A Dose of Timed Logic, in Guarded Measure

We consider interval measurement logic IML, a sublogic of Zhou and Hansen’s interval logic, with measurement functions which provide real-valued measurement of some aspect of system behaviour in a given time interval. We interpret IML over a variety of time domains (continuous, sampled, integer) and show that it can provide a unified treatment of many diverse temporal logics including duration calculus (DC), interval duration logic (IDL) and metric temporal logic (MTL). We introduce a fragment GIML with restricted measurement modalities which subsumes most of the decidable timed logics considered in the literature. Next, we introduce a guarded first-order logic with measurements MGF . As a generalisation of Kamp’s theorem, we show that over arbitrary time domains, the measurement logic GIML is expressively complete for it. We also show that MGF has the 3-variable property. In addition, we have a preliminary result showing the decidability of a subset of GIML when interpreted over timed words. The importance of reasoning about timed systems has led to considerable research on models and logics for timed behaviours. We consider a slightly more general situation where, in addition to time, we can use other measurement functions as well. For instance, instead of saying “during the last 24 hours, the rainfall was 100 mm,” we can say that “the time elapsed for the last 100 mm of rainfall was over 4 months.” We can also have measurements of quantities like “mean value” of a proposition within a time interval. Guelev has shown how probabilities might be incorporated into such a framework [Gue00]. Unlike data languages [BPT03], there is no finite state mechanism associated with the measurement functions. Thus we are in the setting of the interval logic with measurements defined by Zhou and Hansen [ZH04]. There exists quite a menagerie of timed and duration logics. In Section 1 below, we review the literature and define our logic χIML[Σ] over a signature Σ of measurement functions, and parameterised by a set of primitive comparisons χ dependent on Σ. We show that it can provide a unified treatment of many diverse temporal logics including duration calculus (DC), interval duration logic (IDL) and metric temporal logic (MTL). In Section 2, we consider an enrichment of Kamp’s FO[<] with measurements. The undecidability of this logic motivates us to formulate and investigate a E. Asarin and P. Bouyer (Eds.): FORMATS 2006, LNCS 4202, pp. 260–273, 2006. c © Springer-Verlag Berlin Heidelberg 2006 A Dose of Timed Logic, in Guarded Measure 261 fragment χMGF [<,Σ] with χ-guarded measurement quantifiers. The next two sections show that χGIML is expressively complete for χMGF . Kamp’s syntactic techniques were used by Venema [Ven91], and we extend these as well as the pebble games of Immerman and Kozen [IK87] in our proofs. As in Kamp’s result, we show along the way that χMGF has the three-variable property. Thus the expressiveness of our logic is reasonably delineated. We would have liked to have established a connection to aperiodic languages [Bac03] but this must remain future work. We now turn to decidability. We find that IML and GIML are in general undecidable, but for a set of weak comparisons (which disallow equality tests between measurements and constants), we use a result by Hirshfeld and Rabinovich [HR99] and our expressive completeness to show that Weak -GIML[ ] is decidable for continous time. We also prove by translation into one-clock alternating timed automata [LW05, OW05], decidability over timed words of a sublogic Punct-FgIML[ ] of GIML[ ], which only has nesting-free forward guarding. 