Deciding Intuitionistic Propositional Logicvia Translation into Classical Logic

Daniel S. Korn1 Christoph Kreitz2

1 FG Intellektik, FB Informatik, TH-DarmstadtAlexanderstraße 10, D–64238 Darmstadt

e-mail: korn@informatik.th-darmstadt.de, phone: +49-6151-16 66342 Department of Computer Science

Cornell University, Ithaca, NY 14853-7501e-mail: kreitz@cs.cornell.edu, phone: +1-607-255 1068

W. McCune, ed. 14th International Conference on Automated Deduction (CADE-14), LNAI 1249,pp. 131–145, c©Springer Verlag, 1997.

Abstract. We present a technique that efficiently translates proposi-tional intuitionistic formulas into propositional classical formulas. Thistechnique allows the use of arbitrary classical theorem provers for de-ciding the intuitionistic validity of a given propositional formula. Thetranslation is based on the constructive description of a finite counter-model for any intuitionistic non-theorem. This enables us to replace uni-versal quantification over all accessible worlds by a conjunction over theconstructed finite set of these worlds within the encoding of a refutingKripke-frame. This way, no additional theory handling by the theoremprover is required.

1 Introduction

Following the ideas of Ohlbach in [10] deduction systems for classical logic canbe viewed as “processors” for theorem proving environments. In order to obtaina theorem prover for a given non-classical logic (which is then regarded as a“user-friendly surface language”) we simply need to write a “compiler” (i.e.,translation procedure) for that particular logic into classical logic. Once we haveproved soundness and completeness of this translation, we are free to use thevery best of the existing classical theorem provers and let it do the work for us.A basic as well as natural approach to this end is given, for instance, by Moore [9]or van Benthem [17]. This approach essentially encodes the definition of validityw.r.t. the Kripke-semantics (see [3, 16]) of a given logic within the language ofclassical first-order logic. For example the modal formula 2a would be translatedinto A ⇒ ∀v.(w0Rv ⇒ a(v)), where R denotes the accessibility relation betweenpossible worlds, a(v) denotes a being forced at v, w0 denotes some arbitrary“root”-world and A is a conjunction of axiom formulas encoding the logic-specificproperties of R (e.g. reflexivity, transitivity, symmetry, etc.).This technique has two major disadvantages: On the one hand the classical the-orem prover has to reason about the semantics of the given logic over and overagain at run-time. On the other hand decidability is lost even for the proposi-tional fragment of the given logic.There are several approaches to recover from the first problem. The so called“functional” translation proposed in [10], for instance, allows the use of theory

unification and thus to handle the theory knowledge about the given logic on ameta-level rather than on the object level. However, translation methods thatexplicitly deal with the second problem do not exist.In this paper we present a translation procedure which includes both features:preserving decidability and liberating the theorem prover from dealing with the-ory knowledge. To this end, our translation mechanism uses more sophisticatedknowledge about intuitionistic logic (which we will denote as “J” in the follow-ing). Our solution relies on a constructive view of the finite-model property forthe propositional fragment of J (which we will denote as “Jp” in the following).Informally, this means that for any non-Jp-theorem we are able to constructa countermodel with a finite set of possible worlds. In the above example wewould therefore become able to replace ∀v.w0Rv ⇒ a(v) with

∧v∈R(w0)

a(v),where R(w0) denotes the finite set of all possible worlds in the constructedmodel which are accessible from w0.In the following we begin by describing these latter ideas in more detail. Wethen present the translation itself. Finally, some remarks on the complexity ofthe translation as well as some considerations about related and future work willconclude our expositions.

2 Constructing finite potential countermodels

In this section we present the principal semantical ideas on which our transla-tion mechanism is based. We assume the reader to be familiar with the basicconcepts and terminology of classical and intuitionistic logic. Since, however,our definition of intuitionistic interpretation, forcing, model and validity resultsfrom a combination of the respective definitions in [3] and [16] and thus slightlydiffers from them we will repeat it in the following:

Definition 1 Intuitionistic interpretationLet W be a non-empty set, R ⊆ W × W a transitive and reflexive relation(accessibility), and V an evaluation function mapping elements of W to sets ofatomic formulas.Then 〈W,R,V〉 is an intuitionistic interpretation or simply a Jp-interpretation.2

Given 〈W,R,V〉 and a w ∈ W we denote by w∗ an arbitrary v ∈ W with wRv.Thus by saying “P holds for all w∗” we mean “P holds for all v with wRv” and“there is a w∗ such that P” means “there is a v with wRv such that P”.

Definition 2 Intuitionistic forcing, model, validityLet Ij = 〈W,R,V〉 be a Jp-interpretation, w ∈ W.A formula A is called forced (otherwise unforced) at w — denoted as w A (andw 6 A respectively) — iff one of the following conditions holds:

1. A is atomic, and A ∈ V(w∗) for any w∗

2. A = A1 ∧ A2, w A1, and w A2

3. A = A1 ∨ A2, and w A1 or w A2

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4. A = ¬A1, and w∗ 6 A1 for any w∗

5. A = A1 ⇒ A2, and w∗ A2 if w∗ A1 for any w∗

Ij is called an intuitionistic model or simply a Jp-model (otherwise Jp-countermodel) for A — denoted as Ij |=j A (Ij 6|=j A respectively) — iff w Afor any w ∈ W. A is called intuitionistically valid or simply Jp-valid (otherwiseJp-invalid) — denoted as |=j A ( 6|=j A respectively) — iff every Jp-interpretationis a Jp-model for it. 2

