Interval temporal logic HS. Point-based Temporal Logics (PTLs), such as the linear-time temporal logic LTL  and the branching-time temporal logics CTL and CTL  provide a standard framework for the specification of the dynamic behavior of reactive systems that makes it possible to describe how a system evolves state-by-state (“point-wise” view). PTLs have been successfully employed in model checking (MC) [16, 36] for the automatic verification of complex finite-state systems modeled as finite propositional Kripke structures. Interval Temporal Logics (ITLs) provide an alternative setting for reasoning about time [21, 33, 39]
. They assume intervals, instead of points, as their primitive temporal entities allowing one to specify temporal properties that involve, e.g., actions with duration, accomplishments, and temporal aggregations, which are inherently “interval-based”, and thus cannot be naturally expressed by PTLs. ITLs find applications in a variety of computer science fields, including artificial intelligence (reasoning about action and change, qualitative reasoning, planning, and natural language processing), theoretical computer science (specification and verification of programs), and temporal and spatio-temporal databases (see, e.g.,[33, 26, 35]).
The most prominent example of ITLs is Halpern and Shoham’s modal logic of time intervals (HS)  which features one modality for each of the 13 possible ordering relations between pairs of intervals (the so-called Allen’s relations ), apart from equality. The satisfiability problem for HS turns out to be highly undecidable for all interesting (classes of) linear orders . The same happens with most of its fragments [14, 25, 29] with some meaningful exceptions like the logic of temporal neighbourhood , over all relevant (classes of) linear orders , and the logic of sub-intervals , over the class of dense linear orders .
Model checking of (finite) Kripke structures against HS has been investigated only recently [26, 27, 28, 30, 31, 7, 8, 5, 10]. The idea is to interpret each finite path of a Kripke structure as an interval, whose labelling is defined on the basis of the labelling of the component states, that is, a proposition letter holds over an interval if and only if it holds over each component state (homogeneity assumption ). Most of the results have been obtained by adopting the so-called state-based semantics : intervals/paths are “forgetful” of the history leading to their starting state, and time branches both in the future and in the past. In this setting, MC of full HS is decidable: the problem is at least Expspace-hard , while the only known upper bound is non-elementary . The known complexity bounds for full HS coincide with those for the linear-time fragment of HS which features modalities and for prefixes and suffixes. These complexity bounds easily transfer to finite satisfiability, that is, satisfiability over finite linear orders, of under the homogeneity assumption. Whether or not these problems can be solved elementarily is a difficult open question. On the other hand, in the state-based setting, the exact complexity of MC for many meaningful (linear-time or branching-time) syntactic fragments of HS, which ranges from to , Pspace, and beyond, has been determined in a series of papers [31, 7, 9, 11, 13, 10].
The expressiveness of HS with the state-based semantics has been studied in , together with other two decidable variants: the computation-tree-based semantics variant and the traces-based one. For the first variant, past is linear: each interval may have several possible futures, but only a unique past. Moreover, past is finite and cumulative, and is never forgotten. The trace-based approach instead relies on a linear-time setting, where the infinite paths (traces) of the given Kripke structure are the main semantic entities. It is known that the computation-tree-based variant of HS is expressively equivalent to finitary CTL (the variant of CTL with quantification over finite paths), while the trace-based variant is equivalent to LTL. The state-based variant is more expressive than the computation-tree-based variant and expressively incomparable with both LTL and CTL. To the best of our knowledge, complexity issues about MC and the satisfiability problem of HS and its syntactic fragments under the trace-based semantics have not been investigated so far.
