define the degree of a L-formula A as the number of occurrences of connectives in A. Now, in the context of this paper we shall abuse standard terminology and call consequence relation on a language L any relation |∼⊆ 2 F(L) × F(L), satisfying the following conditions: In turn, a Tarskian consequence relation (Tcr for short) on L is a consequence relation on L satisfying the following additional condition: An L-substitution is a function σ : F(L) −→ F(L) such that for every n-ary connective and formulae A 1 , . . ., A n , A Tarskian propositional logic is a pair L= L, |∼ L , where L is a propositional language and |∼ L is a Tcr on L satisfying the following additional condition: for every L-substitution σ . 3

DEFINITIONS 3
Let |∼ be a Tcr for L. |∼ is finitary if for every Γ and every A such that Γ |∼ A, there is a finite Δ ⊆ Γ such that Δ |∼ A. In turn, a Tarskian propositional logic L, |∼ L is finitary if so is |∼ L .
In a finitary Tarskian propositional logic, the following 'restricted' version of transitivity suffices: Since finitariness is essential for practical reasoning-where a conclusion is always derived from a finite set of assumptions-here we are interested only in finitary logics.
In turn, the following generalization of the notion of many-valued matrix, as well as notions thereof, are essential for our investigation and has been extensively studied by Avron and co-authors [e.g. 2, 8-11]: DEFINITION 2.1 A non-deterministic matrix (Nmatrix) for L is a triple M = V, D, O , where: • V is a non-empty set of truth-values; • D is a non-empty proper subset of V (whose elements are called the designated elements of V); • O is a function that associates an n-ary function : V n −→ 2 V \{∅} with every n-ary connective of L.
We say that M is (in)finite if so is V.

DEFINITIONS 4
Let M be an Nmatrix for L.
• A partial M-valuation for L is a function v : F(L) * −→ V for some F(L) * ⊆ F(L) satisfying the following conditions: -The set F(L) * is closed under subformulae; i.e. sub(F(L) * ) = F(L) * .
( 1 ) • A partial M-valuation is a (full) M-valuation if its domain is F(L).

REMARK 2.2
When taking an Nmatrix as a generalization of a matrix, the latter is viewed as a special type of an Nmatrix in which each always returns a singleton.In such a case, each can be treated as a function : V n −→ V. Thus, when there is no risk of confusion, we shall identify singletons of truth-values with the truth-values themselves.

DEFINITIONS 5
Let M be an Nmatrix for L. The Tcr induced by M, M , is defined by: Γ M A if for every partial M-valuation v, if v(B) ∈ D for all B ∈ Γ , then v(A) ∈ D. We denote by L M = L, M the Tarskian propositional logic induced by M.

DEFINITIONS 6
Given a Tarskian propositional logic L = L, |∼ L and a Nmatrix M for L, we say that: • L is characterized by M iff L is both sound and complete for M.
The semantics of Nmatrices shares with the semantics of ordinary (deterministic) matrices important properties such as compactness [10], decidability [see 2] and, even more importantly, analyticity [9].In fact, because of analyticity Γ M A is decidable whenever Γ and M are finite.Namely, when assesing whether or not Γ M A, analyticity allows the search to be restricted to partial M-valuations whose domain is sub(Γ ∪ {A}).This, together with the finiteness of Γ and M, assures that the search space is also finite and, thus, that the corresponding algorithm always terminates. 4ow, the following notion is fundamental throughout our investigation: Let L be a finitary Tarskian propositional logic.An approximation system for L is a triple A = P, , {R α } α∈P , where P, is a directed set, called the parameter set, and {R α } α∈P is a family of consequence relations on L such that: • P, has a minimum element 0; • R 0 is a finitary Tcr; • for each α ∈ P, R α is decidable in polynomial time; We shall call L the limiting logic of the approximation system A , and L, R 0 its base logic.Each relation R α is an approximation to L.
Naturally, approximation systems are of practical and theoretical interest whenever the limiting logic is known or conjectured to be intractable. 5

Belnap's semantics and the need for imprecise values
First Degree Entailment (FDE) captures relevant entailment between implication-free formulae.Based on work of Dunn [e.g. 42], and an observation by Smiley (in correspondence), Belnap [14,15] gave an interesting semantic characterization of FDE in terms of a 4-valued logic, and pointed out its usefulness as the logic in which 'a computer should think'.This characterization has become not only the standard semantics of FDE, but also its standard presentation.It is motivated from the use of deductive reasoning as a basic tool in the area of 'intelligent' database management or questionanswering systems.Databases have a great propensity to be incomplete and become inconsistent: what is stored in a database is usually obtained from different sources which may provide only partial information and may well conf lict with each other.For a matrix to characterize a logic adequate for making deductions with information that might be both inconsistent and partial, at least 4 different values are needed [see 2].An elegant 4-valued matrix is precisely Belnap-Dunn's.
The set of truth-values is {t, f, b, n} and is denoted by 4. These values are interpreted as four possible ways in which an atom p can belong to the present state of information of a computer's database, which in turn is fed by a set Ω of equally 'reliable' sources: t means that the computer is told that p is true by some source, without being told that p is false by any source; f means that the computer is told that p is false but never told that p is true; b means that the computer is told that p is true by some source and that p is false by some other source (or by the same source in different times); n means that the computer is told nothing about the value of p.In essence, each value represents a subset of the set {true, false} of the classical values [42].These four values form two distinct lattices, depending on whether we consider the partial information ordering induced by set-inclusion (approximation lattice) or the partial ordering based on 'closeness to the truth' (logical lattice).The information ordering is the one according to which the epistemic state of the computer concerning an atom can evolve over time.As Belnap points out: When an atomic formula is entered into the computer as either affirmed or denied, the computer modifies its current set-up by adding a 'told True' or 'told False' according as the formula was affirmed or denied; it does not subtract any information it already has [ . . .] In other words, if p is affirmed, it marks p with t if p were previously marked with n, with b if p were previously marked with f; and of course leaves things alone if p was already marked either t or b [14, p. 12].
A set-up is simply an assignment to each of the atoms of exactly one of the values in 4. The values of complex formulae are obtained by means of considerations related to 'Scott's thesis' about approximation lattices [14], resulting in the truth-tables in Table 1.Using these truth-tables, every set-up can be extended to a valuation function v : F(L) −→ 4, where L = {∨, ∧, ¬}, in the usual inductive way.We call this function a 4-valuation.It establishes how the computer is to answer questions about complex formulae based on a set-up.While answering questions on the basis of a given epistemic set up is computationally easy, we do not have a logic yet.As Belnap puts it, we 'want some rules for the computer to use in generating what it implicitly knows from what it explicitly knows', i.e. we need a logic for the computer to reason.This is achieved by turning the truth-tables into a valuation system: Belnap-Dunn's matrix, M 4 , is the matrix for L where V = 4, D = {t, b}, and the functions in O are defined by the truth-tables in Table 1.(Warning: do not confuse the values t and f in 4 with true and false.The latter are local values referring to the information coming from a source, the former are global values, summarinzing the epistemic state of the computer with respect to all the sources.)Thus: The consequence relation is then defined as follows: For the unrestricted language allowing arbitrary formulae involving ∧, ∨ and ¬, the decision problem for this consequence relation is co-NP complete [see 3,75].This fact follows from the celebrated result by Cook [24] showing that CPL is co-NP complete, together with the fact that the decision problem of inconsistency in CPL can be reduced to the decision problem of entailment in FDE.The latter specifically as follows: where p 1 , ..., p n are the atoms occurring in Γ .PROOF.By definition, Γ is classically inconsistent iff there is no classical valuation, v : F(L) −→ {true, false}, such that v(A) = true for all A ∈ Γ .In turn, also by definition, this holds iff for every 4-valuation, v : F(L) −→ {t, f, b, n}, such that v(A) ∈ {t, b} for all A ∈ Γ , v(A) = b for some A ∈ Γ and so, by the FDE-tables, v(p i ) = b for some p i occurring in Γ .Thus, by the FDE-tables for ¬ and ∧, this holds iff for every 4-valuation v such that v(A) ∈ {t, b} for all A ∈ Γ , v(p i ∧ ¬p i ) = b for some p i occurring in Γ and so, by the This situation brings us to the need for tractable approximations.In the next section, we shall present a sort of natural deduction system for FDE based on two key observations.First, as is implicit in the quotation from Belnap above, the values in 4, except for b, cannot be taken as stable.An epistemic set up is just a snapshot of an epistemic state that evolves over time.If we want to consider the truth-values t, f, n as stable we need to assume complete information about the set of sources Ω.Namely, while the meaning of b is 'there is at least a source assenting to p and at least a source dissenting from p' (which is information empirically accessible to x in the sense that x may hold this information without a complete knowledge of Ω), the meaning of t, f and n involves information of the kind 'there is no source such that . . .', and so requires complete information about the sources in Ω, which may not be empirically accessible to x at any given time.What if the agent does not have such a complete knowledge about the sources?For instance, the agent may well be receiving information from an'open' set of sources as they become accessible (even if the information coming from each single source is assumed to be robust).In such a case, the possibility for an agent to come across a source falsifying 'there is no source such that . . .' is always open.Thus, despite their informational nature, three of the values in 4 are information-transcendent when interpreted as timeless.They refer to an 'objective' informational situation concerning the domain of all sources, that may well be inaccessible to the computer at any given time.This motivates the need for a stable imprecise value such as 't or b', which is implicit in the choice of the set of designated values by Belnap.Inspired by work of D'Agostino [26], and Fitting and Avron [7,44,45], we shall address this question by shifting to signed formulae, where the signs express such imprecise values associated with two distinct bipartitions of 4.
A second key observation is that, as suggested by Belnap [14,15], there is no reason to assume that an agent is 'told' about the values of atoms only.As we shift from objective truth and falsity to informational truth and falsity, this is a highly unrealistic restriction.In most practical contexts, we may be told that a certain disjunction is true without being told which of the two disjuncts is the true one, or that a certain conjunction is false without being told which of the two conjuncts is the false one.As a simple example of the former situation, take the information that Alice and Bob are siblings (either they have the same mother or they have the same father); for the latter, take the information that Alice and Bob are not siblings, i.e. for any individual x, the conjunction 'x is a parent of Bob and x is a parent of Alice' must be false, which amounts to saying that either the first or the second conjunct is false, without necessarily knowing which.In the context of CPL, these considerations naturally lead to a non-deterministic 3-valued semantics that was anticipated by Quine.See [29] for further references and a discussion that includes the following quotation from Dummett to the effect that in non-mathematical contexts our information may well be irremediably disjunctive in nature: I may be entitled to assert 'A or B' because I was reliably so informed by someone in a position to know, but if he did not choose to tell me which alternative held good, I could not apply an or-introduction rule to arrive at that conclusion.[...] Hardy may simply not have been able to hear whether Nelson said 'Kismet hardy'or 'Kiss me Hardy', though he heard him say one or the other: once we have the concept of disjunction, our perceptions themselves may assume an irremediably disjunctive form.[...] Unlike mathematical information, empirical information decays at two stages: in the process of acquisition, and in the course of retention and transmission.An attendant directing theatregoers to different entrances according to the colours of their tickets might even register that a ticket was yellow or green, without registering which it was, if holders of tickets of either colours were to use the same entrance; even our observations are incomplete, in the sense that we do not and cannot take in every detail of what is in our sensory fields.That information decays yet further in memory and in the process of being communicated is evident.In mathematics, any effective procedure remains eternally available to be executed; in the world of our experience, the opportunity for inspection and verification is f leeting [41, pp. 266-278].
These two observations prompt us to propose a proof-theoretic approach to depth-bounded reasoning in FDE that is similar to the one taken in [30][31][32] for CPL.Before addressing this issue, however, we shall provide in the next section a proof-theoretic characterization of unbounded reasoning in FDE that will pave the way for defining its tractable approximations.