The elements of W can be regarded as knowledge stages since they are meantto represent collections of formulas for which evidence is known at a particularstage in time. Consequently accessibility is to be seen in a temporal sense: for agiven w any w∗ is supposed to denote a future stage of knowledge. The followinglemma which has been proved in [3] ensures that any evidence for some formulawhich is known at a given stage will remain known at any future stage:

Lemma 1 Heredity of forcingLet Ij = 〈W,R,V〉 be a Jp-interpretation and A be a propositional formula.Then for any w ∈ W: w A ⇒ w∗ A 2

Throughout this paper we will refer to this property as the “heredity condition”or simply as “heredity”.In terms of the above definitions the basic principle of our translation is toconstruct a decidable classical formula which describes the negation of a potentialJp-countermodel for the given input formula. Classical validity of this descriptionwill then ensure the impossibility of a Jp-countermodel for the given formula andhence its Jp-validity. To achieve decidability we must ensure to always construct afinite potential countermodel (i.e. a countermodel with finitely many knowledgestages) for the given formula. In the following we will focus on this issue.According to the definition of Jp-models the construction of every Jp-counter-model for a given formula A must begin with a single root knowledge stagew0 where A itself is not forced. We will then successively have to refine thiscountermodel depending on the principal operator of the formulas forced orunforced at some knowledge stage of the countermodel. If, for instance, A = a∨bthen, since w0 6 A, we need to add the facts w0 6 a as well as w0 6 b.For atomic, negated, and implicative formulas the refinement of our counter-model may include the addition of further knowledge stages. If, for instance,A = a ⇒ b then according to the forcing conditions for implicative formulasthere must be a w∗0 , say w1, with w1 a but w1 6 b. We shall say w1 refutesa⇒ b. Note that we have to assume w1 6= w0 since in general we cannot justifyw0 a (as opposed to w0 6 b which must be the case since otherwise w0 a⇒ b).Hence, in general we must assume a Jp-countermodel for a⇒ b to have at leasttwo knowledge stages as shown in fig. 1. If, however, we could for some reasonjustify w0 a then w0 would already refute a⇒ b by the reflexivity of accessibil-ity. In this case the refinement of our countermodel would only consist of addingw0 6 b to the previous properties. This observation is crucial to ensure finitenessof potential Jp-countermodels. The example in fig. 2 shows a countermodel whichwould have to be infinite if we did not use this observation.

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±°²m¨

§¦1 w0a 6⇒ b

±°²m¨

§¦1 w1a, 6bHHY

Fig. 1: countermodel for a⇒ b

±°²m¨

§¦1 w0I1 ∧ I2 6⇒ c

±°²m¨

§¦1 w1I1, I2, 6c, a 6⇒ b

HHY 6

±°²m¨

§¦1 w2I1,I2, a, 6b, d 6⇒ c

HHY 6

±°²m¨

§¦1 w3I1, I2, a, d, 6c, a 6⇒ b

HHY 6

±°²m¨

§¦1 w4I1,I2, a, d, 6b, d 6⇒ c

HHY 6

· · · HHY

Fig. 2: infinite countermodel for((a⇒ b) ⇒ c)︸ ︷︷ ︸

I1

∧ ((d⇒ c) ⇒ b)︸ ︷︷ ︸I2

⇒ c

As in the previous example we refute the main implication by adding a refutingknowledge stage w1 accessible from w0 with w1 I1 ∧ I2 (and thus w1 I1 aswell as w1 I2) but w1 6 c. From w1 I1, i.e. w1 (a ⇒ b) ⇒ c and w1 6 c weobtain the refinement w1 6 a⇒ b which is indicated by the arrow at w1 in fig. 2.So we need to refine our countermodel by adding another knowledge stage w2

refuting a⇒ b, i.e. w2 a but w2 6 b. However, since by the heredity conditionw2 I2, i.e. w2 (d ⇒ c) ⇒ b we obtain w2 6 d ⇒ c from w2 6 b. Therefore wemust add a fourth knowledge stage w3 refuting d ⇒ c, i.e. w3 d but w3 6 c.Again w3 I1 by heredity so w3 6 a ⇒ b for the same reason as in w1. Hencewe need another stage w4 which again refutes a ⇒ b. Proceeding accordinglywe continuously have to add knowledge stages that alternately refute the twoimplications thus yielding an infinite countermodel.If, however, we take into account that w3 a (which follows from w2 a by theheredity condition) then we only have to refine our countermodel by adding theassumption w3 6 b which must hold since otherwise w3 a ⇒ b. Hence we donot need to add w4 to our countermodel anymore. The same reflection appliesto d⇒ c. Thus the construction of our countermodel actually terminates at w3.¿From these reflections we obtain, that we do not need to consider further acces-sible knowledge stages for refuting a given implication I whenever the currentknowledge stage is accessible from one that already refutes I. Thus, our con-struction technique of adding knowledge stages in this manner while steppingbackwards through implications will always lead to a finite countermodel.For negated formulas we can achieve an even better behavior. Constructing acountermodel for such a formula, say ¬A, amounts to finding a knowledge stagew0 where ¬A is not forced. According to the intuitionistic semantics of thenegation this means that A must be forced at least at one knowledge stagew1 accessible from w0. As in the implicative case we must assume w1 6= w0

unless we can somehow justify w0 A. In particular, if we again continue ourconstruction process until we reach some w∗1 with w∗1 6 ¬A then we can concludew∗1 A by heredity and thus we do not need to add any further knowledge stagerefuting ¬A. Hence, stepping backwards through negations in this manner willalso lead to finite countermodels. As a matter of fact, once we have refuted a

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negated formula we do not need to add any further knowledge stage to ourcountermodel at all. To see why, we introduce the notion of F -maximality :

Definition 3 Maximal knowledge stageLet Ij = 〈W,R,V〉 be an intuitionistic interpretation and F be a propositionalformula. A knowledge stage w ∈ W is said to be F -maximal iff for any subfor-mula A of F the following proposition is true:

(∃w∗.w∗ A) ⇒ w A 2

In other words, w is maximal w.r.t. F if for every subformula A of F eitherevidence for A is known at w or it is impossible to construct evidence for A atall. The following theorem shows that for any propositional formula F and anyknowledge stage an accessible F -maximal knowledge stage must always exist.