Parametric extensions of point-based temporal logics. Traditional PTLs such as standard LTL  allow only to express qualitative requirements on the temporal ordering of events. For example, in expressing a typical request-response temporal requirement, it is not possible to specify a bound on the amount of time for which a request is granted. A simple way to overcome this drawback is to consider quantitative extensions of PTLs where temporal modalities are equipped with timing constraints for allowing the specification of constant bounds on the delays among events. A well-known representative of such logics is Metric Temporal Logic (MTL) . However this approach is not practical in the first stages of a design, when not much is known about the system under development, and is useful for designers to use parameters instead of specific constants. Parametric extensions of traditional PTLs, where time bounds can be expressed by means of parameters, have been investigated in many papers. Relevant examples include parametric LTL , Prompt LTL , and parametric MTL .
Our contribution. In this paper we introduce a parametric extension of the interval temporal logic HS under the trace-based semantics, called parametric HS (PHS). The extension is obtained by means of inequality constraints on the temporal modalities of HS which allow to specify parametric lower/upper bounds on the duration (length) of the interval selected by the temporal modality. Similarly to parametric LTL , we impose that a parameter can be exclusively used either as upper bound or as lower bound in the timing constraints. We address the decision problems of checking the existence of a parameter valuation such that (1) a given PHS formula is satisfiable, and (2) a given Kripke structure satisfies a given PHS formula (MC). By adapting the alternating color technique for Prompt LTL  and by exploiting known results on linear-time hybrid logic HL [19, 38, 4], we show that the considered problems are decidable. Additionally, we consider the syntactic fragment of PHS which allows only temporal modalities for the Allen’s relations meets , started-by and its inverse . We show that subsumes parametric LTL, and its flat fragment is exponentially more succinct than LTL + past. Moreover, we establish that satisfiability and MC of are Expspace-complete, and we provide tight bounds on optimal parameter values for both problems.
We fix the following notation. Let be the set of integers, the set of natural numbers, and . Let be an alphabet and be a non-empty finite or infinite word over . We denote by the length of ( if is infinite). For all , with , is the -th letter of , while is the infix of given by .
We fix a finite set AP of atomic propositions. A trace is an infinite word over . For a logic interpreted over traces and a formula , denotes the set of traces satisfying . The satisfiability problem for is checking for a given formula , whether .
Kripke Structures. In the context of model-checking, finite state systems are usually modelled as finite Kripke structures over a finite set AP of atomic propositions which represent predicates over the states of the system. A (finite) Kripke structure over AP is a tuple , where is a finite set of states, is a left-total transition relation, is a labelling function assigning to each state the set of propositions that hold over it, and is the initial state. An infinite path of is an infinite word over such that and for all . A finite path of is a non-empty infix of some infinite path of . An infinite path induces the trace given by . We denote by the set of traces associated with the infinite paths of . Given a logic interpreted over traces, the (linear-time) model checking problem against is checking for a given Kripke structure and a formula , whether .
Büchi nondeterministic automata. A Büchi nondeterministic finite automaton over infinite words (Büchi NFA for short) is a tuple , where is a finite input alphabet, is a finite set of states, is the initial state, is the transition relation, and is a set of accepting states. Given an infinite word over , a run of over is a an infinite sequence of states such that and for all . The run is accepting if for infinitely many , . The language accepted by is the set of infinite words over such that there is an accepting run of over .
2.1 Allen’s relations and Interval Temporal Logic Hs
An interval algebra to reason about intervals and their relative orders was proposed by Allen in , while a systematic logical study of interval representation and reasoning was done a few years later by Halpern and Shoham, who introduced the interval temporal logic HS featuring one modality for each Allen relation, but equality .
Let be a linear order over the nonempty set , and be the reflexive closure of . Given two elements such that , we denote by the (non-empty closed) interval over given by the set of elements such that and . We denote the set of all intervals over by . We now recall the Allen’s relations over intervals of the linear order :
the meet relation , defined by if (i.e., the start-point of the second interval coincides with the end-point of the first interval);
the before relation , defined by if (i.e., the start-point of the second interval strictly follows the end-point of the first interval);
the started-by relation , defined by if and (i.e., the second interval is a proper prefix of the first interval);
the finished-by relation , defined by if and (i.e., the second interval is a proper suffix of the first interval);
the contains relation , defined by if and (i.e., the second interval is contained in the internal of the first interval);
the overlaps relation , defined by if (i.e., the second interval overlaps at the right the first interval);
for each the relation , defined as the inverse of , i.e. if .