Intelim deduction in FDE
In what follows we shall use signed formulae (S-formulae for short).These are expressions of the form T A, F A, T * A, F * A, where A is an unsigned formula.Denoting an agent with x and a 4valuation with v, their intended interpretation is respectively as follows: 'x holds that A is at least true' (expressing that v(A) ∈ {t, b}); 'x holds that A is non-true' (v(A) ∈ {f, n}); 'x holds that A is non-false' (v(A) ∈ {t, n}); 'x holds that A is at least false' (v(A) ∈ {f, b}).Crucially, S-formulae of the form T A or F * A express information that x may hold even without a complete knowledge of the set sources Ω.However, this is not the case of the other two types of S-formulae which involve complete knowledge of Ω and so can only be assumed hypothetically.Now, we say that the conjugate of T A is F A and vice versa, and that the conjugate of T * A is F * A and vice versa.Let us use S as a variable ranging over {T , F , T * , F * }, and with S denote: and T * if S = F * .Besides, we shall write T Γ for {T A | A ∈ Γ }.Moreover, we shall use ϕ, ψ, θ , . .., possibly with subscripts, as variables ranging over S-formulae; and X , Y , Z, . .., possibly with subscripts, as variables ranging over sets of S-formulae.Further, let us use φ to denote the conjugate of ϕ.Finally, we say that the unsigned part of an S-formula is the unsigned formula that TABLE 2. Introduction rules for the standard FDE connectives results from it by removing its sign.Given an S-formula ϕ, we denote by ϕ u the unsigned part of ϕ and by X u the set {ϕ u | ϕ ∈ X }.Note also that, for the reasons explained in the previous section, an agent may hold the information that T A ∨ B, but neither the information that T A nor that T B. Similarly, she may hold the information that F * A ∧ B, but neither the information that F * A nor that We identify the basic (0-depth) logic of our hierarchy of approximations with the inferences that an agent can draw without making hypotheses about the 'objective' informational situation concerning the whole of Ω.In other words, without making hypothetical assumptions that go beyond the information that he holds.We shall show that a natural proof-theoretic characterization of this basic logic is obtained by means of the set of introduction and elimination (intelim) rules respectively displayed in Tables 2 and 3.The analogous 0-depth system for CPL in [30,31] is characterized by the intelim rules obtained by removing all the starred signs, replacing them with the unstarred signs T and F , interpreted as 'only true' and 'only false', and eliminating duplicates.Note that the characterization of the basic logic bears some resemblance with natural deduction, but does not have discharge rules, since no hypothetical reasoning is involved.Besides, observe that the intelim rules for disjunction and conjunction are dual of each other, and that a sentence and its negation are treated in a symmetric way.Moreover, in the elimination rules, we shall refer to the premise containing the connective that is to be eliminated as major and to the other premise as minor.In turn, given that the intelim rules have all a linear format, their application generates intelim sequences.Namely, finite sequences (ϕ 1 , . . ., ϕ n ) of S-formulae such that, for every i = 0, . . ., n, either ϕ i is an assumption or it is the conclusion of the application of an intelim rule to preceding S-formulae.In Figure 1 we show a simple example of an intelim sequence, where each assumption is marked with an '@'.
The intelim rules are all sound, but not complete for full FDE.Indeed, as we shall show below, these rules just characterize the basic logic in the hierarchy of approximations.Completeness for full FDE is obtained by adding only two branching structural rules, according to which we are allowed to: (i) append both T A and F A as sibling nodes to the last element of any intelim sequence; (ii) append both T * A and F * A in a similar way.The intuitive meaning of these rules is that one of the two cases must obtain considering the whole of Ω even if the agent has no actual information about which is the case.In this sense, we call the information expressed by each conjugate S-formula virtual information; i.e. hypothetical information that the agent does not hold, but she temporarily assumes as if she held it.We respectively call these branching rules PB and PB * as they are closely For CPL only the first rule, with T and F interpreted as 'only true' and 'only false', makes sense and is sufficient for completeness.Given that the sets of starred and unstarred rules look identical, one might wonder why we need both sets and ultimately four signs.The reason is given by the negation connective.For example, according to the FDE-table for ¬ and the intended interpretation of the signed formulae, T ¬A (v(¬A) ∈ {t, b}) implies F * A (v(A) ∈ {f, b}), and not F A as one would expect classically.The other cases involving negation are, of course, analogous [see 46].
With the addition of PB and PB * to the stock of rules, deductions are represented by downwardgrowing trees, which brings the method somewhat closer to tableaux.Each application of PB or PB * stands for the introduction of virtual information about the imprecise value of a formula A, which we shall respectively call the PB-formula or PB * -formula.Note once again that, whereas signed formulae of the form T A and F * A represent information that may be empirically obtained (when A turns out to be b), signed formulae of the form T * A and F A are obtainable only by applying PB or PB * .In turn, any S-formulae appended via those branching rules will be called a virtual assumption.Now, PB and PB * are essentially cut rules that may introduce formulae of arbitrary degree.However, as we will show in Lemma 5.4, their application can be restricted so as to satisfy the subformula property.Moreover, from our informational viewpoint, the main conceptual advantage of this prooftheoretic characterization consists in that it clearly separates the intelim rules that fix the meaning of the connectives in terms of the information that an agent holds from the two structural rules that introduce virtual information (PB and PB * ).
Intuitively, the more virtual information needs to be invoked via PB or PB * the harder the inference is for the agent, both from the computational and the cognitive viewpoint.In this sense, the nested applications of PB and PB * provide a sensible measure of inferential depth.This naturally leads to defining an infinite hierarchy of tractable depth-bounded approximations to FDE in terms of the maximum number of nested applications of PB and PB * that are allowed.In turn, such a hierarchy can be intuitively associated with a hierarchy of increasingly idealized agents with more and morealbeit always bounded-cognitive and computational resources or inferential power.Note, however, that the inferential depth associated with an agent is not intended to be interpreted as an upper bound on her inferential power.Rather, it is understood as the maximum depth for which it is guaranteed that, if she possesses the information explicitly carried by the assumptions, she possesses the information explicitly carried by the conclusion.
Before giving definitions and results, we remark that (i) unlike the branching rules of Smullyanstyle tableaux, our branching rules are structural in that they do not involve any specific logical operator; (ii) the elimination rules, together with the branching rules, were early introduced in [26] as constituting a refutation method for full FDE called RE fde .So, the completeness of RE fde trivially implies the completeness of the system presented in this paper.However, our intelim method can be used as a direct-proof method as well as a refutation method, and leads to more powerful approximations.A direct completeness proof can also be given based on the semantics, which yields the subformula property.In this paper we choose to prove a more general version of the subformula property by means of proof transformations.

