Expressiveness of the Interval Logics of Allen s Relations on the Class of all Linear Orders: Complete Classification

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1 Proeeings of the Twenty-Seon International Joint Conferene on Artifiial Intelligene Expressiveness of the Interval Logis of Allen s Relations on the Class of all Linear Orers: Complete Classifiation Dario Della Monia University of Uine, Italy Valentin Goranko Tehnial University of Denmark Angelo Montanari University of Uine, Italy Guio Siavio University of Muria, Spain, an UIST Ohri, Maeonia Abstrat We ompare the expressiveness of the fragments of Halpern an Shoham s interval logi (HS), i.e., of all interval logis with moal operators assoiate with Allen s relations between intervals in linear orers. We establish a omplete set of interefinability equations between these moal operators, an thus obtain a omplete lassifiation of the family of 2 12 fragments of HS with respet to their expressiveness. Using that result an a omputer program, we have foun that there are 1347 expressively ifferent suh interval logis over the lass of all linear orers. 1 Introution Interval reasoning naturally arises in various fiels of artifiial intelligene, suh as theories of ation an hange, natural language analysis an proessing, an onstraint satisfation problems. Interval temporal logis formalize reasoning about interval strutures over orere omains, where time intervals, rather than time instants, are the primitive ontologial entities. The variety of binary relations between intervals in linear orers was first stuie systematially by Allen [Allen, 1983], who explore their use in systems for time management an planning. The moal logi featuring moal operators orresponing to Allen s interval relations was introue by Halpern an Shoham in [Halpern an Shoham, 1991]; we hereafter all that logi HS. Temporal logis with interval-base semantis have also been propose as a suitable formalism for the speifiation an verifiation of harware [Moszkowski, 1983] an of real-time systems [Zhou an Hansen, 2004]. In [Halpern an Shoham, 1991], it was shown that the satisfiability problem for HS is uneiable in all natural lasses We woul like to aknowlege the Spanish projet TIN C03-01 (G. Siavio), the HYLOCORE projet, fune by the Danish Natural Siene Researh Counil (V. Goranko), the Italian PRIN projet Innovative an multi-isiplinary approahes for onstraint an preferene reasoning (D. Della Monia, A. Montanari), an the Spanish/South-Afrian Projet HS (V. Goranko, G. Siavio). Currently affiliate with University of Salerno, Italy. of linear orers. For a long time, these sweeping uneiability results have isourage attempts for pratial appliations of interval logis. A renewe interest in the area has reently been stimulate by the isovery of several interesting eiable fragments of HS [Bresolin et al., 2007a; 2007b; 2008; 2009; 2010; Montanari et al., 2010a; 2010b]. In that ontext, an for the purpose of ientifying expressive interval logis for various intene appliations, the omparative analysis of the expressiveness of the variety of interval logis is a major researh problem in the area. In partiular, the important problem arises to analyze the mutual efinabilities among the moal operators of the logi HS an to lassify the fragments of HS with respet to their expressiveness. In the present paper we aress an solve that problem, by ientifying a omplete set of inter-efinability formulae among the moal operators of HS an thus proviing a omplete lassifiation of all fragments of HS with respet to their expressiveness for the strit semantis (exl. point intervals) over the lass of all linear orers. Using that result we have foun that there are exatly 1347 expressively ifferent suh fragments out of 2 12 = 4096 sets of moal operators in HS. The hoie of strit semantis, exluing point intervals, instea of inluing them (non-strit semantis), onforms to the efinition of interval aopte by Allen in [Allen, 