Table 1 gives a graphical representation of the Allen’s relations , , , , , and together with the corresponding HS (existential) modalities.
|Allen relation||HS||Definition w.r.t. interval structures||Example|
Syntax and semantics of HS. HS formulas over AP are defined as follows:
where and is the existential temporal modality for the (non-trivial) Allen’s relation , where . The size of a formula is the number of distinct subformulas of . We also exploit the standard logical connectives (disjunction) and (implication) as abbreviations, and for any temporal modality , the dual universal modality defined as: . Moreover, we will also use the reflexive closure of the Allen’s relation (resp., ) and the associated temporal modalities and (resp., and ) where corresponds to and corresponds to . Given any subset of Allen’s relations , we denote by the HS fragment featuring temporal modalities for only.
The logic HS is interpreted on interval structures , which are linear orders equipped with a labelling function assigning to each interval the set of propositions that hold over it. Given an HS formula and an interval , the satisfaction relation , meaning that holds at the interval of , is inductively defined as follows (we omit the semantics of the Boolean connectives which is standard):
It is worth noting that we assume the non-strict semantics of HS, which admits intervals consisting of a single point. Under such an assumption, all HS-temporal modalities can be expressed in terms of , and (see ). As an example, can be expressed in terms of and as , while can be expressed in terms of and as
Interpretation of HS over traces. In this paper, we focus on interval structures over the standard linear order on (-interval structures for short) satisfying the homogeneity principle: a proposition holds over an interval if and only if it holds over all its subintervals. Formally, is homogeneous if for every interval over and every , it holds that if and only if for every . Note that homogeneous -interval structures over AP correspond to traces where, intuitively, each interval is mapped to an infix of the trace. Formally, each trace induces the homogeneous -interval structure whose labeling function is defined as follows: for all with and , if and only if for all . For the given finite set AP of atomic propositions, this mapping from traces to homogeneous -interval structures is evidently a bijection. For a trace , an interval over , and an HS formula , we write to mean that . The trace satisfies , written , if .
Expressiveness completeness and succinctness of the fragment over traces. It is known that HS over traces has the same expressiveness as standard LTL , where the latter is expressively complete for standard first-order logic FO over traces . In particular, the fragment of HS is sufficient for capturing full LTL : given an LTL formula, one can construct in linear-time an equivalent formula . Note that when interpreted on infinite words , modality allows to select proper non-empty prefixes of the current infix subword of , while modality allows to select subwords whose first position coincides with the last position of the current interval. Here, we show that is exponentially more succinct than LTL + past. For each , we denote by the formula capturing the intervals of length : .
For each , let and be the -language consisting of the infinite words over such that any two positions that agree on the truth value of propositions also agree on the truth value of . It is known that any Büchi NFA accepting needs at least states . Thus, since any formula of LTL + past can be translated into an equivalent Büchi NFA with a single exponential blow-up, it follows that any formula of LTL + past capturing has size at least single exponential in . On the other hand, the language is captured by the following formula having size linear in :
Hence, we obtain the following result.
(over traces) is exponentially more succinct than LTL + past.
3 Parametric Interval Temporal Logic
In this section, we introduce a parametric extension of the interval temporal logic HS over traces, called parametric HS (PHS for short). The extension is obtained by means of inequality constraints on the temporal modalities of HS which allow to compare the length of the interval selected by the temporal modality with an integer parameter. Like parametric LTL , the parameterized operators are monotone (either upward or downward) and a parameter is upward (resp., downward) if it is the subscript of some upward (resp., downward) modality.