Theorem 1 Existence of maximal knowledge stageLet Ij = 〈W,R,V〉 be a Jp-interpretation, w ∈ W and F be a propositionalformula. Then there exists an F -maximal w∗.

Proof: Let Φ = {A1, . . . , An} be the (finite) set of all subformulas of F . If thereis no w∗ forcing an Ai which is unforced at w then w is F -maximal. Otherwiseconsider some w∗ where Ai is forced. Then Ai is also forced at any w∗∗ bylemma 1 so we only need to look for some Aj ∈ Φ \ {Ai} to become forced atsome w∗∗. If there is no such w∗∗ then w∗ is F -maximal. Otherwise we proceedin the same manner for w∗∗ and Φ \ {Ai, Aj}. Obviously this process terminatesafter at most k steps at some F -maximal w(∗k) for k ∈ {0, . . . , n}. By transitivityof R, w(∗k) is also a w∗. 2

The chief feature of such an F -maximal knowledge stage is that classical andintuitionistic negation become semantically equivalent for all subformulas of F ,since every subformula which is not forced at this stage will neither becomeforced at any accessible knowledge stage. Hence, ¬A is forced at such a stage ifand only if A is not, for any subformula A of F . So if F is the input formula ofour translation and we have added a knowledge stage w1 to our countermodelin order to refute some negated subformula ¬A of F at some knowledge stagew0 then we know by theorem 1 that there is an F -maximal w∗1 with w∗1 A byheredity. Hence we may refine our countermodel by adding w∗1 instead of w1. Butthen we do not need to add any further knowledge stage to our countermodelanymore since we can be sure that every subformula of F which we might assumeto become forced at some knowledge stage accessible from w∗1 is already forced atw∗1 . Thus our technique for constructing countermodels will only add propertiesto the existing knowledge stages without adding further stages as soon as wehave refuted a negated formula.Eventually, refuting an atomic formula a at some knowledge stage w0 amountsto finding an accessible knowledge stage w1 with w1 6 a. But then also w0 6 aby heredity. Thus it suffices to refine our countermodel by adding w0 6 a so theconstruction technique will not have to add further stages in this case either.These insights provide a basic scheme for the construction of a potential Jp-countermodel for a given formula F :

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1. start with a knowledge stage w0 where F is not forced.2. for any negated or implicative subformula F ′ which is assumed to be un-

forced at some knowledge stage wi and where wi is neither F -maximal noraccessible from some wj already refuting F ′ add an accessible stage wi+1

with the necessary properties to refute F ′ and all properties following fromthe heredity condition. If F ′ is negated then assume wi+1 to be F -maximal.

3. for any other subformula which is possibly not forced at some wi or anysubformula that is possibly forced at wi add the appropriate forcing condi-tions for its immediate subformulas to wi (and possibly all accessible stages)according to the definition of intuitionistic forcing.

4. repeat the two previous steps until no new knowledge stages can be addedto the current countermodel and no new properties can be added to any ofthe knowledge stages in the current countermodel.

In the following sections we will elaborate the technical details for this.

3 The Translation

In this section we describe how we use the construction technique presented inthe previous section in order to generate a classical translation ΨS(F ) for anygiven formula F such that ΨS(F ) is classically valid iff F is Jp-valid. Essentiallythis is achieved by generating a suitable representation for the negation of apossible Jp-countermodel within classical logic. We use unary function symbolsto denote mappings from a given knowledge stage to an accessible one (cf. [10]).As shown above the existence of accessible knowledge stages within a potentialJp-countermodel depends only on negated and implicative subformulas not beingforced somewhere. We will thus associate our function symbols with such sub-formulas. To describe this concept more precisely we introduce some formalism:

Definition 4 Tree path, indexed subformulaLet F be the formula under consideration and p denote a word from {1, 2}∗.Then Fp is the subformula of F that will be reached by traversing the formationtree of F according to p where a “1” means “go left” and a “2” means “go right”.A negation is defined to have a left subformula only. We shall call such a wordp a tree path and Fp an indexed subformula.By op(Fp) we denote the principal operator of Fp and by pol(Fp) we mean thepolarity of Fp where F itself has polarity 0 and the polarity flips between 0 and1 whenever going to the left subformula of an implication or a negation. Byatomic(Fp) we denote that Fp is atomic. Given a tree path p such that Fp doesnot denote a subformula of F (e.g. (¬A)2) then we assume op(Fp) = pol(Fp) tobe ∅ and atomic(Fp) to be false. 2

Fig. 3 shows the formula of fig. 2 where all connectives are marked with the treepath of the respective subformula. In this figure we have, e.g., op(F1) = “ ∧ ”,pol(F11) = 1 but pol(F111) = 0 as well as atomic(F2).The function symbols which represent transitions between knowledge stages ofa potential countermodel will come in the form “wp”, where p denotes the tree