1 A Classification of Timed Behaviours and Logics Timed logics describe the evolution of system behaviour in time. For us, time is a linear order (T,<), and we will further assume that T is a subset of the nonnegative reals (which we designate ) with < the usual ordering. Intv(T ), the set of (closed) intervals of T , is {[b, e] ∈ T×T | b ≤ e}. A time frame TF = (T,<, d) is a subset of the real order ( , <) with d giving the absolute value of the distance on the real line between two real numbers, i.e. d[b, e] = |b− e|. Zhou and Hansen have proposed an interesting interval logic [ZH04] where the variables (measurement functions) denote real-valued measurements of system behaviour in a given time interval. Formally we have a signature Σ = {m1, . . . ,mn} of measurement function symbols (of arity 2), and we assume that it contains the distinguished function which measures the length of the interval. We will often abbreviate the signature { } to . Zhou and Hansen’s logic allows first order real arithmetic over such measurements. In this section, we introduce a restricted version of this logic where a measurement may only be compared with an integer constant. We call this logic interval measurement logic, IML. Let Pvar be a finite set of propositional variables. A behaviour of a system over TF is a pair of maps θ : (Pvar → T → {0, 1})× (Σ → Intv → ), where Σ might depend on Pvar. For convenience we write θ(p) as a boolean function of time and θ(m)[b, e], for m ∈ Σ, as giving the value of the measurement function m on the interval [b, e]. Moreover, we require that the measurement is always interpreted as length of the interval, i.e. θ( )[b, e] = d[b, e] = |b− e|. An interval model is a pair θ, [b, e]. It is useful to consider several classes of time frames TF = (T,<, d) where T ⊆ . In the literature, we find a variety of timed logics which use these different classes as time frames. Some such logics are listed in the next section. 262 K. Lodaya and P.K. Pandya Continuous infinite time. T = . Continuous time finitely variable behaviours. We call a continuous time behaviour θ finitely variable if for any p in Pvar, θ(p) changes only finitely often within a finite interval. Continuous time prefix behaviours. T = [0, r] for some r ∈ where [0, r] denotes the set of reals between 0 and r. Let max(T ) = r give the maximum time-point upto which the behaviour is captured. Sampled time infinite behaviours. T has the form {r0, r1, . . .}, the countably infinite set of sampling points where r0 = 0 and r0, r1, . . . forms an unbounded increasing sequence within . These behaviours are also called timed ω-words. Sampled time prefix behaviours. T has the form {r0, r1, . . . , rk}, the finite set of sampling points where r0 = 0 and ri ∈ . Also ri < ri+1. Let max(T ) = rk. These behaviours are also called finite timed words. Discrete time. This is a subclass of sampled time behaviour (infinite or prefix) where all sampling points have integer values. 1.1 Interval Measurement Logic The formulae of interval measurement logic χIML[Σ] are parameterised by a set χ of atomic measurement comparisons. For concreteness, let us fix Punct(Σ) to be the countable set of comparisons m ∼ c, for all m ∈ Σ, ∼ in {<,=, >} and c in Z, the set of integers. Since punctuality is a strong requirement [AFH96], we also define Weak(Σ) to be the subset of weak comparisons made only using the < symbol, and Test(Σ) to be the set of comparisons of the form m = 0. Boolean combinations of the propositional variables Pvar and 0, 1 (denoting false and true respectively) are called propositions, Prop. Let P,Q range over propositions, m ∼ c over comparisons from a set χ and D1, D2 over formulae. The formulae of χIML[Σ] have the syntax P 0 | P | m ∼ c | D1 D2 | D1 D2 | D1 −D2 | D1 ∧D2 | ¬D When we write IML[Σ] we mean that χ is the full set of comparisons Punct(Σ). Semantics of IML. For a proposition P and time point t, θ, t |= P is defined inductively as usual. Let θ, [b, e] |= D denote that the formulaD ∈ IML evaluates to true in the behaviour θ at interval [b, e] ∈ Intv(θ). Omitting the boolean cases, this is defined as follows. θ, [b, e] |= P 0 iff b = e and θ, b |= P θ, [b, e] |= P iff b < e and for all t : b < t < e. θ, t |= P θ, [b, e] |= m ∼ c iff θ(m)[b, e] ∼ c θ, [b, e] |= D1 D2 iff for some z : b ≤ z ≤ e. θ, [b, z] |= D1 and θ, [z, e] |= D2 θ, [b, e] |= D1 D2 iff for some z : e ≤ z. θ, [b, z] |= D1 and θ, [e, z] |= D2 θ, [b, e] |= D1 −D2 iff for some z : z ≤ b. θ, [z, e] |= D1 and θ, [z, b] |= D2 A Dose of Timed Logic, in Guarded Measure 263 Derived operators. Note that 1 0 holds for all point intervals whereas 1 holds for all extended intervals. The formula P def = P 0 P states that P must hold invariantly over the interval, except