DEFINITIONS 6
• Let X = {ϕ 1 , . . ., ϕ m }.Then T is an intelim tree for X if there is a finite sequence T n = T , and for each i < n, T i+1 results from T i by an application of an intelim rule to preceding S-formulae in the same branch, or by an application of PB or PB * .• A branch of an intelim tree is closed if it contains an S-formula ϕ and its conjugate φ; otherwise, it is open.• An intelim tree is said to be closed when all its branches are closed; otherwise, it is open.
• An intelim proof of ϕ from X is an intelim tree T for X such that ϕ occurs in all open branches of T .• An intelim refutation of X is a closed intelim tree T for X .
Note that every refutation of X is, simultaneously, a proof of ϕ from X , for every ϕ.This because there are no open branches and so the condition that ϕ occurs at the end of all open branches is vacuously satisfied.This is, of course, a kind of explosivity, but it regards signed formulae, and it FIGURE 2. A single-branch open itelim tree is compatible with the non-explosivity regarding formulae in FDE.The reason of that compatibility is that a set consisting of S-formulae all of the form T A cannot lead to explosion because there cannot be an intelim refutation of such a set.To begin with, starting from a set T Γ , there is no way of obtaining S-formulae of the form F A or T * A by applying only intelim rules.Starting from a set T Γ , the only way of obtaining formulae of such forms is by applying PB or PB * and, thus, adding virtual information.Nonetheless, a set T Γ cannot lead to explosion even if we add virtual information when unfolding the information contained in T Γ .In fact, as the following result shows, for a set of S-formulae X to lead to explosion it must contain (in itself) S-formulae of the form F A or T * A, i.e. virtual information.PROPOSITION 4.1 Any intelim tree for a set T Γ has at least a branch containing only S-formulae of the form T A or F * A.
PROOF.By an easy induction on the number of nodes of the intelim tree.
The above proposition implies that any intelim tree for a set T Γ is open and, thus, that there is no refutation of such a set.This fact regarding our proof-theoretic characterization of FDE corresponds to the fact regarding its 4-valued semantics according to which all the elements of any set of formulae can have a designated truth-value.More specifically, it corresponds to the fact that for any formula A there is a 4-valuation v such that v(A) = b; namely, v such that for all p ∈ At(L), v(p) = b.Note that a set T Γ may well contain T A and T ¬A, or either of them may well be obtained from that set by applying the rules, which precisely amounts to the formula A having the truth-value b.However, that pair of S-formulae does not close a branch.
Moreover, as it is usual in refutation methods, our intelim method can be used to obtain counterexamples from open branches [see also 46].For instance, in Figure 2 Now, as mentioned above, PB and PB * may introduce formulae of arbitrary degree.However, the set of formulae that can be used as PB-formulae or PB * -formulae can be bounded in a variety of ways without loss of completeness.We call this set virtual space and define it as a function f of the set Γ ∪{A}, consisting of the premises Γ and of the conclusion A of the given inference.The strictest way of bounding the virtual space consists in allowing as PB-formulae and PB * -formulae only atomic formulae that occur in Γ ∪ {A}.A more liberal option is allowing only subformulae of the formulae in Γ ∪{A}.More generally, let F be the set of all functions f on the finite subsets of F(L) such that: 7(i) for all Δ, at(Δ) ⊆ f (Δ); (ii) f (Δ) is closed under subformulae, i.e. sub(f (Δ)) = f (Δ); (iii) the size of f (Δ) is bounded above by a polynomial in the size of Δ, i.e. |f (Δ)| ≤ p(|Δ|) for some fixed polynomial p. (This last requirement will be essential in order to define tractable approximations below.)The choice of an specific function to yield suitable values of the virtual space for each particular deduction problem is the result of decisions that are conveniently made by the system designer, depending on the intended application. 8In turn, the functions in F are partially ordered by the relation such that Distinguished examples of functions in F are the identity function f (Δ) = Δ, sub and at.However, in general, f (Δ) may contain formulae that are not in sub(Δ).For instance, the operation f that maps Δ to the set of all formulae of bounded degree that can be built out of sub(Δ) and at(Δ) is also in F. Thus, our intelim method allows for (possibly shorter) deductions that do not have the subformula property simply by permitting applications of PB or PB * to formulae that are not subformulae either of the premises or of the conclusion.However, even in this latter deductions the virtual space is still bounded.
The branching rules are not the unique rules of our intelim method that may bring about violations of the subformula property.The introduction rules could in principle be indefinitely applied, leading to ever more complex formulae.Nonetheless, as we shall show below, the application of both kind of rules can be restricted so as to satisfy the subformula property.More specifically, we shall show that every intelim proof of ϕ from X (intelim refutation of X ) can be transformed into an intelim proof of ϕ from X (an intelim refutation of X ) with the subformula property.

Subformula property
The subformula property (SFP, for short) is a key property of logical systems in that it allows us to search for proofs or refutations by analytic methods; i.e. by considering solely deduction steps involving formulae that are 'contained' in the assumptions, or also in the conclusion in the case of proofs.This implies a drastic reduction of the search space which is crucial for the purpose of automated deduction.When it comes to propositional logics, this search space is finite for each putative inference, paving the way for decision procedures.Particularly, in our intelim method, the SFP guarantees that we can impose a bound on the applications of PB and PB * , that could in principle be applied to arbitrary formulae, with no loss of deductive power.Similarly, it guarantees that we can impose a bound on the sensible applications of introduction rules, which could in principle be indefinitely applied, yielding ever more complex formulae.DEFINITION 5.1 An intelim proof T of ϕ from X (an intelim refutation of X ) has the subformula property if, for every S-formula Now, consider the intelim sequences of Figure 3.The first one is a proof of T * ¬s from {T * ¬p, T q, T * ¬r ∨ ¬s, T r}.The second one is a proof of (an arbitrary) T q from {T p, F p}; i.e. an instance of the explosivity of our intelim method.Note that both proofs are redundant.In the first FIGURE 3. Redundant itelim sequences proof, the S-formula T r ∨ p is first introduced (from premise T r) and then eliminated (using the minor premise F p) to re-obtain the S-formula T r which was already contained in the sequence; i.e. this proof contains circular reasoning.In the second proof, the S-formula T p ∨ q is first introduced (from premise T p) and then eliminated (using F p as minor premise); yet, the sequence was already closed before the application of the disjunction introduction and so, by Definitions 6, the closed sequence T p, F p was already a proof of T q from T p and F p.
The same kind of redundancy is present whenever a formula is, simultaneously, the conclusion of an introduction and the major premise of an elimination.