1983]. It has at least two strong motivations. First, a number of representation paraoxes arise when the non-strit semantis is aopte, ue to the presene of point intervals, as pointe out in [Allen, 1983]. Seon, when point intervals are inlue, there seems to be no intuitive semantis for interval relations that makes them both pairwise isjoint an jointly exhaustive. The struture of the paper: after the preliminary Setion 2, in Setion 3 we state the main result of the paper, an we prove that the propose set of inter-efinability equations is orret. The muh more iffiult proof of ompleteness is given in Setion 4. Setion 5 provies an assessment of the work one an it outlines future researh iretions. 2 Preliminaries Let D = D, < be a linearly orere set. An interval over D is an orere pair [a, b], where a, b D an a b. Intervals of the type [a, a] are alle point intervals, while the others are alle strit intervals. There are 12 ifferent non-trivial relations (exluing the equality) between two strit intervals 845

2 A L B E D O [a, b]r A[, ] b = [a, b]r L[, ] b< [a, b]r B[, ] a =, < b [a, b]r E[, ] b =, a < [a, b]r D[, ] a<,<b [a, b]r O[, ] a<<b< a b Table 1: Allen s interval relations an the orresponing HS moalities. in a linear orer, often alle Allen s relations [Allen, 1983]: the six relations epite in Table 1 an the inverse relations. We treat interval strutures as Kripke strutures an Allen s relations as aessibility relations in them, thus assoiating a moal operator X with eah Allen s relation R X. For eah operator X, its transpose, enote by X, orrespons to the inverse relation R X of R X (that is, R X =(R X ) 1 ). Halpern an Shoham s logi HS is a multi-moal logi with formulae built over a set AP of propositional letters, the propositional onnetives an, an a set of moal operators assoiate with all Allen s relations. With every subset {R X1,...,R Xk } of these relations, we assoiate the fragment X 1 X 2...X k of HS, the formulae of whih are efine by the grammar: ϕ ::= p ϕ ϕ ϕ X 1 ϕ... X k ϕ. The other propositional onnetives, an, an the ual operators [X] are efine as usual, e.g., [X]ϕ X ϕ. For a fragment F = X 1 X 2...X k an a moal operator X, we write X Fif X {X 1,...,X k }. Given two fragments F 1 an F 2, we write F 1 F 2 if X F 1 implies X F 2, for every moal operator X. The semantis of HS is given in terms of interval moels M = I(D),V, where I(D) is the set of all (strit) intervals over D. The valuation funtion V : AP 2 I(D) assigns to every p APthe set of intervals V (p) on whih p hols. The truth of a formula on a given interval [a, b] in an interval moel M is efine by strutural inution on formulae: M,[a, b] p iff [a, b] V (p), for all p AP; M,[a, b] ψ iff it is not the ase that M,[a, b] ψ; M,[a, b] ϕ ψ iff M,[a, b] ϕ or M,[a, b] ψ; M,[a, b] X ψ iff there exists an interval [, ] suh that [a, b]r X [, ] an M,[, ] ψ, where R X is any of Allen s relations. A formula φ of HS is vali, enote = φ, if it is true on every interval in every interval moel. Two formulae φ an ψ are equivalent, enote φ ψ, if = φ ψ. Definition 2.1. A moal operator X of HS is efinable in an HS-fragment F, enote X F,if X p ψ for some formula ψ = ψ(p) F, for any fixe propositional variable p. In suh a ase, the equivalene X p ψ is alle an inter-efinability equation for X in F. It is known from [Halpern an Shoham, 1991] that, in the strit semantis, all moal operators in HS are efinable in the fragment ontaining the moalities A, B, an E, an their transposes A, B, an E (In the non-strit semantis, the four moalities B, E, B, an E suffie, as shown in [Venema, 1990]). In this paper, we ompare an lassify the expressiveness of all fragments of HS on the lass of all interval strutures over linear orers. Formally, let F 1 an F 2 be any pair of suh fragments. We say that: F 2 is at least as expressive as F 1, enote F 1 F 2,if every operator X F 1 is efinable in F 2. F 1 is stritly less expressive than F 2, enote F 1 F 2, if F 1 F 2 but not F 