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F :⇒

F1 : ∧ F2 : c

F11 :⇒ F12 :⇒

F111 :⇒ F112 : c F121 :⇒ F122 : b

F1111 : a F1112 : b F1211 : d F1212 : c

PPPi ³³³1PPPi ³³³1

PPPi 6 6 ³³³1

6 ³³³1 6 ³³³1

Fig. 3: formula tree for ((a⇒ b) ⇒ c) ∧ ((d⇒ c) ⇒ b) ⇒ c

path of the subformula Fp which wp is meant to be associated with. Thus, we willhave to deal with the set W(F ) = {wp|Fp is special}, where we call a subformulaFp special , iff op(Fp) ∈ {⇒,¬} and pol(Fp) = 0. In the above example we obtainW(F ) = {w, w111, w121}. The respective subformulas appear boxed in fig. 3.Recall that the elements wp of W(F ) denote functions which map a given knowl-edge stage w to an accessible one w∗ refuting Fp. For fig. 2 and 3 we have w111

mapping w1 to w2 or w121 mapping w2 to w3. Thus the set knowledge stageswithin a potential Jp-countermodel for the given formula F will be denoted byappropriate concatenations of the symbols in W(F ) applied to some constant w0

representing the root knowledge stage where F itself is not forced. In fig. 2, forinstance, w0 would denote w0 and w would denote a function mapping w0 tow1 (thus denoted by w(w0)). From there, we can possibly advance to w111(w(w0))denoting w2 or to w121(w(w0)), denoting another knowledge stage which couldexist by refuting implication F121. Eventually from w2, i.e. w111(w(w0)) we couldgo to w121(w111(w(w0))) denoting w3. Conversely, from w121(w(w0)) we can go tow111(w121(w(w0))). Any other possible application of function symbols to one ofthese six terms would denote a multiple refutation of the same implication whichis unnecessary according to our reflections in the previous section. Hence thesesix terms constitute the desired set of all necessary knowledge stages in a pos-sible countermodel for F , which we call the accessibility set for w0 (denoted asRF (w0)). Fig. 4 illustrates the construction process we have just described.In the following w0 and any application of a concatenation of function symbolsfrom W(F ) to w0 will be called knowledge stage terms or simply k-terms.The example in fig. 4 yields the basic principle of constructing the accessibilityset for a given k-term t: first of all, set RF (t) = {t}. Then for any wp occurring int, look for a special subformula Fp′ of Fp such that wp′ does not already occur int and add the term wp′(t) to RF (t). This latter step will be called an accessibilityextension of t′ (at p over F ). Eventually, RF (t) is defined as the fixpoint of

Fig. 4: construction of accessibility set

extensiontermvia w via w111 via w121

w0 w(w0) — —w(w0) — w111(w(w0)) w121(w(w0))w111(w(w0)) — — w121(w111(w(w0)))w121(w(w0)) — w111(w121(w(w0))) —w111(w121(w(w0))) — — —w121(w111(w(w0))) — — —

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recursively constructing every accessibility extension for all its terms. However,terms of the form wp(t) with op(Fp) = “¬” do not have to be extended, sincethey denote F -maximal knowledge stages. In the following we shall call thesek-terms F -maximal, as well. The formal definition of RF reads as follows:

Definition 5 Accessibility extension, accessibility setLet F be an arbitrary propositional formula, p be a tree path for F and t be aknowledge stage term, where wεt denotes that w occurs somewhere within t.The accessibility extension for t at p over F (denoted as E(Fp, t)) is definedinductively as follows:

1. E(Fp, t) = {wp(t)}, iff Fp is special and wp 6 εt2. E(Fp, t) = E(Fp1, t) ∪ E(Fp2, t), iff either op(Fp) ∈ {¬,⇒} and pol(Fp) = 1

or op(Fp) ∈ { ∧ , ∨ }3. E(Fp, t) = ∅, otherwise

The accessibility set for t over F (denoted as RF (t)) is defined as the smallestset satisfying

1. t ∈ RF (t)2. if t is a constant symbol then E(F, t) ⊆ RF (t)3. if wp(t′) ∈ RF (t) and op(Fp) = “⇒” then for any wqεwp(t′):

E(Fq1, wp(t′)) ⊆ RF (t) and E(Fq2, wp(t′)) ⊆ RF (t) 2

Knowing how to construct the accessibility set we can now solve our initial prob-lem, i.e. to translate a given propositional formula F into another propositionalformula ΨS(F ) which will be classically valid if and only if F is Jp-valid. Theessential part of this translation is given by a morphism ΨF which encodes withinclassical logic whether a subformula Fp is forced at a knowledge stage denotedby a k-term t. The result of ΨF(t, Fp) will thus be a classical formula describingthe negation of a partial Jp-countermodel for Fp if pol(Fp) = 0 or a partialJp-model for Fp if pol(Fp) = 1 starting from the knowledge stage denoted by t.In the example of fig. 2 and fig. 3 we have F11 = (a ⇒ b) ⇒ c (wherepol(F11) = 1) and hence translate it into ΨF(t, F11) =

∧t′∈RF (t)(ΨF(t′, F111) ⇒

ΨF(t′, F112)), meaning that if F11 is forced at t then F112 is forced at every t′

accessible from t where F111 is forced. Conversely, F111 = a ⇒ b has polarity0 and hence ΨF(t′, F111) means there is some knowledge stage w111(t′) accessi-ble from t′ refuting F111. So basically we should expect ΨF(t′, F111) to reduceto ΨF(w111(t′), a) ⇒ ΨF(w111(t′), b). But this is only the case if w111 6 εt′ (e.g.t′ = w(w0)), since otherwise t′ denotes a knowledge stage accessible from onewhich already refutes F111 and hence by the heredity condition a is alreadyforced at t′ (e.g. t′ = w121(w111(w(w0)))). In this case the translation would eval-uate only to ΨF(t′, b) thus adding the fact that b is not forced at t′.Likewise, for an F -maximal k-term t the translation will switch to a mode whereno extensions of t are generated anymore. This translation will come in the formΨF-max(t, Fp), essentially encoding that Fp is classically satisfied by the set of allsubformulas of F that are forced at the knowledge stage denoted by t.