Syntax and semantics of PHS Let be a finite set of upward parameter variables and be a finite set of downward parameter variables such that and are disjunct. The syntax of PHS formulas over AP and the set of parameter variables is given in positive normal form as follows:
where , , , , , and . We denote by PromptHS the fragment of PHS where the unique parameterized temporal modalities are of the form . Moreover, given any subset of Allen’s relations , we denote by (resp., ) the PHS (resp., PromptHS) fragment featuring temporal modalities for only. We will focus on PHS and the fragment .
For an interval over , we denote by the length of , given by . The semantics of a PHS formula is inductively defined with respect to a trace , an interval over , and a parameter valuation assigning to each parameter variable a positive integer. We write to mean that holds at the interval of under the valuation . The interpretation of all temporal operators of HS and connectives is identical to their HS interpretations. The parameterized operators are interpreted as follows, where and :
We say that the trace is a model of formula under the parameter valuation , written , if . For a PHS formula and a Kripke structure over AP, we consider:
the set consisting of the parameter valuations such that for each trace of , , and
the set consisting of the valuations such that for some trace .
The (linear-time) model-checking problem against PHS is checking for a given Kripke structure and PHS formula whether . The satisfiability problem against PHS is checking for a given PHS formula whether .
Given two valuations and , we write to mean that for all . A parameterized operator is upward-monotone (resp., downward-monotone) if for all formulas , valuations and such that , entails that (resp., entails that ). By construction, all the parameterized operators are monotone. In particular, being and disjunct, by increasing (resp., decreasing) the values of upward (resp., downward) parameters, the satisfaction relation is preserved.
The operators in PHS parameterized by variables in are upward-closed, while those parameterized by variables in are downward-closed.
Let be a PHS formula and let and be variable valuations satisfying for every and for every . Then entails that .
Note that if we also allow for all and , the parameterized modalities , , , and , then the modalities and , for and are dual and have opposite kind of monotonicity. It easily follows that the logic is indeed closed under negation.
Given a PHS formula with upward (resp., downward) parameters in (resp., ), one can construct in linear time a PHS formula with upward (resp., downward) parameters in (resp., ) corresponding to the negation of , i.e. such that for each parameter valuation and trace over , iff .
We now show that parametric LTL (PLTL)  can be easily expressed in . Recall that PLTL formulas over AP and the set of parameters are defined as:
where , , , X, U, and G are the standard next, until, and always modalities, respectively, and and are parameterized versions of the always and eventually modalities. Other parameterized modalities such as or can be easily expressed in the considered logic . For a PLTL formula , a trace , a parameter valuation , and a position , the satisfaction relation is defined by induction as follows (we omit the semantics of LTL constructs which is standard):
For a PLTL formula , one can build in linear time a formula such that for all traces , , and parameter valuations , iff .
The mapping , homomorphic with respect to atomic propositions and Boolean connectives, is defined as follows:
Note that by Proposition 3 and the results in , the relaxation of the assumption or the adding of parameterized operators of the form would lead to an undecidable model-checking problem already for the parameterized extension of by just one parameter.
Expressively complete fragments. Two PHS formulas and are strongly equivalent, denoted by , if for all traces, intervals over , and parameter valuations , we have that iff . We show that the fragment consisting of formulas with no occurrences of parameterized operators is sufficient to capture the full logic PHS.
Given a PHS formula , one can build in linear time a strongly equivalent formula with no occurrences of the parameterized operators .
We first show that the fragment is expressively complete for PHS. The strong equivalences exploited for expressing all the HS modalities in terms of the modalities in the fragment can be trivially adapted to the parameterized setting. Here, we illustrate the equivalences for the existential parameterized operators where and :
It remains to show that for the fragment , the universal upward parameterized operators can be expressed in terms of the other modalities. One can easily show that the following strong equivalences hold, where is (resp., is ) and is (resp., is ). Hence, the result follows.