2 F 1. F 1 an F 2 are equally expressive (or, expressively equivalent), enote F 1 F 2,ifF 1 F 2 an F 2 F 1. F 1 an F 2 are expressively inomparable, enote F 1 F 2, if neither F 1 F 2 nor F 2 F 1. In orer to show non-efinability of a given moal operator in a given fragment, we use a stanar tehnique in moal logi, base on the notion of bisimulation an the invariane of moal formulae with respet to bisimulations (see, e.g., [Blakburn et al., 2002]). Let F be an HS-fragment. An F-bisimulation between two interval moels M = I(D),V an M = I(D ),V over AP is a relation Z I(D) I(D ) satisfying the following properties: loal onition: Z-relate intervals satisfy the same propositional letters over AP; forwar onition: if ([a, b], [a,b ]) Z an ([a, b], [, ]) R X for some X F, then there exists [, ] suh that ([a,b ], [, ]) R X an ([, ], [, ]) Z; bakwar onition: likewise, but from M to M. The important property of bisimulations use here is that any F-bisimulation preserves the truth of all formulae in F. Thus, in orer to prove that an operator X is not efinable in F, it suffies to onstrut a pair of interval moels M an M an a F-bisimulation between them, relating a pair of intervals [a, b] M an [a,b ] M, suh that M,[a, b] X p, while M, [a,b ] X p. 3 Comparing the expressiveness of the fragments of HS In orer to lassify all fragments of HS with respet to their expressiveness, it suffies to ientify all efinabilities of moal operators X in fragments F, where X / F. A efinability X F is optimal if X F for any fragment F suh that F F. A set of suh efinabilities is optimal if it onsists of optimal efinabilities. The main result of the paper is the following theorem. Theorem 3.1. The set of inter-efinability equations given in Table 2 is soun, omplete, an optimal. Most of the equations in Table 2 are known from [Halpern an Shoham, 1991], exept the efinability L BE an its symmetri, L BE, whih are new. We will first prove the sounness of the given set of inter-efinability equations. Lemma 3.2. The set of inter-efinability equations given in Table 2 is soun. 846

3 L p A A p L p A A p O p E B p O p B E p D p E B p D p E B p L p B [E] B E p L p E [B] E B p L A L A O BE O BE D BE D BE L BE L BE Table 2: The omplete set of inter-efinability equations Proof. We only nee to prove the sounness for the new interefinability equations L p B [E] B E p an its symmetri for L. The proofs are analogous, so we only prove the former. First, we prove the left-to-right iretion. Suppose that M,[a, b] L p for some moel M an interval [a, b]. This means that there exists an interval [, ] suh that b<an M,[, ] p. We exhibit an interval [a, y], with y > b suh that, for every x (stritly) in between a an y, the interval [x, y] is suh that there exist two points y an x suh that y >y, x<x <y, an [x,y ] satisfies p. Let y be equal to. The interval [a, ], whih is starte by [a, b], is suh that for any of its ening intervals, that is, for any interval of the form [x, ], with a<x, we have that x<< an M,[, ] p. As for the other iretion, we must show that B [E] B E p implies L p. To this en, suppose that M,[a, b] B [E] B E p for a moel M an an interval [a, b]. Then, there exists an interval [a, ], for some >b, suh that [E] B E p is true on [a, ]. As a onsequene, the interval [b, ] must satisfy B E p, that means that there are two points x an y suh that y>, b<x<y, an [x, y] satisfies p. Sine x>b, then M,[a, b] L p. Proving ompleteness is the har task; optimality will be establishe together with it. The ompleteness proof is organize as follows. For eah HS operator X, we show that X is not efinable in any fragment of HS that oes not ontain as efinable (aoring to Table 2) all operators of some of the fragments in whih X is efinable (aoring to Table 2). More formally, for eah HS operator X, the proof onsists of the following steps: 1. using Table 2, fin all fragments F i suh that X F i ; 2. ientify the list M 1,...,M m of all -maximal fragments of HS that ontain neither the operator X nor any of the fragments F i ientifie by the previous step; 3. for eah fragment M i, with i {1,...,m}, provie a bisimulation for M i whih is not a bisimulation for X. Details of the ompleteness proof will be provie in a series of lemmas (of inreasing omplexity) in the next setion. 