possibly at the last point. – D def = true D true holds provided D holds for some subinterval. – D def = ¬ ¬D holds provided D holds for all subintervals. – → D def = D true holds provided some forward extension of the interval satisfies D. Symmetrically ← D def = true −D. Validity. As usual D is valid iff for all behaviours θ, θ |= D, where – For prefix behaviours, θ |= D iff θ, [0,max(θ)] |= D – For infinite behaviours, θ |= D iff θ, [0, e] |= D for all e ∈ dom(θ). Example 1. The formula ( P ⇒ ≤ c) states that P can be continuously true for at most c time units. Various sublogics of IML have appeared in the literature. We use different signatures to relate our work to a few of these. (The original versions of some of these logics do not include the modalities D1 D2 and D1 −D2 which were introduced by Venema [Ven90].) Duration calculi. Let the signature Duration(Pvar) = { }∪{P | P ∈ Prop}. The term ∫ P is interpreted to measure the accumulated amount of time for which proposition P is true in an interval. Thus, we obtain the logic PunctIML[Duration(Pvar)]. This logic is called duration calculus, DC, when interpreted over continuous time finitely variable models [ZH04]; interval duration logic, IDL, when interpreted over sampled time prefix models [Pan02]; and DDC when interpreted over integer time prefix models. Mean value calculus. Let the signatureMean(Pvar) = { }∪{P | P ∈ Prop}. The term P is interpreted to measure the mean value of proposition P in an interval [b, e]. The logic Punct-IML[Mean(Pvar)] is the mean value calculus, MVC, interpreted over continuous time finitely variable models [ZL94]. CDT. Consider a signatureΣ of measurement functions without . If we only allow comparisons with zero—that is, χ is Test(Σ)—effectively we are restricting from real-valued measurements to boolean-valued ones. Such a measurement function is nothing but an atomic proposition (such as “did it rain?”) evaluated at every interval. This is an idea which has been long studied by philosophers of time. The corresponding logic Test -IML[Σ] was called CDT [Ven91], as the modalities , −, + are named C, D and T respectively. Interval length logic. On the other hand, we can consider the signature { } without any other measurement functions. The logic Punct-IML[ ] is called interval length logic. As in most real-time logics, it only includes the measurement of time distance using the distinguished function . Interval temporal logic. Finally, the trivial logic IML[∅] with the empty signature is called interval temporal logic, ITL. This logic has been studied over all the classes of models discussed above. 264 K. Lodaya and P.K. Pandya The discrete time logic DDC has been shown to be decidable using an automata-theoretic decision procedure [Pan02]. The general situation is bleaker. Proposition 1. The logic Punct-IML[ ] is undecidable for continuous, finitely variable and sampled behaviours whether infinite or prefix. Test-IML[Σ] is undecidable for infinite time. Proof. As in the undecidability proof for DC [ZH04], for each 2-counter machine M we can define a formula D(M) of Punct-IML[ ] which is satisfiable iff M has a halting run. The nonhalting problem is encoded using a very narrow subset of CDT, with unary modalities definable from , in [Lod00]. 1.2 Guarded Measurement Faced with the strong undecidability results described above, we restrict the logic by permitting only guarded use of measurement formulae. A χ-guarded modality has the syntax G → D | G ← D, where the guard G is a boolean formula over the set of comparisons χ. The meaning of guarded modalities is as follows: θ, [b, e] |= G → D iff b = e and for some z : b ≤ z. θ, [b, z] |= G ∧D θ, [b, e] |= G ← D iff b = e and for some z : z ≤ b. θ, [z, b] |= G ∧D Formally, G → D def = 1 0∧ → (G ∧D) and G ← D def = 1 0∧ ← (G ∧D). Example 2. The formula (¬( ≤ c) → ¬ P ) is Weak -guarded and states that P can be continuously true only for at most c time units. χGIML[Σ] is the sublogic of IML[Σ] where measurements only appear in χguards. Thus, Punct-GIML[Σ] guards use boolean combinations of comparisons from Punct(Σ). If only forward (resp. backward) measurement guards are used, we call the logic FGIML (resp. BGIML). If