We investigate the following “epistemic” extensions of (fragments of) first order logics: if phi is a formula, then box_i phi (i in I) is also a formula, where I is a fixed finite set. The intended meaning of box_i phi is “the i-th agent (ith participant of the model) knows phi”. The main result of the paper is Theorem 1: if L is such a fragment of first order logic whose consequence relation is weakly decidable, then the consequence relation of the epistemic extension of L remains weakly decidable, as well

Sági, Gábor

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Designing Decidable Logics of EpistemologyGa´bor Sa´gi ?Alfre´d Re´nyi Institute of Mathematics,Hungarian Academy of Sciences,Budapest P.O.B. 127, H-1364 Hungarysagi@renyi.huandDepartment of Algebra, BUTE,Budapest, Egry J. u. 1,H-1111 HungaryAbstract. We investigate the following “epistemic” extensions of (frag-ments of) first order logics: if ϕ is a formula, then iϕ is also a formula,where I is a fixed finite set. The intended meaning of iϕ is “the ithagent (ith participant of the model) knows ϕ”. The main result of thepaper is Theorem 1: if L is such a fragment of first order logic whoseconsequence relation is weakly decidable, then the consequence relationof the epistemic extension of L remains weakly decidable, as well.1 IntroductionDefinition 1. Let L be a fragment of first order logic and let I be any finite set.The set EL of elementary epistemic formulas over L is defined to be the smallestset satisfying the following two stipulations:• EL contains all formulas of L and• for any i ∈ I and ϕ ∈ EL we have iϕ ∈ FormE,I(L) (that is, EL is closedfor the operations i, for any i ∈ I).In addition, FormE,I(L) is defined to be the set of all Boolean combinations ofEL.The intended meaning of iϕ is “the ith agent (ith participant of the model)knows ϕ”, where ϕ is a formula that may also contain j operations.Logics of epistemology has been studied intensively, for related investigationswe refer to [1], [2] and the references therein.Our main aim is to provide semantics for the formulas FormE,I(L) in such away, that the consequence relation of our semantics remains decidable, wheneverthe consequence relation of L is decidable. For a quite expressive fragment offirst order logic with (weakly) decidable consequence relation, we refer to [3].To achieve our goal, we need further preparations. In Section 2 we are sum-ming up the preliminaries and definitions we need, in Section 3 we present theproofs.? Supported by Hungarian National Foundation for Scientific Research grant K83726and by the Ja´nos Bolyai Research Scholarship of the Hungarian Academy of Sciences.2 Ga´bor Sa´gi2 Technical IntroductionOur notation is standard, however, the following short list may help the reader.ThroughoutN denotes the set of natural numbers. Let L be a logic. Then FormLand |=L denote respectively, the set of formulas of L and the consequence relationof L (as usual, |=L also denotes the satisfaction relation of L). If A is a modelfor L then Th(A) denotes the theory of A which is defined to beTh(A) = {ϕ ∈ FormL : A |=L ϕ}.Throughout, by Go¨del numbering we mean an injective functiong : FormE,I(L)→ Nsuch that both g and g−1 is computable. It is well known, that such a g func-tion exists (in fact, there exists a primitive recursive such g with g−1 primitiverecursive, as well). We do not specify g further, because below we will use thefact only, that such a g exists (and we do not use the particular form, or furtherproperties of such a g).Definition 2. Let ϕ ∈ FormE,I(L). Then the tautological skeleton taut(ϕ) isdefined inductively as follows.if ϕ is a formula of L, then