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ΨS(F )

= ΨF(w0, F )

= ΨF(w0, F1) ∨ ΨF(w0, F2)

= a(w0) ∨ ¬ΨF(w2(w0), F21)

= a(w0) ∨ ¬ΨF-max(w2(w0), F21)

= a(w0) ∨ ¬a(w2(w0))

ΨS(G)

= ΨF(w0, G)

= ¬ΨF(w(w0), G1)

= ¬ΨF-max(w(w0), G1)

= ¬¬ΨF-max(w(w0), G11)

= ¬¬(ΨF-max(w(w0), G111) ∨ ΨF-max(w(w0), G112))

= ¬¬(a(w(w0)) ∨ ¬ΨF-max(w(w0), G1121))

= ¬¬(a(w(w0)) ∨ ¬a(w(w0)))

Fig. 5: translation examples

Definition 6 Formula morphismLet F be an arbitrary propositional formula, p be a tree path for F and t bean arbitrary knowledge stage term. Then the formula morphisms ΨF(t, Fp) andΨF-max(t, Fp) are defined as follows:

ΨF(t, Fp) =ΨF-max(t, Fp), iff t is F -maximal. Otherwise:

Fp(t), iff atomic(Fp) and pol(Fp) = 0∧t′∈RF (t) Fp(t′), iff atomic(Fp) and pol(Fp) = 1

ΨF(t, Fp1)op(Fp)ΨF(t, Fp2), iff op(Fp) ∈ { ∨ , ∧ }ΨF(wp(t), Fp1) ⇒ ΨF(wp(t), Fp2), iff op(Fp) = “⇒”, pol(Fp) = 0, wp 6 εt

ΨF(t, Fp2), iff op(Fp) = “⇒”, pol(Fp) = 0, wpεt∧t′∈RF (t)(ΨF(t′, Fp1) ⇒ ΨF(t′, Fp2)), iff op(Fp) = “⇒” and pol(Fp) = 1

¬ΨF(wp(t), Fp1), iff op(Fp) = “¬”, pol(Fp) = 0∧t′∈RF (t)(¬ΨF(t′, Fp1)), iff op(Fp) = “¬” and pol(Fp) = 1

ΨF-max(t, Fp) =Fp(t), iff atomic(Fp)

ΨF-max(t, Fp1)op(Fp)ΨF-max(t, Fp2), iff op(Fp) ∈ { ∨ , ∧ ,⇒}¬ΨF-max(t, Fp1), iff op(Fp) = “¬” 2

The initial call of ΨF is defined through the following specification morphism:

Definition 7 Specification morphismLet F be an arbitrary propositional formula and w0 be a constant symbol. Thenthe specification morphism ΨS(F ) is defined as ΨF(w0, F ). 2

Fig. 5 demonstrates the translation process for the two formulas F = a ∨ ¬aand G = ¬¬(a ∨ ¬a). Obviously ΨS(F ) is classically invalid whereas ΨS(G)is classically valid. This amounts to the well-known Jp-invalidity of F and theJp-validity of G respectively.For ΨS the following theorem holds trivially:

Theorem 2 Decidability of codomain of ΨS

If F is a propositional formula of then ΨS(F ) is classically decidable.

Proof: Obviously every predicate symbol in ΨS(F ) is ground-instantiated andno quantifiers occur in ΨS(F ). Therefore ΨS(F ) is a propositional formula andhence classically decidable. 2

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In order to complete our logic morphism we define bidirectional interpretationmorphisms ΨJ, Ψ−1

J which map a Jp-countermodel for a given formula to a clas-sical countermodel for its translation and vice versa. Above we have describedthe intended meaning of the syntactical elements of the target language. Theinterpretation morphism amounts to formalizing these notions. We start withthe unary function symbols wp which are supposed to denote mappings from agiven knowledge stage w to some w∗ which refutes the associated subformulaFp. The following definition first introduces the notion of knowledge extension ofa knowledge stage w indexed by a tree path p which is the set of all w∗ refutingFp. The respective function symbol wp will be interpreted as an extension succes-sor function which maps w to an arbitrary element of its knowledge extensionindexed by p if it is non-empty or to an arbitrary F -maximal w∗ otherwise:

Definition 8 Indexed knowledge extension, extension successorLet Ij = 〈W,R,V〉 be a Jp-interpretation w ∈ W be an arbitrary knowledgestage, F be an arbitrary propositional formula and p be a tree path for F .The knowledge extension of w (over F indexed by p), denoted as EX(Fp, w), isdefined as follows:

EX(Fp, w) =

{w∗|w∗ Fp1 and w∗ 6 Fp2}, if op(Fp) = “⇒”{w∗|w∗ Fp1 and w∗ is F -maximal}, if op(Fp) = “¬”

∅, otherwise

The extension successor function succ(Fp) : W −→ W over F indexed by p isdefined as follows:

succ(Fp, w) ={

some arbitrary w′ ∈ EX(Fp, w), if EX(Fp, w) 6= ∅some arbitrary F -maximal w∗ otherwise 2