4 The ompleteness proof In this setion, we will prove that, for eah moal operator X of HS, the set of inter-efinability equations for X in Table 2 is omplete for that operator, that is, X is not efinable in any fragment of HS that oes not ontain (as efinable) all operators of some of the fragments liste in Table 2 in whih X is efinable. Due to spae limitations, we will not prove in etail all the ases. A etaile proof an be foun in the extene tehnial report an it will appear in a future journal version of the present paper. 4.1 Completeness for L an L Lemma 4.1. The set of inter-efinability equations for L an L given in Table 2 is omplete. Proof. Aoring to Table 2, L is efinable in terms of A an BE. Hene, the fragments BEDOALEDO an BDOALBEDO are the only -maximal ones not featuring L an ontaining neither A nor BE. To prove the thesis, it suffies to exhibit a bisimulation for eah one of these two fragments that oes not preserve the relation inue by L. Thanks to Lemma 3.2, BEDOALEDO an BDOALBEDO are expressively equivalent to BEOAED an BDOABE, respetively. Thus, to all our purposes, we an simply refer to the latter ones instea of the former ones. As for the first fragment, let M 1 = I(N),V 1 an M 2 = I(N),V 2 be two moels an let V 1 an V 2 be suh that V 1 (p) ={[2, 3]} an V 2 (p) =, where p is the only propositional letter of the language. Moreover, let Z be a relation between (intervals of) M 1 an M 2 efine as Z = {([0, 1], [0, 1])}. It an be easily shown that Z is a BEOAEDbisimulation. The loal property is trivially satisfie, sine all Z-relate intervals satisfy p. As for the forwar an bakwar onitions, it suffies to notie that, starting from the interval [0, 1], it is not possible to reah any other interval using any of the moal operators of the fragment. At the same time, Z oes not preserve the relation inue by the moality L. Inee, ([0, 1], [0, 1]) Z an M 1, [0, 1] L p, but M 2, [0, 1] L p. Therefore, L is not efinable in BEDOALEDO. As for the seon fragment, let M 1 = I(Z ),V 1 an M 2 = I(Z ),V 2 be two moels base on the set Z = {..., 2, 1}, an let V 1 an V 2 be suh that V 1 (p) = {[ 2, 1]} an V 2 (p) =, where p is the only propositional letter of the language. Moreover, let Z be a relation between (intervals of) M 1 an M 2 efine as follows: ([x, y], [w, z]) Z ef [x, y] =[w, z] an [x, y] [ 2, 1]. We prove that Z is a BDOABE-bisimulation. First, the loal property is trivially satisfie, sine all Z-relate intervals satisfy p. Moreover, starting from any interval, the only interval that satisfies p, that is, [ 2, 1], annot be reahe using the set of moal operators feature by our fragment. At the same time, Z oes not preserve the relation inue by L, as([ 4, 3], [ 4, 3]) Z an M 1, [ 4, 3] L p, but M 2, [ 4, 3] L p. Therefore, L is not efinable in BDOALBEDO. A ompletely symmetri argument an be applie for the ompleteness proof of L. 4.2 Completeness for E, E, B, an B Lemma 4.2. The set of inter-efinability equations for E, E, B, an B given in Table 2 is omplete. Proof. Aoring to Table 2, we will show that E is not efinable in terms of the only -maximal fragment not fea- 847