taut(ϕ) = ϕ;taut(¬ψ) = ¬taut(ψ);taut(ψ ∧ %) = taut(ψ) ∧ taut(%);taut(iψ) = Zn where Zn is the nth propositional variable and n is theGo¨del-number of iψ.In addition, if X ⊆ FormE,I(L), then taut(X) is defined to betaut(X) = {taut(ϕ) : ϕ ∈ X}.Remark 1. It is easy to see, that taut is a computable function, that is, thereexists an algorithm computing taut(ϕ) from ϕ. Moreover, ϕ is also computablefrom taut(ϕ), because each propositional variable Zn corresponds at most oneformula ψ ∈ FormE,I(L), namely, Zn corresponds to that ψ (if any) whose Go¨delnumber is n.Definition 3. Let X ⊆ FormE,I(L) and let i ∈ I be fixed. Then cli(X) isdefined to becli(X) = {ϕ,iϕ : taut(X) |=L taut(ϕ)}.Definition 4. By an 〈E , I〉-structure we mean a pair 〈A, f〉 where A is an L-structure and f : I → P(FormE,I(L)) is a function, such that for any i ∈ I• If ϕ ∈ FormL and A |=L ϕ then ϕ ∈ f(i);• cli(f(i)) = f(i).Designing Decidable Logics of Epistemology 3Definition 5. Let 〈A, f〉 be an 〈E , I〉-structure and let k be an evaluation overA. Then the satisfaction relation 〈A, f〉 |=E,L ϕ[k] is defined recursively on thecomplexity of ϕ ∈ FormE,I(L) as follows.• for an atomic (first order) formula 〈A, f〉 |=E,L ϕ[k] iff A |= ϕ[k];• 〈A, f〉 |=E,L ¬ϕ[k] iff 〈A, f〉 6|=E,L ϕ[k];• 〈A, f〉 |=E,L ϕ ∧ ψ[k] iff 〈A, f〉 |=E,L ϕ[k] and 〈A, f〉 |=E,L ψ[k];• 〈A, f〉 |=E,L ∃vnϕ[k] iff there exists an evaluation k′ such that 〈A, f〉 |=E,Lϕ[k′] and for any m 6= n we have k(m) = k′(m);• 〈A, f〉 |=E,L iϕ[k] iff ϕ ∈ f(i).Finally, 〈A, f〉 |=E,L ϕ iff 〈A, f〉 |=E,L ϕ[k] for any evaluation k over A.Using the notation of the previous definition, it is easy to see, that for afirst order formula ϕ ∈ Form(L) the assertion 〈A, f〉 |=E,L ϕ is equivalent withA |= ϕ.Definition 6. Let Σ ⊆ FormE,I(L) and let ϕ ∈ FormE,I(L). Then Σ |=E,L ϕiff for any 〈E , I〉-structure 〈A, f〉 the following holds:if for all ψ ∈ Σ we have 〈A, f〉 |=E,L ψ then 〈A, f〉 |=E,L ϕ.We say, that the consequence relation of a logic L is weakly decidable iff thereexists an algorithm pi whose input is a finite set Σ ⊆ Form(L) and a formulaϕ ∈ Form(L) and pi always stops after a finite number of steps and provides acorrect answer for the question “Σ?|=L ϕ”.Our main result is as follows.Theorem 1. Suppose the consequence relation |=L of L is weakly decidable.Then |=E,L is also weakly decidable.The rest of this paper is devoted to prove this theorem. To do so, we needfurther preparations.3 ProofsLemma 1. Assume, that the consequence relation |=L of L is decidable. LetX ⊆ FormE,I(L) be a decidable subset of FormE,I(L) and let i ∈ I be fixed.Then cli(X) is a decidable subset of FormE,I(L).Proof. Clearly, if X is decidable, then so is taut(X) (because by Remark 1,taut−1 is computable and X is assumed to be decidable). Combining this withthe assumption, that the consequence relation |=L of L is decidable, the state-ment follows immediately.Now we will define a relation Ded and show, that this relation is decidable.Finally, we show, that Ded and the consequence relation |=E,L coincide, thusthe algorithm deciding Ded also witnesses, that the consequence relation |=E,Lis weakly decidable.4 Ga´bor Sa´giDefinition 7. Let X ⊆ FormE,I(L) and let ϕ ∈ FormE,I(L). ThenDed0(X) = {ψ ∈ FormL : X ∩ FormL |=L ψ}.Now suppose, that Dedn has already been defined for some n ∈ N. Then Dedn+1(X)is defined by recursion as follows.Dedn(X) ⊆ Dedn+1(X);if ϕ = iψ then ϕ ∈ Dedn+1(X) iff ψ ∈ cli(Dedn(X));if ϕ = ¬ψ then ϕ ∈ Dedn+1(X) iff ψ 6∈ Dedn+1(X);if ϕ = ψ ∧ % then ϕ ∈ Dedn+1(X) iff ψ ∈ Dedn+1(X) and % ∈ Dedn+1(X).Finally, letDed(X) =⋃n∈NDedn(X).Theorem 2. Assume, that the consequence relation |=L of L is weakly decid-able. Let X ⊆ FormE,I(L) be a finite subset of FormE,I(L). Then