in the modality G → D of FGIML, we do not allow guarded modalities in D, we get a logic with nesting-free forward guarding, which we denote Punct -FgIML. Guarded modalities exist in the literature, though not in direct fashion. Relative distance. A subset of interval duration logic IML[Duration(Pvar)] where measurements only occur within the guard G of a modality P G (originally due to [Wil94]) has been called LIDL [Pan02]. This logic can be encoded in the backwards guarded logic BGIML[Duration(Pvar)] by encoding the LIDL formula P G as the BGIML formula G ← P 0 ¬P . All the other constructs of LIDL are already available in BGIML. Metric temporal logic. The logic MTL [Koy90] can be encoded in PunctFGIML[ ] as follows (see [Pan96] for details). For every MTL formula φ we define a translation α(φ): Let α(p) = p 0 true. Let BP (D) def = A Dose of Timed Logic, in Guarded Measure 265 ( 1 0∧ → D) true. Then, the constrained until modality of MTL is encoded as follows. α(φ UI ψ) = (I( ) → ¬(true BP (α(φ) true)) BP (α(ψ))) true Here I( ) is the constraint corresponding to the interval I, e.g. for [3, 5) we get ≥ 3 ∧ < 5. It can be shown that for all θ, b, e, φ we have θ, b |=mtl φ iff θ, [b, e] |=iml α(φ). By a variation of this construction, we can model MTL with both past and future modalities in Punct-GIML[ ]. Guarding in first order logic has been an important tool for obtaining decidability. Unfortunately, we can show that in the presence of punctual measurements guarding does not guarantee decidability. MTL with future operators is undecidable over continuous infinite time and MTL with past and future operators is undecidable over sampled prefix time [OW05] and the second author and Vijay Suman have recently shown that LIDL is undecidable over sampled time, prefix or infinite. (However, LIDL with only length measurements is decidable over sampled prefix time [Pan02].) All these logics can be encoded within fragments of GIML giving the following results. Proposition 2. 1. The logic Punct-FGIML[ ] is undecidable for continuous infinite time. 2. The logic Punct-GIML[ ] is undecidable for sampled time. 3. The logic Punct-BGIML[Duration(Pvar)] is undecidable for sampled time. 2 First Order Logics with Measurement In place of interval measurement logic, we can specify a behaviour θ using the first order logic with measurement MFO [Σ̂]. This is the first order logic with equality over the signature Σ̂ = (Pvar, {<}, Σ) where each p ∈ Pvar denotes a monadic predicate. Example 3. The formula ∀x, y. x < y ∧ (∀z. x < z < y ⇒ P (z)) ⇒ (x, y) ≤ c states that P cannot be true continuously for more than c time units. We can associate a classical first order structure θ interpreting Σ̂ with a given behaviour θ. The domain of θ is T with linear order <. For each p ∈ Pvar there is a monadic predicate p(x) which is interpreted as the set θ(p). The functions m ∈ Σ are interpreted as θ(m). The semantics of MFO [Σ̂] is given as usual and omitted here. Proposition 3. There is a bijection (θ, θ) between the IML[Σ] behaviours and the first-order structures interpreting Σ̂. While the monadic theory of linear order MonFO[<] is decidable [LL66], introduction of even the basic measurement Σ = { } makes the logic MFO [Σ̂] 266 K. Lodaya and P.K. Pandya undecidable [ZH04]. Hence we resort to a notion of guarded use of measurements. Let χMGF [Σ] be the measurement-guarded fragment of MFO [Σ̂] which extendsMonFO[<] by the χ-guarded quantifier φ(t0) = ∃t(G(t0, t)∧ψ(t0, t)), where ψ is a formula with at most two free variables t0 and t, and the guard G is a boolean combination of comparisons from the set χ over the signature Σ. Thus the measurement terms appear in a very restricted context. We now translate our interval measurement logics into measurement guarded first order logics. The notation φ(x, y) indicates a formula with at most two free variables x and y. We will use notation such as FO(x, y) to indicate a logic with formulas with at most two free variables x and y. Superscripts as in FO designate k-variable fragments of a logic (now including bound as well as free variables). STz( P )(x, y) def = x = y ∧ P (x) STz( P )(x, y) def = x < y ∧ ∀z(x < z < y ⇒ P (z)) STz(G → D)(x, y) def = x = y ∧ ∃z(ST (G)(y, z) ∧ y ≤ z ∧ STx(D)(y, z)) STz(G ← D)(x, y) def = x = y ∧ ∃z(ST (G)(z, x) ∧ z ≤ x ∧ STy(D)(z, x)) STz(D1 D2)(x, y) def = ∃z(x ≤ z ≤ y ∧ STy(D1)(x, z) ∧ STx(D2)(z, y)) STz(D1 −D2)(x, y) def = ∃z(z ≤ x ∧ STy(D1)(z, x) ∧ STx(D2)(z, y)) STz(D1 D2)(x, y) def = ∃z(y ≤ z ∧ STy(D1)(x, z) ∧ STx(D2)(y, z)) The translation of guards is obvious: ST (m ∼ c)(x, y) = m(x, y) ∼ c. The translation uses the standard trick of reusing variables. Thus STz(D)(x, y) produces a MGF (x, y) formula using at most the variables {x, y, z}. Proposition 4. There is a standard translation from χGIML[Σ] to χMGF [Σ] which has the property that θ, [b, e] |= D iff θ |= STz(D)[b/x, e/y]. 3 Expressive Completeness of GIML for MGF 3 Without loss of generality we assume the logic χMGF 3 consists of formulae with variables x1, x2, x3. In this section, following the proof of Kamp’s theorem [Kamp68] as used by Venema [Ven91], we show that the measurement logic χGIML has the same expressive power as χMGF . The first lemma is routine [GO]. Let Li,j , i = j, i, j ∈ {1, 2, 3}, be the subset of χMGF (xi, xj) consisting of boolean combinations of quantifier-free formulas of MGF 3 and quantified MGF 3 formulas with one free variable in {xi, xj}. Li,j is the same as Lj,i. Lemma 1. Any MGF 3 formula is equivalent to a boolean combination of formulae from L1,2 ∪ L2,3 ∪ L3,1. Now we translate Li,j to GIML. Following [Ven91], we use a forward translation α : Li,j → GIML and a backward translation α− : Li,j → GIML. The boolean cases are routine. We assume the measurement functions are symmetric. A Dose of Timed Logic, in Guarded Measure 267 α(x = x) = true α−(x = x) = true α(xi = xj) = 1 0 α−(xi = xj) = 1 0 α(xi < xj) = ¬ 1 0 α−(xi < xj) = false α(xj < xi) = false α−(xj < xi) = ¬ 1 0 α(x < x) = false α−(x < x) = false α(P (xi)) = P 0 true α−(P (xi)) = true P 0 α(P (xj)) = true P 0 α−(P (xj)) = P 0 true α(m(x, y) ∼ c) = m ∼ c α−(m(x, y) ∼ c) = m ∼ c α(φ1(xi, xj) ∧ φ2(xi, xj)) α−(φ1(xi, xj) ∧ φ2(xi, xj)) = α(φ1(xi, xj) ∧ α(φ2(xi, xj)) = α−(φ1(xi, xj)) ∧ α−(φ2(xi, xj)) α+(¬φ(xi, xj)) = ¬α+(φ(xi, xj)) α−(¬φ(xi, xj)) = ¬α−(φ(xi, xj)) The translation of a quantifier uses the fact that the , − , + modalities cover all cases in which a third time point can be oriented with respect to two points. α+(∃xk. φ1(xi, xk) ∧ φ2(xk, xj)) = α(φ1(xi, xk)) α(φ2(xk, xj)) ∨ α(φ1(xi, xk)) +α−(φ2(xk, xj)) ∨ α−(φ1(xi, xk)) −α+(φ2(xk, xj)) α−(∃xk. φ1(xi, xk) ∧ φ2(xk, xj)) = α−(φ2(xk, xj)) α−(φ1(xi, xk)) ∨ α−(φ2(xk, xj)) α(φ1(xi, xk)) ∨ α(φ2(xk, xj)) −α−(φ1(xi, xk)) The translation of a measurement guarded formula uses the forward and backward guarded modalities to cover the way the quantified variable is oriented with respect to the free variable of the formula. α+(∃xk. G(xi, xk) ∧ ζ(xi, xk)) = [α(G(xi, xk)) → α(ζ(xi, xk)) ∨ α−(G(xi, xk)) ← α−(ζ(xi, xk))] true α−(∃xk. G(xi, xk) ∧ ζ(xi, xk)) = true [α(G(xi, xk)) → α(ζ(xi, xk)) ∨ α−(G(xi, xk)) ← α−(ζ(xi, xk))] α(∃xk. G(xj , xk) ∧ ζ(xj , xk)) = true [α(G(xj , xk)) → α(ζ(xj , xk)) ∨ α−(G(xj , xk)) ← α−(ζ(xj , xk))] α−(∃xk. G(xj , xk) ∧ ζ(xj , xk)) = [α(G(xj , xk)) → α(ζ(xj , xk)) ∨ α−(G(xj , xk)) ← α−(ζ(xj , xk))] true By a careful case analysis over the syntax of Li,j , we can show that the translations α and α− preserve the semantics in the expected way. Lemma 2. For all θ and all [b, e] ∈ Intv(θ), θ, [b, e] |= α(ζ(xi, xj)) iff θ |= ζ[b/xi, e/xj ] and θ, [b, e] |= α−(ζ(xi, xj)) iff θ |= ζ[e/xi, b/xj ]. By combining Lemmas 1 and 2, and observing that the translation above can be parameterised by the set of guards χ, we get the following theorem. Theorem 1. The logic χGIML[Σ] is expressively complete for the three-variable measurement guarded fragment with two free variables χMGF [Σ](x, y). A referee reminded us that our proof works for an even larger family of logics: χIML[Σ] is expressively complete for χFO[Σ](x, y). 