4 turing it, namely, ALBDOALBEDO. (The inverse moality E an the symmetri moalities B an B an be ealt with using similar arguments.) Thanks to Lemma 3.2, it atually suffies to provie a bisimulation for ABDOABE. Let M 1 = I(R),V 1 an M 2 = I(R),V 2, where p is the only propositional letter of the language, the valuation funtion V 1 : AP 2 I(R) is efine as: [x, y] V 1 (p) ef x Q iff y Q, an the valuation funtion V 2 : AP 2 I(R) as: [w, z] V 2 (p) ef w Q iff z Q, an ([0, 3], [w, z]) / R E. Moreover, let Z be a relation between (intervals of) M 1 an M 2 efine as follows: ([x, y], [w, z]) Z ef [x, y] V 1 (p) iff [w, z] V 2 (p). We show that Z is an ABDOABE-bisimulation between M 1 an M 2. The satisfation of the loal onition immeiately follows from the efinition. The forwar onition an be heke as follows. Let Q = R\Q an let [x, y] an [w, z] be two Z-relate intervals. For eah moal operator X of the language, let us assume that [x, y]r X [x,y ].Wehaveto exhibit an interval [w,z ] suh that [x,y ] an [w,z ] are Z- relate, an [w, z] an [w,z ] are R X -relate. We proee ase-by-ase. Let X = A (an thus y = x ). Suppose that [x,y ] V 1 (p) (resp., [x,y ] / V 1 (p)). We an always fin a point z > z suh that [z,z ] V 2 (p) (resp., [z,z ] / V 2 (p)), inepenently from z belonging to Q or Q (sine both Q an Q are right-unboune). This implies that [x,y ] an [z,z ] are Z-relate. Sine [w, z] an [z,z ] are obviously R A -relate, we have the thesis. If X = B, the argument is similar to the previous one, but, in this ase, the ensity of Q an Q is exploite. If X = D, it suffies to hoose two points w an z suh that w<w <z <z, z 3, w belongs to Q if an only if x oes, an z belongs to Q if an only if y oes. As in the previous ase, the existene of suh points is guarantee by the ensity of Q an Q. If X = O, w an z are require to be suh that w<w <z<z, an both ensity an right-unbouneness of Q an Q must be exploite. The remaining ases as well as the bakwar onition an be verifie in a very similar way. At the same time, Z oes not preserve the relation inue by E : we have that ([0, 3], [0, 3]) Z, M 1, [0, 3] E p, but M 2, [0, 3] E p. Therefore, E annot be efine in the fragment ALBDOALBEDO. 4.3 Completeness for A an A In the proofs of Lemma 4.3 an Lemma 4.4, in orer to get the bisimulation we want, we nee to exploit a well-known property of the set of real numbers R: R (resp., Q, Q) an be partitione into a ountable number of pairwise isjoint subsets, eah one of whih is ense in R. More formally, there are ountably many nonempty sets R i (resp., Q i, Q i ), with i N, suh that, for eah i N, R i (resp., Q i, Q i ) is ense in R, R = i N R i (resp., Q = i N Q i, Q = i N Q i), an R i R j =, (resp., Q i Q j =, Q i Q j = ), for eah i, j N with i j. Lemma 4.3. The set of inter-efinability equations for A an A given in Table 2 is omplete. Proof. Aoring to Table 2, it suffies to show that A is not efinable in the only -maximal fragment not ontaining it, namely, LBEDOALBEDO, whih, by Lemma 3.2, turns out to be equivalent to LBEABE. Let M 1 = I(R),V 1 an M 2 = I(R),V 2 be two moels built on the only propositional letter p. In orer to efine the valuation funtions V 1 an V 2, we take avantage of two partitions of the set R, one for M 1 an the other one for M 2, eah of them onsisting of exatly four sets that are ense in R. Formally, for j =1, 2 an i =1,...,4, let R i j be ense in R. Moreover, for j =1, 2, let R = 4 i=1 Ri j an Ri j Ri j = for eah i, i {1, 2, 3, 4} with i i. For j =1, 2, we fore points in R 1 j (resp., R2 j, R3 j, R4 j ) to behave in the same way with respet to the truth of p/ p over the intervals they initiate an terminate by imposing the following onstraints: x, y( if x R 1 j, then M j, [x, y] p); x, y( if x R 2 j, then M j, [x, y] p); x, y( if x R 3 j, then (M j, [x, y] p iff y R 1 j R3 j )); x, y( if x R 4 j, then (M j, [x, y] p iff y R 2 j R4 j )). It an be easily shown that, from the given onstraints, it immeiately follows that: x, y( if y R 1 j, then (M j, [x, y] p iff x R 3 j )); x, y( if y R 2 j, then (M j, [x, y] p iff x R 4 j )); x, y( if y R 3 j, then (M j, [x, y] p iff x R 3 j )); x, y( if y R 4 j, then (M j, [x, y] p iff x R 4 j )). The above onstraints univoally inues the following efinition of the valuation funtions V j (p) :AP 2 I(R) : { [x, y] V j (p) ef (x R 3 j y R 1 j R3 j ) (x R 4 j y R2 j R4 j ). Now, let Z be the relation between (intervals of) M 1 an M 2 efine as follows. Two intervals [x, y] an [w, z] are Z- relate if an only if at least one of the