For the formula in fig. 2 and 3 we have, e.g, EX(F121, w2) = {w3} and thussucc(F121, w2) = w3.Within ΨJ we interpret any function symbol wp by succ(Fp). Hence, the universeUc of our classical interpretation Ic = 〈ιc,Uc〉 will be the set of knowledge stagesW of the given Jp-interpretation. The unary predicate symbols of our classicallanguage then denote the set of all w ∈ W where the respective 0-ary predicatesymbol of the intuitionistic language is forced.Conversely, given a classical interpretation for the translated formula we willconstruct the set W by a proper choice from the elements of Uc. More precisely,W will contain ιc(w0) as well as ιc(t) for any k-term t in RF (w0) such that anysubterm wp(s) of t actually refutes the subformula Fp, i.e. Ic 6|=c ΨF(s, Fp). Bythis restriction we are enabled to conclude that the left hand subformula of thesubformula associated with some function symbol in a given knowledge stageterm t is forced at ιc(t). For the formula of fig. 2, e.g., having w3 = ιc(t) witht = w121(w111(w(w0))) ∈ W we want to determine from w111εt that F1111 = a isforced at w3, so we may drop any consideration of further accessible knowledgestages refuting F111 = a⇒ b again.Finally, we want any atomic subformula Fp to be forced at ιc(t) if Ic |=c ΨF(t, Fp).But in order to preserve the heredity condition we must restrict ourselves to po-

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larity 1 subformulas, since only then we know the conjunction over all accessibleknowledge stages to be satisfied by Ic. However, if we deal with an F -maximalk-term then we know that the respective knowledge stage has no successor otherthan itself. Hence, the above restriction will become obsolete in this case:

Definition 9 Interpretation morphismLet F be an arbitrary propositional formula, Ij = 〈W,R,V〉 be a Jp-interpretation and w0 ∈ W be an arbitrary knowledge stage.Then the interpretation morphism, ΨJ(Ij , F, w0) = 〈ιc,Uc〉 is defined as follows:

1. Uc = W2. ιc(wp) = succ(Fp) for any wp ∈ W(F )3. ιc(f) = w0 for arbitrary f 6∈ W(F )4. ιc(p) = ⊥ for every 0-ary predicate symbol p5. ιc(p) = {w ∈ W|w p} for every unary predicate symbol p6. ιc(p) = ∅ for every other predicate symbol p.

Conversely, given a classical interpretation Ic = 〈ιc,Uc〉 then the reverse inter-pretation morphism Ψ−1

J (Ic, F ) = 〈W,R,V〉 is defined as follows:

1. W is the smallest set such that(a) ιc(w0) ∈ W(b) If ιc(t) ∈ W then also ιc(wp(t)) ∈ W, provided t is not F -maximal,

Ic 6|=c ΨF(t, Fp) and wp(t) ∈ E(Fq1, t) ∪ E(Fq2, t) for some wqεt

2. R = {(ιc(t), ιc(t′))|ιc(t), ιc(t′) ∈ W and t′ ∈ RF (t)}3. p ∈ V(w), iff there is a tree path p and a term t such that p = Fp,

atomic(Fp), w = ιc(t) ∈ W and either(a) t is F -maximal and Ic |=c ΨF-max(t, Fp), or(b) t is not F -maximal, pol(Fp) = 1 and Ic |=c ΨF(t, Fp)

The interpretation of terms and formulas is then defined for both directions ofthe interpretation morphism in the usual way. 2

As an example consider a classical countermodel Ic = 〈ιc,Uc〉 for ΨS(F ) as infig. 5, i.e. Uc = {w0, w1}, ιc(w0) = w0, ιc(w2(w0)) = w1 and ιc(a) = {w1}. ThenΨ−1

J (Ic, F ) = 〈W,R,V〉 with W = {w0, w1}, R = {(w0, w0), (w0, w1), (w1, w1)}and V(w1) = {a} whereas V(w0) = ∅In [7] we have proved ΨS to be sound and complete in the sense that there isa classical countermodel for ΨS(F ) whenever there is a Jp-countermodel for F(soundness) as well as vice versa (completeness). The existence of these counter-models is ensured via ΨJ, Ψ−1

J which map a given countermodel from source totarget logic and vice versa respectively. In the following we give a short surveyof these theorems.

Theorem 3 Soundness of the translationLet F be a propositional formula and Ij = 〈W,R,V〉 a Jp-interpretation suchthat Ij 6|=j F , i.e. w0 6 F for some w0 ∈ W. Then ΨJ(Ij , F, w0) 6|=c ΨS(F ).

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Proof: the proof is split into two lemmata. The first one proves w Fp ⇔Ic[t\w] |=c ΨF-max(t, Fp) for a given F -maximal knowledge stage w ∈ W anarbitrary F -maximal knowledge stage term t and any subformula Fp of F , whereIc = ΨJ(Ij , F, w0) = 〈ιc,Uc〉. This is carried out by a straightforward inductionover the degree of Fp (i.e. the number of logical connectives in Fp).Based on this lemma, the second one proves the following for any knowledgestage w ∈ W, any k-term t and any subformula Fp of F :1. if pol(Fp) = 0 and w 6 Fp then Ic[t\w] 6|=c ΨF(t, Fp)2. if pol(Fp) = 1 and w Fp then Ic[t\w] |=c ΨF(t, Fp)

This is also carried out by simultaneous induction over the degree of Fp. Since bydefinition ιc(w0) = w0 and pol(F ) = 0, we then have Ic 6|=c ΨF(w0, F ) = ΨS(F ).2

Theorem 4 Completeness of the translationLet F be a propositional formula and Ic = 〈ιc,Uc〉 a classical interpretation suchthat Ic 6|=c ΨS(F ). Then Ψ−1

J (Ic, F ) 6|=j F .