DEFINITIONS 7
An occurrence of an S-formula ϕ in an intelim tree T is: (i) a detour if ϕ is both the conclusion of an introduction and the major premise of an elimination; (ii) idle if it is not the terminal node of its branch, it is not used as premise of some application of an intelim rule, and it is not the conjugate of some S-formula occurring in the same branch.

DEFINITIONS 8
Given an intelim tree T , a path in T is a finite sequence of nodes such that the first node is the root of T and each of the subsequent nodes is an immediate successor of the previous one.A path is closed if it contains both ϕ and φ for some ϕ.
Note that, according to the above definition, every branch is a maximal path.DEFINITION 5.2 Let T be an intelim proof of ϕ from X (an intelim refutation of X ).T is non-redundant if it satisfies the following conditions: 1. it contains no idle occurrences of S-formulae; 2. none of its branches contains more than one occurrence of the same S-formula; 3. none of its branches properly includes a closed path.
Observe that if an intelim proof or refutation contains a detour, then either condition 2 or 3 above is violated.Thus: LEMMA 5.3 If an intelim proof or refutation T is non-redundant, then it contains no detours.PROOF.By the definitions above and inspection of the intelim rules.Every detour makes the tree redundant.Now, turning an intelim proof or refutation T into a non-redundant one (with no increase in the size of the proof or refutation) is computationally easy, in that it only involves the following pruning steps: 1. check if there are closed paths and remove whatever follows after them; 2. remove any repetition of S-formulae in the same branch; 3. check if there are idle occurrences of S-formulae, and 4. for each idle occurrence of an S-formula ϕ: • if ϕ is the conclusion of an application of an intelim rule, just remove ϕ from T ; • if ϕ is a virtual assumption introduced by an application of PB or PB * , remove both ϕ and the whole subtree generated by its conjugate S-formula φ introduced in the same application of PB or PB * ; then attach the subtree below ϕ to the immediate predecessor of ϕ.
It is easy to verify that the result of this procedure is still an intelim proof of the same conclusion from the same premises, or an intelim refutation of the same assumptions.
In turn, given a proof T of ϕ from X (a refutation of X ), and any operation f ∈ F, we say that an application of PB or PB * in T is f -analytic if its PB-formula or PB * -formula is in f (X u ∪ {ϕ u }) (f (X u )); i.e. in the virtual space defined by the function f .Recall that the latter is, by definition, closed under subformulae and polynomially bounded.When f = sub, i.e. the virtual space consists exactly of the subformulae of X u ∪{ϕ u } (X u ), we just say that the application of PB or PB * is analytic.Thus: LEMMA 5.4 Given any f ∈ F, every intelim proof T of ϕ from X (intelim refutation of X ) can be transformed into an intelim proof of T ϕ from X (intelim refutation T of X ) such that every application of PB and PB * in T is f -analytic.
PROOF.We use the notation T ψ to denote either an empty intelim tree or a non-empty intelim tree such that ψ is one of its terminal nodes.The proof is by lexicographic induction on γ (T ), κ(T ) , where γ (T ) denotes the maximum degree of a PB-formula or a PB * -formula in T that is not fanalytic, and κ(T ) denotes the number of occurrences of such non-f -analytic PB-formulae or PB *formulae of maximal degree.Let γ (T ) = m > 0 and let A be a PB-formula or a PB * -formula of degree m.There are several cases depending on the logical form of A and on whether A is PBformula or a PB * -formula.We sketch only the case where A = B ∧ C and A is a PB * -formula; the other cases being similar.So, T has the following form: where T b and T c are intelim trees such that each of their open branches contains ϕ, or are both closed intelim trees.Let T be the following intelim tree: Clearly, T is an intelim proof of ϕ from X (an intelim refutation of X ).Moreover, either γ (T ) < γ (T ), or γ (T ) = γ (T ) and κ(T ) < κ(T ).
In fact, the transformations used in the proof of the above lemma show that every intelim tree can be turned into an equivalent one in which all the PB-formulae and PB * -formulae are atomic.Thus, in principle, we could reformulate the notion of intelim tree in such a way that PB and PB * are applied only to atomic formulae without loss of completeness.Nevertheless, if we demand that the applications of PB and PB * be restricted to atomic formulae, the property of being an intelim tree is no longer preserved under uniform substitutions of the atomic formulae occurring in the tree with arbitrary formulae. 9On the other hand, if we require that the notion of intelim tree be restricted so as to permit only analytic applications of PB and PB * (i.e.f -analytic applications with f = sub), the property of being an intelim tree is indeed invariant under uniform substitutions.
The following theorem states the SFP of our intelim method when f = sub:10 THEOREM 5.5 (Generalized SFP).
For every f ∈ F, if T is an intelim proof of ϕ from X (an intelim refutation of X ) such that (i) T is non-redundant, and (ii) every application of PB and PB * in T is f -analytic, then for every S-formula ψ occurring in T , PROOF.Let T be a intelim proof of ϕ from X (refutation of X ) satisfying (i) and (ii), and suppose that there are S-formulae ω in T such that Let us call such S-formulae spurious.Let ψ be a spurious formula such that ψ u is of maximal degree in T .Then ψ cannot result from the application of an elimination rule, otherwise T would contain an spurious formula whose unsigned part is of strictly greater degree; namely, the major premise of this elimination.Moreover, given that T contains only f -analytic applications of PB and PB * according to (ii), no spurious S-formula can occur in it as a virtual assumption introduced by an application of PB or PB * .Therefore, ψ must be the conclusion of an introduction.Since, according to (i), T is non-redundant, it contains no idle occurrences of S-formulae and so either (a) ψ = θ for some θ occurring in the same branch or (b) ψ is used as a premise of a rule application.However, both cases are impossible.Regarding (a), by the same arguments just used for ψ, θ (the conjugate of ψ) can only be the conclusion of an introduction.Then, it is not difficult to see, by inspection of the introduction rules, that case (a) implies that one of the premises of the introduction of ψ = θ must be the conjugate of one of the premises of the introduction of θ .So, one of the branches of T properly contains a closed path, against the assumption that T is non-redundant.As for (b), first note that ψ cannot be the minor premise of an elimination, otherwise there would be again an spurious formula whose unsigned part is of greater degree in T ; namely, the major premise of this elimination.Moreover, ψ cannot be used in T as major premise of an elimination, otherwise ψ would be a detour and, by Lemma 5.3, T would be redundant, against hypothesis (i).
6 Depth-bounded approximations to FDE DEFINITIONS 9 The depth of an intelim tree T is the maximum number of virtual assumptions occurring in a branch of T .An intelim tree T is a k-depth intelim proof of ϕ from X (a k-depth intelim refutation of X ) if T is an intelim proof of ϕ from X (an intelim refutation of X ) and T is of depth k.
Note that a 0-depth intelim tree is nothing but an intelim sequence.Examples of, respectively, two proofs of depth 1, a refutation of depth 1 and a proof of depth 2, all with the SFP, are given in Figure 4. Again, each assumption is marked with an '@'.

DEFINITIONS 10
For all X , ϕ, • ϕ is 0-depth deducible from X , X 0 ϕ, iff there is a 0-depth intelim proof of ϕ from X ; • X is 0-depth refutable, X 0 , iff there is a 0-depth intelim refutation of X .