following onitions hols: 1. x R 1 1 R 2 1 an w R 1 2 R 2 2; 2. x R 3 1, w R 3 2, an (y R 1 1 R 3 1 iff z R 1 2 R 3 2); 3. x R 3 1, w R 4 2, an (y R 1 1 R 3 1 iff z R 2 2 R 4 2); 4. x R 4 1, w R 3 2, an (y R 2 1 R 4 1 iff z R 1 2 R 3 2); 5. x R 4 1, w R 4 2, an (y R 2 1 R 4 1 iff z R 2 2 R 4 2). We show that the relation Z is an LBEABE-bisimulation. It an be easily heke that every pair ([x, y], [w, z]) of Z-relate intervals is suh that either [x, y] V 1 (p) an [w, z] V 2 (p) or [x, y] V 1 (p) an [w, z] V 2 (p). In orer to verify the forwar onition, let [x, y] an [w, z] be two Z-relate intervals. For eah moal operator X of the language an eah interval [x,y ] suh that [x, y]r X [x,y ], we have to exhibit an interval [w,z ] suh that [x,y ] an [w,z ] are Z-relate, an [w, z] an [w,z ] are R X -relate. We proee ase-by-ase. Let X = L. We must onsier five sub-ases epening on the sets x an y belong to: (i) if x R 1 1 R 2 1, then for eah w R 1 2 suh that w >z, we have that, for every z >w, ([x,y ], [w,z ]) Z an [w, z]r L [w,z ] (the existene of w is guarantee by rightunbouneness of R 1 2); (ii) if x R 3 1 an y R 1 1 R 3 1, then 848

5 for eah w,z suh that z<w <z an w,z R 3 2,we have that ([x,y ], [w,z ]) Z an [w, z]r L [w,z ] (rightunbouneness of R 3 2); (iii) if x R 3 1 an y R 2 1 R 4 1, then for eah w,z suh that z<w <z, w R 3 2, an z R 4 2, we have that ([x,y ], [w,z ]) Z an [w, z]r L [w,z ] (right-unbouneness of R 3 2 an R 4 2); (iv) if x R 4 1 an y R 1 1 R 3 1, then for eah w,z suh that z<w <z, w R 4 2, an z R 3 2, we have that ([x,y ], [w,z ]) Z an [w, z]r L [w,z ] (right-unbouneness of R 3 2 an R 4 2); (v) if x R 4 1 an y R 2 1 R 4 1, then for eah w,z suh that z< w <z an w,z R 4 2, we have that ([x,y ], [w,z ]) Z an [w, z]r L [w,z ] (right-unbouneness of R 4 2). Assume now X = B. If x R 1 1 R 2 1 an w R 1 2 R 2 2, then for any w < z < z, both ([x, y ], [w, z ]) Z an [w, z]r B [w, z ] hol. If x R i 1 an w R i 2, for some i {3, 4}, an y R k 1, for some k {1, 2, 3, 4}, then for any w < z < z suh that z R k 2, it hols that ([x, y ], [w, z ]) Z an [w, z]r B [w, z ] (the existene of z is guarantee by ensity of R k 2 in R). Finally, if x R i 1 an w R i 2 for i, i {3, 4} with i i, then if y R 1 1 R 3 1 (resp., y R 2 1 R 4 1) for any w<z <zsuh that z R 2 2 R 4 2 (resp., z R 1 2 R 3 2), it hols that ([x, y ], [w, z ]) Z an [w, z]r B [w, z ] (ensity of R 2 2 an R 4 2, resp., R 1 2 an R 3 2,inR). The remaining ases an be ealt with in a similar way. Let us onsier now two intervals [x, y] an [w, z] suh that x R 1 1, w R 1 2, y R 3 1, an z R 1 2. By efinition of Z, [x, y] an [w, z] are Z- relate, an by efinition of V 1 an V 2, there exists y >y suh that M 1, [y, y ] p, but there is no z >zsuh that M 2, [z,z ] p. This allows us to onlue that Z oes not preserve the relation inue by A, an thus A is not efinable in LBEDOALBEDO. A ompletely symmetri argument an be applie for the ompleteness proof of A. 4.4 Completeness for D, D, O, an O To eal with the moalities D, D, O, an O, we proee as follows. We first introue the notion of f-moel, that is, for any given funtion f : R Q, we efine a moel M f, alle f-moel, whose valuation is base on f. Then, for any given pair of funtions f 1 an f 2, we efine a suitable relation Z f2 f 1 between the moels M f1 an M f2 (from now on, we will simply write Z when there is no ambiguity about the involve moels). Finally, we speify the requirements that f 1 an f 2 must satisfy to make Z the bisimulation we want (these requirements vary from one moality to the other). Lemma 4.4. The set of inter-efinability equations for D, D, O, an O given in Table 2 is omplete. Proof. We will etail the ase of the moality D. The other ases an be prove using similar arguments. Aoring to Table 2, D is efinable in terms of BE. The fragments ALBOALBEDO an ALEOALBEDO are thus the only -maximal ones not featuring D an not ontaining BE. We shoul provie a bisimulation, not preserving the relation inue by D, for eah