Proof: the proof is again split into two lemmata. The first one proves Ic |=c

ΨF-max(t, Fp) ⇔ ιc(t) Fp for any F -maximal k-term t with ιc(t) ∈ W and anysubformula Fp of F , where Ij = Ψ−1

J (Ic, F ) = 〈W,R,V〉. This is again carriedout by a straightforward induction over the degree of Fp.For the second lemma we need the definition of accessibility-set compatibility :A knowledge stage term t is accessibility-set compatible to a tree path p w.r.t. agiven formula F (denoted as compF (p, t)) if t is F -maximal or if E(Fp, t) ⊆ RF (t).This ensures ιc(t′) ∈ W for extensions t′ of t which may occur within ΨF(t, Fp).Based on this definition, the previous lemma, and some lemmata which ensurethe necessary properties for W and R, the second lemma proves the following forany k-term t such that ιc(t) ∈ W and any subformula Fp of F with compF (p, t):1. if pol(Fp) = 0 and Ic 6|=c ΨF(t, Fp) then ιc(t) 6 Fp

2. if pol(Fp) = 1 and Ic |=c ΨF(t, Fp) then ιc(t) Fp

This is also carried out by simultaneous induction over the degree of Fp. Theonly non-trivial case is for op(Fp) = “⇒”, pol(Fp) = 0 and wpεt. Then therestrictive definition of W allows to deduce Ic |=c ΨF(t, Fp1) and hence by in-duction hypothesis ιc(t) Fp1 from wpεt which otherwise does not follow fromIc 6|=c ΨF(t, Fp) = ΨF(t, Fp2).Finally, knowing by definition that ιc(w0) ∈ W, pol(F ) = 0 and compF (∅, w0),we obtain ιc(w0) 6 F and hence Ij 6|=j F . 2

4 Complexity of the morphismAs mentioned above the set of knowledge stages in a potential Jp-countermodelfor a given formula only depends on its special subformulas. Therefore we con-sider the complexity of a countermodel compared to the number of these specialsubformulas. By means of the notation introduced in the previous section wewill have to consider the cardinality of RF (w0) for the root knowledge stage w0

compared to the one of W(F ) for a given formula F . The following theorem dealswith the worst-case complexity:

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No. Res. Tran. DP Tot. ft

1 inv. 100 17 117 02 inv. 17 0 17 103 val. 16 0 16 04 inv. 50 17 67 05 inv. 33 17 50 06 inv. 0 0 0 07 inv. 16 0 16 08 inv. 0 0 0 09 val. 17 0 17 010 val. 33 67 100 011 val. 16 0 16 0

No. Res. Tran. DP Tot. ft

12 inv. 684 483 1167 013 val. 50 0 50 014 inv. 67 16 83 015 inv. 34 16 50 016 inv. 0 0 0 017 inv. 84 116 200 071a val. 17 0 17 071b inv. 83 34 117 072a val. 116 134 250 45072b val. 250 583 833 1202074 val. 116 17 133 3540

No. =Pelletier’s problem no.

Res. =validity result

Tran. =translation runtime

DP =Davis-Putnam runtime

Tot. =Tran.+DP

ft =ft runtime

Fig. 6: Runtime comparison for propositional Pelletier problems

Theorem 5 Worst-case complexity of countermodelsLet n = |W(F )| for a propositional formula F . Then |RF (w0)| ∈ O(n!). 2

The formula in fig. 2 and fig. 3 shows an example for the worst-case complexity.The two special subformulas F111 and F121 are not subformulas of each other.We shall say that they are unordered w.r.t. the given formula. Hence, any pos-sible extension ordering between these two must be considered for w1 = w(w0)(cf. fig. 4). In general, having n unordered implicative special subformulas whichare all left hand subformulas of the same implicative special subformula Fp andnot of any negated special subformula we must consider O(n!) different reductionorderings and hence O(n!) elements of RF (wp(t)) for a suitable k-term t.This situation becomes less severe the more the implicative special subformulasare ordered and the more negated special subformulas occur in the given formulaF . If, for instance, F = ¬F1 then the cardinality of RF (w0) = {w0, w(w0)} is just2 and hence in O(1) (cf. formula G in fig. 5).Therefore the average-case cardinality of RF (w0) is as difficult to predict as theaverage shape of a formula. Moreover the relation between |RF (w0)| and the ac-tual computation time for the translation as well as the size of its output isnot a trivial one since it heavily depends on various structural features of thegiven formula. It is obvious from the definition of our translation, however, thatin the worst-case the translation output encodes the maximum search space fora contraction-free sequent derivation. Yet our technique is constructively ade-quate in the sense that search space complexity increases along with the degreeof constructivity of the possible proof. Fig. 6 shows a practical runtime com-parison for the propositional problems presented in [13] between a prototypi-cal Eclipse-PROLOG implementation of our technique together with the non-normalform Davis-Putnam prover presented in [12] and the C implementationof the ft-prover [14] (version 1.23). All runtimes were measured on a SPARC 10in microseconds where “0” means “not measurable”. This comparison actuallyconfirms the claim of constructive adequateness:

– We could beat ft for formulas involving negated special subformulas in theimmediate vicinity of the root (72a,72b,74).

– We were beaten for the above-mentioned worst-case type formulas (12,17).– For intermediate cases we achieved comparable results (1-11,13-16,71a,71b).

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We regard these evaluations as promising inasmuch as neither our translationmethod nor the Davis-Putnam prover have yet been optimized or encoded in anefficient programming language (as opposed to ft for which both has been done).