Ded(X) is adecidable subset of FormE,I(L).Proof. A simple inspection of Definition 7 together with Lemma 1 shows, thatDedn(X) is decidable for all n ∈ N, in addition, (the Go¨del number of) analgorithm deciding Dedn(X) may be computed from n. Moreover, ϕ ∈ Ded(X)iff ϕ ∈ Dedn(X), where n is the number of all occurrences of -operations in ϕ.It follows, that Ded(X) is decidable, as desired.Now we are ready to prove Theorem 1. We will split the proof into two parts.Theorem 3. Assume, that the consequence relation |=L of L is weakly decidable.Let X ⊆ FormE,I(L) be a finite subset of FormE,I(L) and let ϕ ∈ FormE,I(L).Thenϕ ∈ Ded(X) implies X |=E,L ϕ.Proof. Suppose ϕ ∈ Ded(X). Then there exists n ∈ N such that ϕ ∈ Dedn(X).So it is enough to show(∗) ϕ ∈ Dedn(X) implies X |=E,L ϕ and〈A, f〉 |= X impies (∀i ∈ I)Dedn(X) ⊆ f(i).We apply induction on n. If n = 0, then (∗) holds, obviously. Now assume,that (∗) holds for 0, ..., n; we shall show, that it remains true for n + 1. To doso, let ϕ ∈ Dedn+1(X).If ϕ ∈ FormL then, in fact, ϕ ∈ Ded0(X), hence (∗) follows for ϕ from then = 0 case.If ϕ = iψ, then, according to Definition 7, we have ψ ∈ cli(Dedn(X)). Byinduction, we have X |=E,L Dedn(X). Assume 〈A, f〉 |=E,L X. It follows, thatDesigning Decidable Logics of Epistemology 5〈A, f〉 |=E,L Dedn(X). So, again by induction, we have Dedn(X) ⊆ f(i). Com-bining this with the second stipulation of Definition 4, we obtain 〈A, f〉 |=E,L ϕ.This shows, that (∗) remains true for ϕ.If ϕ = ¬ψ or ϕ = ψ ∧ % then (∗) for ϕ may be derived from Definition 5 andDefinition 7 in the usual way.This completes the induction, and we are done.Now we prove the converse of Theorem 3.Theorem 4. Assume, that the consequence relation |=L of L is weakly decidable.Let X ⊆ FormE,I(L) be a finite subset of FormE,I(L) and let ϕ ∈ FormE,I(L).ThenX |=E,L ϕ implies ϕ ∈ Ded(X).Proof. Assume, ϕ 6∈ Ded(X); it is enough to show, that X 6|=E,L ϕ. Do doso, we shall construct an 〈E , I〉-structure 〈A, f〉 such that 〈A, f〉 |=E,L X but〈A, f〉 6|=E,L ϕ.First we show, that X ∩ FormL is consistent (in the sense of usual firstorder logic). Indeed, if X ∩ FormL would be inconsistent, then it would follow,that Ded0 = FormL, consequently, we would have Ded(X) = FormE,I(L);particularly we would have ϕ ∈ Ded(X). Thus, there exists a first order structureA such that A |=L X ∩ FormL.Now, for any i ∈ I, let f(i) = cli(ThL(A)). Clearly, 〈A, f〉 is an 〈E , I〉-structure. Observe, that for any ψ ∈ FormE,I(L) we have 〈A, f〉 |=E,L ψ iffψ ∈ Ded(X). Particularly, 〈A, f〉 |=E,L X and 〈A, f〉 6|=E,L ϕ, as desired.Now we are ready to prove the main result of the paper which is a moredetailed version of Theorem 1.Theorem 5. Suppose the consequence relation |=L of L is weakly decidable. LetX ⊆ FormE,I(L) be a finite subset of FormE,I(L) and let ϕ ∈ FormE,I(L).Then we have(1) X |=E,L ϕ iff ϕ ∈ Ded(X);(2) |=E,L is weakly decidable, too.Proof. Combining Theorems 3 and 4, (1) follows immediately. To prove (2) wenote, that according to (1), for any finite X ⊆ FormE,I(L) and ϕ ∈ FormE,I(L)we have X |=E,L ϕ iff ϕ ∈ Ded(X). But, by Theorem 2 Ded(X) is decidable forany decidable X (in addition, an algorithm deciding Ded(X) may be effectivelyconstructed from an algorithm deciding X).References1. Ertola B., Rodolfo, C. On some extensions of intuitionistic logic, Bull. Sect. Logic,41 (2012), no. 1-2, 17-22.2. Kooi, B., Pacuit, E., Logics of rational interaction. Dynamic formal epistemology,5-32, Synth. Libr., 351, Springer, Dordrecht, 2011.3. Sa´gi, G., A note on algebras of substitutions, Studia Logica, Vol. 72, No. 2, 265-284,(2002).

Designing Decidable Logics of Epistemology

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