NOTATION 11
We shall abuse of the same relation symbol ' 0 ' to denote 0-depth deducibility and refutability.PROPOSITION 6.1 L, 0 is a (finitary) Tarskian propositional logic; i.e. 0 satisfies ref lexivity, monotonicity, cut and structurality.PROOF.By the definitions above.Now, the following definition will be useful for showing the tractability of our approximations below: DEFINITION 6.2 A set X is intelim saturated if it satisfies the following conditions: 1. for no A, T A and F A, nor T * A and F * A, are both in X ; Downloaded from https://academic.oup.com/logcom/article/34/5/815/7198103 by guest on 23 July 2024 FIGURE 4. k-depth intelim proofs and refutations 2. for every ϕ, if ϕ follows from some subset of X by an application of an intelim rule, then ϕ ∈ X .
In turn, the notion of k-depth deducibility depends not only on the depth at which the use of virtual information is recursively allowed, but also on the virtual space discussed above.So, finally: DEFINITIONS 11 For all X , ϕ and for all f ∈ F, When X f k ϕ, we say that ϕ is deducible at depth k from X over the f -bounded virtual space.The above definition covers also the case of k-depth refutability by assuming X f k as equivalent to X f k ϕ for all ϕ.When X f k , we say that X is refutable at depth k over the f -bounded virtual space.

NOTATION 12
We shall abuse of the same relation symbol ' f k ' to denote k-depth deducibility and refutability over the f -bounded virtual space.
Observe that in the above definitions the pair of S-formulae, ψ and ψ, denote a pair of (conjugate) virtual assumptions introduced by respectively PB or PB * .Thus, according to the definitions, X f k ϕ iff the conclusion ϕ is obtained at depth k − 1 by introducing either T A and F A, or T * A and F * A, as virtual assumptions-for some A in the virtual space defined by f .This corresponds to the fact that, in our intelim method, a formula ϕ may be obtained at a certain depth by introducing whichever T A or F A by an application of PB but not by introducing T * A or F * A by an application of PB * and vice versa.Now, it follows immediately from Definitions 9 and 11 that: However, it is easy to verify that the relations f k satisfy the following version of cut: Moreover, the relations f k may not be structural in that structurality depends on the function f that defines the virtual space.For example, sub k is structural, while at k is not.In general, structurality can be imposed by restricting the operations in F to those such that, for all substitution σ and all Δ, is the set of all formulae of given bounded degree that can be built out of sub(Δ).Further, since 0 is monotonic, its successors are ordered: corresponds to an increase in the depth at which the nested use of virtual information (restricted to formulae in the virtual space defined by f ) is allowed.Note also that

Tractability
We now show that the decision problem for the k-depth logics is tractable.Theorem 5.5 immediately suggests a decision procedure for k-depth deducibility: to establish whether ϕ is k-depth deducible from a finite set X we apply the intelim rules, together with PB and PB * up to a number k of times, in all possible ways starting from X and restricting to applications which preserve the SFP.If the resulting intelim tree is closed or ϕ occurs at the end of all its open branches, then ϕ is k-depth deducible from X , otherwise it is not.We shall first show the tractability of the 0-depth logic, and then the tractability of the k-depth logics, k > 0. DEFINITION 6.5 The subformula graph for X u , G, is the oriented graph V , E such that V = sub(X u ) and A, B ∈ E iff A is an immediate subformula of B. DEFINITION 6.6 A module is a set consisting of a non-atomic formula, called the top formula of the module, and of its immediate subformulae, called the secondary formulae of the module.DEFINITION 6.7 A G-module is any subgraph M of G whose set of vertices is a module; i.e., consists of a formula with all its immediate subformulae.The top formula of M is the top formula of the underlying module.DEFINITION 6.8 A labelled subformula graph for X u is a pair G, λ , where G is the subformula graph for X u , and λ is a relation λ ⊆ V × {T , F , T * , F * }, called the labelling relation and such that: i. each A ∈ V is related to at most two elements in {T , F , T * , F * }; ii. for no A ∈ V , A, T and A, F are both in λ; iii.for no A ∈ V , A, T * and A, F * are both in λ.

DEFINITIONS 12
An intelim graph for X u based on Y u , with Y u ⊆ X u , is a labelled subformula graph G, λ for X u satisfying the following conditions: 1. for every SA ∈ Y , S ∈ λ(A); 2. for every A ∈ sub(X u ) such that A / ∈ Y u , λ(A) is defined and equal to {T }, {F }, {T * }, {F * }, {T , T * }, {T , F * }, {F , T * }, or {F , F * } only if there are B 1 , . . ., B k ∈ sub(X u ) such that T A, F A, T * A, or F * A follows from by an application of an intelim rule. 11λ(A) = ∅ whenever the image of A under λ is undefined, and we say that the initial intelim graph for X u based on Y u is the intelim graph for X u based on Y u such that λ(A) = ∅ for all A / ∈ Y u .An intelim graph for X u is completed if it satisfies also the converse of 2. Now, let the symbol ' ' stand for the 'inconsistent labeling relation'.This is only a way of speaking to mean that there is no labelling relation consistent with the intelim rules.Simple decision procedures for 0-depth refutability and deducibility are illustrated in Algorithms 1 and 2, and consist in building the initial intelim graph for the set of formulae that are mentioned in the specification of the problem-i.e.just assumptions for refutation problems, and assumptions plus the conclusion for deduction problems-and turning it into a completed intelim graph in accordance with the intelim rules.Both algorithms call the subroutine Expand described in Algorithm 3. The latter, in turn, calls the subroutine Apply Intelim described in Algorithm 4. ALGORITHM 1 Decision procedure for 0-depth refutability Input: A finite set X of S-formulae; 1: push all formulae A such that λ(A) = ∅ into formula stack; 2: while λ = and formula stack is not empty do 3: pop a formula A from formula stack; 4: Apply Intelim(A, G, λ ) 5: end while The correctness of both decision procedures follows from the fact that they return true iff the set if Z contains all the premises for an application of an introduction rule with conclusion SB then 5: if We omit a detailed proof and just brief ly discuss their complexity.The input for Algorithm 1 is a list of all the S-formulae in X , while the input for Algorithm 2 is a pair consisting of a list of all the S-formulae in X and the S-formula ϕ.Let n be the total size of the input; namely O(|X |) for Algorithm 1 and O(|X ∪ {ϕ}|) for Algorithm 2. Observe that: 1.The number of nodes in the subformula graph G (line 1 of both decision procedures) is O(n).

The cost of building the initial labeled subformula graph
3. The while loop in the Expand subroutine, in Algorithm 3, is executed at most as many times as there are nodes in G; namely O(n) times.Here the key observation is that, in line 3, the formula A can be safely removed from formula stack; i.e. each formula in formula stack needs to be visited at most twice.4. The essential cost of each run of the while loop consists in the cost of the Apply Intelim subroutine (Algorithm 4). 5.For each formula A in G there are at most O(n) G-modules containing A (line 1 of Algorithm 4).6.The maximum number of nodes in a G-module is a + 1, where a is the maximum arity of a connective.

The cost of each run of the forall loop in the Apply Intelim subroutine is O(a).
It follows from ( 1)-( 7) that: THEOREM 6.9 Whether or not X 0 ϕ (X 0 ) can be decided in time O(n 2 ), where n = |X ∪ {ϕ}| (n = |X |).
Now, a decision procedure for X sub k ϕ can be obtained from that of X 0 ϕ by generalizing the Expand sobroutine (Algorithm 3) into the one displayed in Algorithm 5.
ALGORITHM 5 Depth-Expand(k, G, λ ) Expand( G, λ ); 3: else 4: push all formulae A such that λ(A) = ∅ into undefined formulae; 5: while λ = and undefined formulae is not empty do 6: pop a formula A from undefined formulae; 7: let λ 1 be such that T ∈ λ 1 (A) (alternatively, set λ = λ 2 ; 13: remove from undefined formulae the formulae A such that λ(A) = ∅; HINT.From Definitions 11 and the observation that there are O(n) distinct subformulae of X u ∪ {ϕ u } (X u ).It can be shown that the procedure terminates in a number of steps less than or equal to so that the upper bound for the unbounded system is just exponential.
The above theorem refers to the basic case where the function that defines the virtual space is sub, but is not difficult to generalize it for any polynomially bounded virtual space [see 30,31].More precisely, when f sub, the complexity of the decision problem is O(n k+2 ).In general, the complexity is O(p(n) k+2 ) where p is a polynomial depending on f (recall that, by definition, the virtual space is polynomially bounded).