5 Conclusion, related and further work

In this paper we have presented a procedure which translates propositional in-tuitionistic formulas into propositional classical ones and thus preserves decid-ability. The output of this procedure neither encodes theory knowledge withinthe object language such that prover would have to derive it at run-time, nordoes it yield the use of theory unification. Rather, it incorporates insights onhow to construct finite countermodels for any non-Jp-theorem. In addition to acomprehensive presentation of the principal ideas and the technical details wehave briefly considered the main aspects of the complexity of our method.There are various approaches which deal with intuitionistic theorem-proving —theoretical as well as practical ones. We have mentioned the general translationmethods for modal logics which (explicitly or implicitly) cover Jp in a very unspe-cific manner via the embedding into S4 (e.g. [9, 10, 15, 17]). The matrix methodpresented in [18] amounts to the functional translation principle described in [10]and has been implemented for intuitionistic logic as described in [11]. All theseapproaches share the disadvantage of neither preserving decidability (at leastnot explicitly) nor of liberating the theorem prover of dealing somehow withtheory knowledge at run-time.Contraction-free tableau calculi for intuitionistic logic have been presented in[1, 5, 8], where the one in [1] is implemented as described in [4]. They explicitlypreserve decidability but since they are connective-driven they cannot competewith matrix or resolution based calculi which are based on connecting comple-mentary literals. Nevertheless, they provided us with the key ideas for our trans-lation method. The refutation system for first-order intuitionistic logic which wehave formulated in [6] is based on the ideas presented here and allows a directcomparison between our approach and the contraction-free calculi. A practicallyefficient implementation of the contraction-free techniques is presented in [14].Decidability is achieved there by a kind of loop checking that resembles the ideasmentioned in section 2. However, it does not incorporate any concept comparableto our maximality principles which would allow unrestricted classical reasoning.Yet, it turns out to have an astonishing practical performance (see fig. 6) butthis may — at least partly — be explained by the efficient encoding in C.In [2] a technique to construct finite Jp-countermodels for non-Jp-theorems basedon a refutation calculus (“CRIP”) has been presented. For implicative specialsubformulas the potential countermodels constructed by our approach are per-fectly equivalent to the ones constructed by CRIP disregarding the provisosfor rules axiom and (11). However, since negated subformulas are treated thereonly as special cases of implications we are able to construct considerably smallercountermodels for negated special subformulas.The present stage of our translation leaves open a lot of space for optimizations.In particular, this includes further reductions of the generated accessibility sets.

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Some of these optimizations are obvious but we have not mentioned them herefor the sake of simplicity. Yet, our long-term goal is to extend our ideas to fullfirst-order logic. Work on this issue is currently in progress (cf. [6]).

Acknowledgments

We would like to thank Heiko Mantel for helping us to put the text into areadable form as well as Jens Otten for providing us with a non-normalformDavis-Putnam prover which is as fast as its source code is short and cryptic.

References

1. R. Dyckhoff: Contraction-Free Sequent Calculi for Intuitionistic Logic, Journalof Symbolic Logic, Vol. 57, No. 3, pp. 795–807, 1992

2. R. Dyckhoff, L. Pinto: Loop-free construction of counter-models for intuitionisticpropositional logic, In Behara, Fritsch, Lintz (eds.): Symposia Gaussiana, Conf. A,pp. 225–232, de Gruyter, 1995

3. M. Fitting: Intuitionistic Logic Model Theory and Forcing, Amsterdam, 119694. A. Heuerding, e. a.: Propositional Logics on the Computer, Proceedings of the

4th TABLEAUX ’95, LNAI 918, pp. 310–323, 19955. J. Hudelmaier: An O(n log n)-space decision procedure for intuitionistic proposi-

tional logic, In: Journal of Logic and Computation 3(1)1, pp. 63–75, 19936. D. Korn, C. Kreitz.: A Constructively Adequate Refutation System for Intuition-

istic Logic, Tech. Rep. AIDA-96-14, TH Darmstadt, 1996.7. D. Korn, C. Kreitz.: Efficiently Deciding Intuitionistic Propositional Logic via

Translation into Classical Logic., Tech. Rep. AIDA-96-10, TH Darmstadt, 1996.8. P. Miglioli, U. Moscato, M. Ornaghi: An Improved Refutation System for

Intuitionistic Predicate Logic, Journal of Automated Reasoning 13, pp 361–373, 19949. Robert C. Moore: Reasoning about Knowledge and Action, Proceedings of the

IJCAI-77, pp 223–227, Stanford, California 94305, 197710. H. J. Ohlbach: Semantics–Based Translation Methods for Modal Logics, Journal

of Logic and Computation, Vol. 1, no. 6, pp 691–746, 199111. J. Otten, C. Kreitz: A Connection Based Proof Method for Intuitionistic Logic,

Proceedings of the 4th TABLEAUX ’95, LNAI 918, pp. 122–137, 199512. J. Otten: On the Advantage of a Non-Clausal Davis-Putnam Procedure, Tech.

Rep. AIDA-97-1, TH Darmstadt, 199713. F. J. Pelletier: Seventy-Five Problems for Testing Automatic Theorem Provers,

Journal of Automated Reasoning 2, pp 191–216, 198614. D. Sahlin, T. Franzen, S. Haridi: An Intuitionistic Predicate Logic Theorem

Prover, Journal of Logic and Computation, Vol. 2, no. 5, pp 619–656, 199215. R. B. Scherl: A Constraint Logic Approach to Automated Modal Deduction,

Report UIUCDCS-R-93-1803, University of Illinois at Urbana-Champaign, 199316. K. Schutte: Vollstandige Systeme modaler und intuitionistischer Logik, Springer,

196817. J. Van Benthem: Correspondence Theory, In: D. Gabbay and F. Guenther (eds.):

Handbook of Philosophical Logic, Vol. II, pp. 167–247, Reidel, 1198418. L. Wallen: Automated Proof Search in Non–Classical Logics, MIT Press, 11990

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