268 K. Lodaya and P.K. Pandya 4 Games and the 3-Variable Property Next we would like to show that the full logic χMGF has the three-variable property, that is, χMGF 3 is expressively equivalent to it. To do this, we set up Ehrenfeucht-Fräıssé games for the k-variable guarded fragments, which are an extension of the k-pebble games for FO [IK87]. Our game is played for n rounds by two players, Spoiler and Duplicator, on a board consisting of a pair of structures A and B. Spoiler is trying to distinguish the two structures, Duplicator to hide the distinctions. Each player uses k pebbles for the syntactic restriction to k variables and a measuring tape and meters to check lengths and measurement values. These devices work in integer units. A k-configuration of consists of the positions of the k pebbles on each structure, which we represent by a pair of partial functions (which are defined where the corresponding pebbles are on the board) α : {1..k} → A and β : {1..k} → B. The k-pebble n-round game on structures A,B with k-configurations α, β is denoted Gn(A, α,B, β). Two configurations are said to be order isomorphic if the sequence of pebble positions, seen as linear orders, are order isomorphic. More precisely, Spoiler’s pebble i is on the board on one structure if and only if Duplicator’s pebble i is present on the other structure, and for each pair of pebbles i, j on the board, both structures satisfy the same formulas from the set {i < j, i = j, i > j}. By linearity, they will satisfy exactly one formula from this set. Two configurations are said to be χ-measurement isomorphic if they are order isomorphic and for each pair of pebbles i, j on the board, both structures satisfy the same measurement formulas from the set χ. If α, β are not order isomorphic, Spoiler wins Gn(A, α,B, β) immediately (after 0 rounds). Each round has one of two kinds of moves. In a pebble move, Spoiler can place his pebble i on an element of one of the structures. Duplicator responds by placing her pebble i on an element of the other structure. After the move, if the resulting configurations α′, β′ are not order isomorphic, Spoiler wins the game. In a measuring move, Spoiler removes all pebbles but one, say pebble i (we call this the free pebble of this move), then places another pebble j, using the measuring tape and meters to set its length and other measurement functions to some desired value. Duplicator has to follow suit on the other structure: she removes all pebbles except pebble i, then places her pebble j using the measuring tape and meters. After the move, if the resulting configurations α′, β′ are not measurement isomorphic, Spoiler wins the game. If Spoiler has not won the game after any of the n rounds, Duplicator wins Gn(A, α,B, β). Following [Imm98], we now relate our games to logical types. The proof relies on the fact that the set of measurement formulas {m(i, j) ∼ c | c ∈ Z} satisfied by a configuration is logically equivalent to a finite conjunction of such formulas, since each value is either the point c or inside an open interval (c, c+ 1). A Dose of Timed Logic, in Guarded Measure 269 Theorem 2 (E-F characterization). Given two time frames A,B and a kconfiguration α0, β0 over them, Duplicator wins an n-move game Gn(A, α0,B, β0) if and only if the configurations (A, α0) and (B, β0) are indistinguishable by a χMGF k formula of quantifier depth n. We now show that for time domains, three variables suffice to express all MGF properties. The proof closely follows the (admittedly tricky) one of [Imm98, Theorem 6.32], which combines winning strategies from simpler games. The measuring move does not yield any difficulty since it always reduces the board to 2-configurations for which a winning strategy exists by supposition. Theorem 3 (3-variable property). Every χMGF formula is equivalent to an χMGF 3 formula over time domains. Putting together the 3-variable property with the expressive completeness result of the previous section, we get a proper generalization of Kamp’s theorem. Venema has shown that χFO [Σ] does not have the 3-variable property [Ven90], so the result cannot be extended to full first order logic with measurements. Corollary 1. The logic χGIML[Σ] is expressively complete for χMGF [Σ]. Hirshfeld and Rabinovich conjectured that there is no finite expressively complete temporal logic for a logic L2 which subsumes Weak -MGF [ ] by having a more generous set of comparisons [HR99]. We observe that since our logic uses countably many constants c, it is not finite according to their definition.