5-valued non-deterministic semantics
The signs of our intelim method can be taken as imprecise truth-values that intuitively encode partial information about the standard truth-values in 4 [see 7,9]; namely, two-element sets of the standard truth-values: Note that t ∩ t * = t and f ∩ f * = f.Let us denote the set of these imprecise truth-values, {t, f, t * , f * }, by 4. Now, we can take the elements of 4 as primitive, and use Dunn-style relational semantics [42] to define analogous notions to those of set-up and 4-valuation: i. for no p, p, t and p, f are both in η; ii. for no p, p, t * and p, f * are both in η.
Given a 4-valuation, η, this is extended to a relation η ⊆ F(L) × 4 by recursive clauses: Intuitively, our imprecise values model a database which has only partial information of (or access to) a set of sources Ω.Let us say that an agent's database is a pair D = Ω, φ , where Ω is a set of information sources and φ is a partial function At(L) × Ω −→ {true, false}.So, given a 4-valuation and denoting an agent with x: • pηt iff x holds that there is s ∈ Ω such that φ(p, s) = true but does not hold that there is no s ∈ Ω such that φ(p, s) = false; • pηf iff x holds that there is no s ∈ Ω such that φ(p, s) = true but does not hold that there is no s ∈ Ω such that φ(p, s) = false; • pηt * iff x holds that there is no s ∈ Ω such that φ(p, s) = false but does not hold that there is no s ∈ Ω such that φ(p, s) = true; • pηf * iff a holds that there is s ∈ Ω such that φ(p, s) = false but does not hold that there is no s ∈ Ω such that φ(p, s) = true. 12he notion of 4-valuation would yield yet another alternative semantics for full FDE.However, we introduce it here only as a first step towards devising a semantics for the depth-bounded approximations defined proof-theoretically above.

The 0-depth logic
Within our conceptual framework, the value of formulae may be completely undefined when the agent's information of Ω is insufficient to even establish any of the imprecise values.As mentioned above, there is no reason to assume that an agent is 'told' about the values of atoms only.In most practical contexts, it may well be that the sources inform the agent that a certain disjunction is true without informing her about which of the two disjuncts is the true one, or, analogously, that a certain conjunction is false without informing her about which of the two conjuncts is the false one.
We shall denote Aη⊥ whenever η is undefined for A. It is technically convenient to treat ⊥ as a fifth value and take it as equivalent to the set {t, b, f, n}.Intuitively, this undefined value stands for full indeterminacy or ignorance about the standard defined value of a formula, which amounts to the situation where all those defined values are admissible.This in the sense the agent's information is not sufficient to discard even a single defined value.An important intuition here is that ⊥ may eventually turn into an imprecise or even an standard truth-value by the development of the agent's reasoning or querying process.Thus, let us denote by 5 the set consisting of the elements of 4 together with ⊥.

DEFINITION 7.2 A 5 non-deterministic valuation is a relation
For no formula A, and S 1 , S 2 ∈ 2 5 , it is the case that: i. Aη S 1 , Aη S 2 and {t, f} ⊆ S 1 ∪ S 2 ; ii.Aη S 1 , Aη S 2 and {t * , f * } ⊆ S 1 ∪ S 2 .TABLE 4. 5N-tables obtained, form their union, which is equivalent to either an imprecise value or again to a singleton set.In the former case, take such an imprecise value as an element of the image of η ; in the latter case, take such a singleton set as equivalent to the image of η .
These computations are justified by our intelim rules which, as explained above, are semantic in nature.For example, when ∨ is the connective at issue, and f and ⊥ are respectively the values of the first and the second 'arguments' according to η, the elements of the image of η can be only t * and ⊥.For the other values are excluded on the basis of the rules: t cannot be in the image of η since, otherwise, the value of the second 'argument' should have been t; f is also excluded because, otherwise, the values of both 'arguments' should have been f; similarly, f * is excluded since, otherwise, the values of both 'arguments' should have been f * .So, t * and ⊥ are the only admissible values in the image of η .In general, the elements of the images of η can be obtained by excluding values in 5 on the basis of the rules, as the reader can verify.Now, this 5-valued relational semantics à la Dunn can be summarized by the 5-valued nondeterministic truth-tables (5N-tables, for short) in Table 4. Thereby, we are in a position to introduce the following notion: DEFINITION 7.4 Let M dfde be the Nmatrix for L, where V = 5, D = {t} and the functions in O are defined by the 5N-tables in Table 4. 13Therefore, using the 5N-tables, a 5 non-deterministic valuation can be defined as a function.Namely, a 5-valuation for L is a function v : F(L) −→ 5.Then, we pick out from the set of all 5-valuations those which agree with the intended meaning of the connectives via M dfde : DEFINITION 7.5 A 5N-valuation is a 5-valuation v such that for all A, B ∈ F(L): Where • is ∨ or ∧.
REMARK 7.6 A 5N-valuation can be seen as describing an information state that is closed under the implicit information that depends only on the informational meaning of the connectives.This is information that the agent holds and with which he can operate, in the precise sense that he has a feasible procedure to decide, for every A, whether the information that A is t or f (analogously, t * or f * ), or neither of them actually belongs to his information state.
DEFINITION 7.7 Given a 5N-valuation v, we say that a formula A is: A set X is said to be 5N-realizable if there is a 5N-valuation v which realizes every element of X .

DEFINITIONS 13
For all X , ϕ, • ϕ is a 0-depth consequence of X , X 0 ϕ, iff for every 5N-valuation v, v realizes ϕ whenever v realizes all the elements of X ; • X is 0-depth inconsistent, X 0 , iff it is not 5N-realizable.
Analogously to the explosivity of the proof-theoretic characterization above, this explosivity regards inconsistent sets of signed formulae.Recall our explanation above of why this is compatible with the non-explosivity regarding (unsigned) formulae.EXAMPLE 1 {T ¬(A ∨ B), T ¬C} 0 T ¬(A ∨ C) By inspection of the 5N-tables, it is easy to check that for any 5N-valuation v s.t.v realizes both T ¬(A ∨ B) and T ¬C, then v also realizes T ¬(A ∨ C).
Let us now show the adequacy of our informational 5-valued non-deterministic semantics with respect to the relation 0 .PROPOSITION 7.9 For all X and ϕ, X 0 ϕ iff X 0 ϕ.
PROOF.The soundness of the intelim rules can be immediately verified by inspection of the 5N-tables: every 5N-valuation which realizes the premise(s) of an intelim rule realizes also the conclusion of the rule.For example, if an agent holds the information that both A and B are t * , then he also holds the information that A ∧ B is t * , since the 5N-table for ∧ excludes the other imprecise values.Thereby, it follows by an elementary inductive argument that, if a 5N-valuation v realizes all the initial S-formulae of a 0-depth intelim tree T (i.e. an intelim sequence), then v realizes all the S-formulae occurring in T .But, of course, no 5N-valuation can realize two conjugate S-formulae simultaneously.Thus, if T is a closed intelim tree, no 5N-valuation can realize all the initial S-formulae of T .Therefore, on the one hand, if T is a 0-depth proof of ϕ from X , then for every 5N-valuation v, v realizes ϕ whenever v realizes all the elements of X .On the other hand, if T is a 0-depth refutation of X , then no 5N-valuation v realizes all the elements of X .
As for completeness, suppose that X 0 ϕ.Then X is not 0-depth refutable; otherwise, by definition of 0-depth intelim proof, it should hold that X 0 ϕ, contrary to our hypothesis.Next, consider the set Y = {ψ|X 0 ψ}.Since X is not 0-depth refutable, for no A, SA and SA are both in Y .Then, it is not difficult to verify that the function v defined as follows: Finally, observe that: (i) ψ ∈ Y for all ψ ∈ X and so, by definition of v, v realizes all ψ ∈ X ; (ii) by the hypothesis that X 0 ϕ, ϕ / ∈ Y and so v does not realize ϕ.Therefore, X 0 ϕ.
COROLLARY 7.10 For all X , X 0 iff X 0 .