of these fragments, but, thanks to the symmetry of the operators, it suffies to onsier only one of them, say ALBOALBEDO (by Lemma 3.2, we have that ALBOALBEDO is expressively equivalent to ABOABE). Given a funtion f : R Q, we efine the f-moel M f, over a language with one propositional letter p only, as the pair I(R),V f, where V f : I(R) 2 AP is efine as follows: [x, y] V f (p) ef y f(x). For any given pair of funtions f 1 an f 2 (from R to Q), the relation Z is efine as follows: ([x, y], [w, z]) Z ef x w, y z, an [x, y] l [w, z], where u v ef u Q iff v Q an [u, u ] l [v, v ] ef u f 1 (u) an v f 2 (v), for {<, =,>}. Finally, the following onstraints are impose on f (if we replae D by one of the other moalities, the onstraints must be suitably replae as well): (i) for every x R, f(x) >x, (ii) for every x Q, both f 1 (x) Q an f 1 (x) Q are left-unboune (notie that surjetivity of f immeiately follows), an (iii) for every x, y R, ifx<y, then there exists u 1 Q (resp., u 2 Q) suh that x<u 1 <y(resp., x<u 2 <y) an y<f(u 1 ) (resp., y<f(u 2 )). Now, we show that if both f 1 an f 2 satisfy the above onitions, then Z is an ABOABE-bisimulation between M f1 an M f2. Let [x, y] an [w, z] be two Z-relate intervals. By efinition, y f 1 (x) an z f 2 (w) for some {<, =, >}. If {=,>}, then both [x, y] an [w, z] satisfy p; otherwise, both of them satisfy p. The loal onition is thus satisfie. As for the forwar onition, let [x, y] an [x,y ] be two intervals in M f1 an [w, z] an interval in M f2.wehave to prove that if [x, y] an [w, z] are Z-relate, then, for eah moal operator X of ABOABE suh that [x, y]r X [x,y ], there exists an interval [w,z ] suh that [x,y ] an [w,z ] are Z-relate an [w, z]r X [w,z ]. One more, we proee ase-by-ase. For the sake of brevity, we only etail the ase of A. The other moalities an also be ealt with by exploiting the requirements for the funtions f 1 an f 2 in a suitable way. Let X = A. By efinition of A, x = y an we are fore to hoose w = z. By y z, it immeiately follows x w. We must fin a point z >zsuh that y z an both y f 1 (y) an z f 2 (z) for some {<, =,>}. Let us suppose that y < f 1 (y). In suh a ase, we hoose a point z suh that z<z <f 2 (z) an y z. The existene of suh a point is guarantee by onition (i) on f 2 an by the ensity of Q an Q in R. Otherwise, if y = f 1 (y), we hoose z = f 2 (z). By efinition of f 1 an f 2 (the oomain of f 1 an f 2 is Q), both y an z belong to Q an thus y z. Finally, if y >f 1 (y), we hoose z >f 2 (z) suh that y z. The existene of suh a point is guarantee by right-unbouneness of Q an Q. Satisfation of the bakwar onition for all moalities an be heke in a similar way. To omplete the proof, we exhibit two funtions that meet the requirements we have impose to f 1 an f 2, but o not preserve the relation inue by D. Let P(Q) ={Q q q Q} an P(Q) ={Q q q Q} be infinite an ountable partitions of Q an Q, respetively, suh that for every q Q, both Q q an Q q are ense in R. For every q Q, let R q = Q q Q q. We efine a funtion g : R Q that maps every 849

6 real number x to the inex q (a rational number) of the lass R q it belongs to. Formally, for every x R, g(x) =q, where q Q is the unique rational number suh that x R q. The two funtions f 1 : R Q an f 2 : R Q are efine as follows: { g(x) if x<g(x), x 1, an x 0 f 1 (x) = 2 if x =1 x +3 otherwise { g(x) if x<g(x) an x [0, 3) f 2 (x) = x +3 otherwise It is not iffiult to hek that the above-efine funtions meet the requirements for f 1 an f 2, an thus Z is an ABOABE-bisimulation. On the other han, Z oes not preserve the relation inue by D. Consier the interval [0, 3] in M f1 an the interval [0, 3] in M f2. It is immeiate to see that these two intervals are Z-relate. However, M f1, [0, 3] D p (as M f1, [1, 2] p), but M f2, [0, 3] D p. This allows us to onlue that D is not efinable in the fragment ALBOALBEDO. 