k-depth logics
Examples 2 and 3 above are valid inferences in FDE that are not so in the 0-depth approximation.Again, the latter is simply the logic of deductive reasoning restricted to the use of actual information.
For those valid inferences that cannot be justified solely by the meaning of the connectives-i.e. by the 5N-tables-the incorporation of virtual information is required.This is information that is not even potentially contained in the current information state.Accordingly, the k-depth logics, k > 0, require the simulation of virtual extensions of the current information state.These extensions are formally defined via the notion of 5-refinement below, where we take the values in 5 as partially ordered by two relations: (i) a such that x a y (read 'x is less defined than, or equal to, y') iff DEFINITION 14 For all X , ϕ, and for all f ∈ F, As Definition 11, the above covers the case of k-depth inconsistency by assuming X f k as equivalent to X f k ϕ for all ϕ.Moreover, according with the above definitions, X f k ϕ iff by simulating either a pair of refinements in which the truth-value of some A (in the virtual space defined by f ) is respectively t or f, or a pair of refinements in in which the truth-value of some A is respectively t * or f * , the conclusion ϕ is realized by either of the members of the pair at depth Now, the next proposition follows from the fact that RE fde is sound and complete for full FDE [26]: PROPOSITION 7.13 For all X , ϕ, and all f ∈ F, The above given that RE fde is a subsystem our intelim method for unbounded k; i.e. a subsystem of the system constituted by the intelim rules together with an arbitrary number of applications of PB and PB * .Indeed, the elimination rules together with PB and PB * can be used to simulate any of the introduction rules.Conversely, the introduction rules together with PB and PB * can be used to simulate any of the elimination rules.This clearly implies that the direct-proof system constituted by the introduction rules together with PB and PB * -let us call it RI fde -is also complete for full FDE.Nevertheless, there are two reasons for using both introduction and elimination rules: (i) it allows for more natural and shorter proofs, although not essentially shorter because the corresponding simulation is polynomial; (ii) it reduces the number of applications of PB and PB * that, as stated above, is key to define the depth of an inference.In fact, regarding (i), it is not difficult to show that: 15PROPOSITION 7.14 ([72]).RE fde and RI fde can linearly simulate each other.Moreover, the simulation preserves the subformula property. of contradictory information is discarded); whereas the value i is information-transcendent when interpreted as timeless.Much as in the case of FDE, this situation motivates the need for stable imprecise values.Again, we address this question by shifting to signed formulae, where the signs express such imprecise values associated with two distinct bipartitions of the corresponding set of standard values.

Intelim deduction in LP and K 3
We use the same signed formulae used for FDE but, of course, interpret them differently.To state such a re-interpretation, we use x to refer to an agent and v to denote respectively a M b 3 -valuation or a M n 3 -valuation.For LP, we interpret: T A as "x holds that A is at least true" (expressing that v(A) ∈ {true, i}); F A as 'x holds that A is false only' (v(A) ∈ {false}); T * A as 'x holds that A is true only' (v(A) ∈ {true}); F * A as 'x holds that A is at least false' (v(A) ∈ {false, i}).As for K 3 , we interpret: T A as "x holds that A is true" (v(A) ∈ {true}); F A as "x holds that A is non-true" (v(A) ∈ {false, i}); T * A as 'x holds that A is non-false' (v(A) ∈ {true, i}); F * A as 'x holds that A is false' (v(A) ∈ {false}).Crucially, according to the respective informational interpretation of the values of LP and K 3 , whereas S-formulae of the form T A and F * A involve only information that does not require complete knowledge of the sources, S-formulae of the form T * A and F A involve information that does require such a complete knowledge.
Thereby, by making minor modifications to our proof system for FDE, we can obtain systems for LP and K 3 which naturally lead to defining analogous hierarchies of tractable depth-bounded approximations to the latter logics.Namely, those systems are obtained by enriching the rules of the FDE's intelim method with the intelim rules for implication displayed in Table 6, together with, respectively, the following structural rules which clearly do not involve introducing virtual information: In turn, hierarchies of depth-bounded approximations can be defined in terms of the maximum number of nested applications of PB and PB * , exactly as before.Although PB and PB * are not the unique structural rules in the case of LP and K 3 , they are the only rules involving the introduction of virtual information.Moreover, it is straightforward to adapt the proofs of Theorems 6.9 and 6.10 to show the tractability of the approximations at issue.Further, by making minor modifications to our semantical framework for the hierarchy of approximations to FDE, it can be shown that the hierarchies for the 3-valued logics also admit of a 5-valued non-deterministic semantics.

Related and future work
In this section, we brief ly recall some proof systems and a cluster of ideas closely related to our approach in this paper.However, in [72] it is shown that a crucial difference between the tableau methods to be recalled below and the intelim methods introduced above is that the latter have an exponential speed-up on the former.Namely, there we introduce a new class of examples which we prove to be hard for all tableau systems sharing the ∨/∧ rules with the classical one [see 7]-which include tableaux for the three many-valued logics treated in this paper-but easy for TABLE 6. Intelim rules for the implication of LP and K 3 their analogous intelim systems.Roughly, the reason of this is that while the tableau methods have operational branching rules that imply a good deal of redundant branchings in the corresponding tree, the intelim methods have only structural branching rules that reduce the amount of branching to a minimum by making all branches mutually exclusive.So, the intelim systems introduced in this paper are interesting independently of the depth-bounded approach.In turn, besides computational efficiency issues and within the context of the approach, another important difference between the tableau methods to be recalled below and the intelim methods introduced above is that since in the former cuts are eliminable, no approximations can be defined by controlling the application of the cut rules.Further, the natural deduction systems recalled below do not comply with a key idea underlying our depth-bounded approximations, according to which the meaning of a logical operator is fixed only in terms of actual information.For in the natural deduction systems, some of the (operational) 'discharge' rules make essential use of virtual information.

Tree methods, signs and natural deduction
The first tableaux for FDE are due to Dunn [42].He introduced a direct-proof tableau system based on a modification of Jeffrey's method of (classical) 'coupled trees'.Dunn's system rules are syntactically identical to the rules of the unsigned version of Smullyan's classical tableaux [71].Now, to the best of our knowledge, the first systems for FDE based on S-formulae-in which the signs stand for sets of values instead of single values-are two refutation tree systems due to D'Agostino [26].Soon after, Fitting [45] introduced a direct-proof tableau system for FDE which is based on the same use of S-formulae.In fact, in the same period, a general method to use signs as sets of truth-values suitable for any finite-valued propositional logic was provided by Hähnle [48].The key idea underlying that use is to increase the expressivity of the signs and, thus, significantly decrease the number of new branches per rule application.Later on, Avron [7] used four signs-interpreting them as intuitively corresponding to positive/negative information concerning truth/falsity-to provide tableaux for a diversity of logics; including FDE, all 3-valued logics, and some logics that do not have finite characteristic matrix.
The S-formulae underlying D'Agostino's systems are the same that we used for the systems introduced in this paper, with the caveat that in D'Agostino's those S-formulae are interpreted in terms of information that an agent possesses without any requirement of such an information being actual-in accordance with the interpretation of the standard 4 values.That is, T A is interpreted as 'v(A) = t or v(A) = b', F A as 'v(A) = f or v(A) = n', and so on.Thereby, the rules of the first system introduced by D'Agostino are formally analogous to the rules of the signed version of Smullyan's classical tableaux.Now, essentially the same tableau refutation system was reintroduced by Bloesch [18], who used it for both FDE and LP.Besides, a system differing only notationally and used for those paraconsistent logics as well as for K 3 , was given by Priest [64].Further, the same system was used for FDE by Fitting in [46].In turn, Fitting's first system for FDE in [45], although formulated using essentially the same S-formulae of the previous sections, is closer to Dunn's coupled trees method.Besides, in [45] only two signs, T and F , are used (explicitly) and a different convention on them is followed.Namely, a signed formula is an expression of the form T A or F A, where A is a formula and which is intuitively and respectively interpreted as 'A is at most true'-meaning that v(A) = t or v(A) = n-and 'A is at most false'-meaning that v(A) = f or v(A) = n.Since only the signs T and F are used, the corresponding rules are syntactically identical to the rules of the signed version of Smullyan's classical tableaux.
As for D'Agostino's second refutation system for FDE in [26], it was introduced as a more computationally efficient alternative to other proof systems.It was baptized RE fde since it is based on KE [see 26,37].The hallmark of KE-inherited by RE fde -is the reduction of the amount of branching to a minimum by making all branches mutually exclusive.Accordingly, RE fde has only two branching rules expressing a generalized rule of Bivalence (our PB and PB * above) and the rest of its rules have all a linear format (our elimination rules in Table 3).In fact, as Propositions 7.13 and 7.14, as well as comments thereof suggest, RE fde (equivalently, RI fde ) may well serve as the basis for defining depth-bounded approximations to FDE.The reason for using both introduction and elimination rules were stated above; however, as a refutation method, RE fde may be still preferred for applications in automated reasoning.Further, of course, KE-like systems serving as basis for approximations to LP and K 3 can be straightforwardly obtained by discarding the corresponding introduction rules.Now, in the literature there are also natural deduction systems that are closely related to the systems that we introduced in this paper.However, all those natural deduction systems are formulated in terms of unsigned formulae.A Gentzen-Prawitz style system for FDE, LP and K 3 was given by Priest in [65].Alternative natural deduction systems for FDE were given by Voishvillo [76], and Tamminga and Tanaka [74].Regarding LP, essentially the same system was reintroduced by Kooi and Tamminga [51].As for K 3 , essentially the same system was introduced again by Tamminga [73].Besides, alternative Fitch-style systems for the three logics were given by Roy [68].

Depth-bounded reasoning and efficient proof systems
Approximations to (fragments, full, or extensions of) CPL via tractable subsystems of increasing inferential power have been investigated since 1990s [e.g.19,25,39,53,69].A hierarchy of tractable depth-bounded approximations to CPL, based on an intelim method analogous to the systems presented in this paper, was widely studied in [e.g.30,31].Such a hierarchy, as well as ideas thereof, have been further explored and applied in diverse works.For example: in [26,37], proof-theoretic and computational advantages of the 'KE fragment' of the corresponding classical intelim method over tableaux and cut-free sequent calculus are studied; in [28], it is outlined how depth-bounded classical reasoning provides means to solve the Bar-Hillel-Carnap paradox, the enduring scandal of deduction, and the problem of logical omniscience; in [36], some applications of the depth-bounded approach to classical logic in formal argumentation theory are studied; semantic bases to defining similar hierarchies of approximations to epistemic logics are put forward in [54]; a preliminary step for an extension to classical first-order logic is given in [35]; a multiagent setting has been explored in [23,55]; an application for defining depth-bounded belief functions for reasoning with uncertainty was given in [13]; an application in the context of answer set programming was introduced in [12]; and a hierarchy based solely on KE has been put forward in [37,72].
The novelty of the contribution of this paper-together with its preliminary version [38]-is that it starts exploring the extension of the depth-bounded approach to propositional non-classical logics.Namely, we defined hierarchies of tractable depth-bounded approximations to three many-valued logics which are closely related to each other.These hierarchies are analogous to the hierarchy of approximations to CPL and may be similarly further explored and applied.To begin with, as mentioned above, it is shown in [72] that the proof-systems underlying the hierarchies are more efficient than their tableau systems counterparts.Besides, extending the methodology used in this paper to a variety of finite-valued logics in the spirit of [20,21,49,62] will be the topic of a subsequent paper.In fact [20] already paves the way for carrying out such an extension since it introduces a general method for extracting KE-style systems for any finite-valued logic.Those systems, as expected, are interesting in their own right because they outperform their tableau systems counterparts.Moreover, there is ongoing work on defining-both proof-theoretically and semantically-a hierarchy of depth-bounded approximations to intuitionistic propositional logic [72].The latter will presumably pave the way for defining similar hierarchies for a wide variety of logics characterized by Kripke-style semantics.
Further, the approximations to the simple paraconsistent logics FDE and LP defined above might serve-both conceptually and technically-as a starting point to extend the approach to more refined paraconsistent logics such as the logics of formal inconsistency (LFIs) [22] and the logics of evidence and truth (LETs) [67].For these extensions, the KE-style systems for mbC, mCi and C 1 given in [59][60][61] could be useful.What is more, hierarchies of depth-bounded approximations to modal and substructural logics could be worked out via KE-style systems already provided for them in the literature [5,33,34,43,47].In fact, depth-bounded approximations to normal modal logics would simultaneously constitute approximations to LFIs, given that the former can be rewritten as the latter [56].In parallel to these extensions, major results would be showing that the underlying KEstyle systems are indeed more efficient that their tableu systems counterparts [see 20], as such results are so far only conjectured.These major results seem attainable since, for instance, the examples given in [72], which are hard for all tableau systems sharing the ∨/∧ rules with the classical one but easy for their analogous intelim systems, should also work for the 'additive fragment' of substructural logics.
Thus, we envisage extensions of the depth-bounded approach covering an ample variety of nonclassical logics.As well-known, these logics find natural applications in Computer Science, AI, Philosophy and Cognitive Science.Depth-bounded approximations to non-classical logics could then be used to provide models of tractable reasoning and computation in, for instance, medical and juridical expert systems [see 4,61] as well as databases and knowledge-bases management [see 40].In particular, the approximations introduced in this paper might be used to provide tractable models of inconsistency search [see 52] and of the deductive behavior of evidence [see 67].Moreover, the systems underlying our hierarchies may be implemented in automated or interactive theorem provers-presumably, being more suitable the KE-style systems for the former [see 4, 58] and intelim-style systems for the latter [see 27].Further, either kind of systems can yield proof search heuristics based on the choice of the formulae on which the corresponding branching rules are applied [see 37,72].
, we can extract counterexamples for disjunctive syllogism out of the open branch of the tree as follows.The branch contains F B, so B can be either f or n.It also contains both T A and F * A, so A should be one of t or b, or A should be one of f or b.Then, A should be exactly b.We thus obtain two counterexamples for disjunctive syllogism.Namely, a 4-valuation v such that v(A) = b and v(B) = f, and another 4-valuation v such that v(A) = b and v(B) = n.

PROPOSITION 6. 4
The k-depth deducibility relations f k satisfy ref lexivity, monotonicity, but not cut.PROOF.By the definitions above.

ALGORITHM 4
Apply Intelim(A, G, λ ) 1: for all the G-modules M containing A do 2: let B be the top formula of M; 3: consider the set Z = {T C | C ∈ M and T ∈ λ(C)} ∪ {F C | C ∈ M and F ∈ λ(C)} ∪ {T * C | C ∈ M and T * ∈ λ(C)} ∪ {F * C | C ∈ M and F * ∈ λ(C)}; 4: is a 5N-valuation.Here we just outline a typical case.Supposev(A) = v(B) = ⊥.Then, F A ∨ B / ∈ Y .Otherwise, if F A ∨ B ∈ Y then,by definition of Y and by the corresponding elimination rule for ∨, F A and F B should also be in Y .Hence, by definition of v, v(A) = v(B) = f, against our assumption.Thus, by the 5N-table for ∨, v(A ∨ B) = f.Analogously,F * A ∨ B / ∈ Y .Otherwise, if F * A ∨ B ∈ Y then, F * A and F * B should also be in Y .So, by definition of v, v(A) = v(B) = f * ,against our assumption.Then, by the 5N-table for ∨, v(A ∨ B) = f * .On the other hand, T A ∨ B or T * A ∨ B, may or may not belong to Y , and so x = ⊥ or x = y for x, y ∈ {t, f, ⊥}; (ii) b such that x b y iff x = ⊥ or x = y for x, y ∈ {t * , f * , ⊥}.DEFINITION 7.11 Let v, w be 5N-valuations.Then, w is a 5-refinement of v, v 5 w, iff v(A) a w(A) or v(A) b w(A) for all A. Now, the following definitions mimics Definitions 11:

TABLE 3 .
Elimination rules for the standard FDE connectives FIGURE 1.An itelim sequence related to a generalized Principle of Bivalence: 6 Downloaded from https://academic.oup.com/logcom/article/34/5/815/7198103 by guest on 23 July 2024Tractable depth-bounded approximations to FDE and its satellites 845 k −1.14That use of a defined truth-value for A, which is not even potentially contained in the current information state, is what we call virtual information.It is easy to check that {T (A∨B)∧¬A}∪{SB} 0 T B∨(A∧¬A), and {T