4.5 Harvest The proof of Theorem 3.1 follows now immeiately. We have use the equations in Table 2 as the basis of a simple program that ientifies an ounts all expressively ifferent fragments of HS with respet to the strit semantis. Using that program, we have foun that, uner our assumptions (strit semantis, over the lass of all linear orers) there are exatly 1347 genuine, that is, expressively ifferent, fragments out of 2 12 = 4096 ifferent subsets of HS-operators. 5 Conlusions In this paper, we have obtaine a soun, omplete, an optimal set of inter-efinability equations among all moal operators in HS, thus proviing a haraterization of the relative expressive power of all interval logis efinable as fragments of HS. Suh a lassifiation has a number of important appliations. As an example, it allows one to properly ientify the (small) set of HS fragments for whih the eiability of the satisfiability problem is still an open problem. It shoul be emphasize that the set of inter-efinability equations liste in Table 2 an the resulting lassifiation o not apply if the non-strit semantis is onsiere. For instane, if the non-strit semantis is assume, then, as shown in [Venema, 1990], A (resp., A ) an be efine in BE (resp., BE). Also, if the semantis is restrite to speifi lasses of linear orers, the ompleteness of the set of equations in Table 2 is no longer guarantee. For instane, in isrete linear orers, A an be efine in BE: A p ϕ(p) E ϕ(p), where ϕ(p) is a shorthan for [E] B ([E][E] E (p B p)); likewise, A is efinable in BE. As another example, in ense linear orers, L an be efine in DO: L p O ( O [O]( O p D p D O p)); likewise, L is efinable in DO. (In view of these inter-efinabilities, Lemma 4.1 annot be prove by using bisimulation between moels over the reals.) The lassifiation of the expressiveness of HS fragments with respet to the non-strit semantis, as well as over speifi lasses of linear orers, is urrently uner investigation an will be reporte in a forthoming publiation. Referenes [Allen, 1983] J. F. Allen. Maintaining knowlege about temporal intervals. Communiations of the ACM, 26(11): , [Blakburn et al., 2002] P. Blakburn, M. e Rijke, an Y. Venema. Moal Logi. CUP, [Bresolin et al., 2007a] D. Bresolin, A. Montanari, an P. Sala. An optimal tableau-base eision algorithm for Propositional Neighborhoo Logi. In Pro. of STACS 2007, volume 4393 of LNCS, pages Springer, [Bresolin et al., 2007b] D. Bresolin, A. Montanari, an G. Siavio. An optimal eision proeure for Right Propositional Neighborhoo Logi. Journal of Automate Reasoning, 38(1-3): , [Bresolin et al., 2008] D. Bresolin, A. Montanari, P. Sala, an G. Siavio. Optimal Tableaux for Right Propositional Neighborhoo Logi over Linear Orers. In Pro. of the 11th European Conferene on Logis in AI, volume 5293 of LNAI, pages Springer, [Bresolin et al., 2009] D. Bresolin, V. Goranko, A. Montanari, an G. Siavio. Propositional interval neighborhoo logis: Expressiveness, eiability, an uneiable extensions. Annals of Pure an Applie Logi, 161(3): , [Bresolin et al., 2010] D. Bresolin, V. Goranko, A. Montanari, an P. Sala. Tableaux for logis of subinterval strutures over ense orerings. Journal of Logi an Computation, 20: , [Halpern an Shoham, 1991] J. Halpern an Y. Shoham. A propositional moal logi of time intervals. Journal of the ACM, 38(4): , [Montanari et al., 2010a] A. Montanari, G. Puppis, an P. Sala. Maximal eiable fragments of Halpern an Shoham s moal logi of intervals. In Pro. of ICALP 2010, volume 6199 of LNCS, pages Springer, [Montanari et al., 2010b] A. Montanari, G. Puppis, P. Sala, an G. Siavio. Deiability of the interval temporal logi AB B on natural numbers. In Pro. of STACS 2010, pages , [Moszkowski, 1983] B. Moszkowski. Reasoning about igital iruits. Teh. rep. stan-s , Dept. of Computer Siene, Stanfor University, Stanfor, CA, [Venema, 1990] Y. Venema. Expressiveness an ompleteness of an interval tense logi. Notre Dame Journal of Formal Logi, 31(4): , [Zhou an Hansen, 2004] C. Zhou an M. R. Hansen. Duration Calulus: A Formal Approah to Real-Time Systems. Springer,