For the logic , we obtain a similar result.
Given a formula , one can build in linear time a strongly equivalent formula with no occurrences of the parameterized operators .
The result directly follows from the strong equivalences provided in the proof of Proposition 4 and the following one, where is (resp., is ) and is (resp., is ): ∎
By the monotonicity of the parameterized modalities and Propositions 4 and 5, we can eliminate all the parameterized modalities, but the existential upward ones, for solving the model-checking and satisfiability problems against PHS (resp., ).
Model checking PHS (resp., ) can be reduced in linear time to model checking PromptHS (resp., ). Similarly, satisfiability of PHS (resp., ) can be reduced in linear time to satisfiability of PromptHS (resp., ).
Let be a PHS (resp., ) formula. By Propositions 4 and 5, we can assume that does not contain occurrences of parameterized operators of the form . Let be the PromptHS (resp., ) formula intuitively obtained from by replacing each occurrence of a downward parameter with the constant . Formally, is homomorphic w.r.t. all the constructs but the downward parameterized modalities and:
As for the model checking problem, we show that iff for each Kripke structure . Let be a parameter valuation such that for each downward parameter . By construction, for all traces , iff . Hence, implies that . On the other hand, if , there is a parameter valuation such that for each trace of , . Let be defined as: for each , and for each . By Proposition 1, it follows that for each trace of , . Thus, we obtain that as well, and the result for the model-checking problem follows. The result for the satisfiability problem is similar. ∎
4 Decision procedures for Phs
In this section, we first provide a translation of HS formulas into equivalent Büchi NFA (asymptotically optimal for formulas), by exploiting as an intermediate step a translation of HS formulas into equivalent formulas of linear-time hybrid logic HL [19, 38, 4] (Subsection 4.1). Then, in Subsection 4.2, we apply the results of Subsection 4.1 and the alternating color technique for Prompt LTL  in order to solve satisfiability and model checking against PHS and . In particular, for the logic , we show that the considered problems are Expspace-complete.
4.1 Translation of Hs in linear-time Hybrid Logic
In this section, we recall the linear-time hybrid logic HL [19, 38, 4], which extends standard LTL + past by first-order concepts. We show that while HS can be translated into the two-variable fragment of HL, for the logic , it suffices to consider the one-variable fragment of HL. Thus, by exploiting known results on [38, 4], we obtain an asymptotically optimal automata-theoretic approach for of elementary complexity.
Syntax and semantics of HL. Given a set of (position) variables, the set of HL formulas over AP and is defined by the following syntax:
, , Y and P are the past counterparts of the next modality X and the eventually modality F, respectively, and is the downarrow binder operator which assigns the variable name to the current position. We denote by (resp., ) the one-variable (resp., two-variable) fragment of HL. An HL sentence is a formula where each variable is not free (i.e., occurs in the scope of a binder modality ). The size of an HL formula is the number of distinct subformulas of .
HL is interpreted over traces . A valuation is a mapping assigning to each variable a position . The satisfaction relation , meaning that holds at position along w.r.t. the valuation , is inductively defined as follows (we omit the semantics of LTL constructs which is standard):
where and for . Thus, binds the variable to the current position. Note that the satisfaction relation depends only on the values assigned to the variables occurring free in the given formula . We write to mean that , where maps each variable to position 0, and to mean that . Note that HL formulas can be trivially translated into equivalent formulas of first-order logic FO over traces and LTL formulas can be trivially translated into equivalent HL formulas. Thus, by the first-order expressiveness completeness of LTL, HL and LTL have the same expressiveness .
Translation of HS into HL. We establish the following result.
Given an HS (resp., ) formula , one can construct in linear-time a two-variable (resp., one-variable) sentence HL such that .
We first consider full HS. We can restrict ourselves to consider the fragment of HS since all temporal modalities in HS can be expressed in by a linear-time translation. Fix two distinct variables and . We define a mapping assigning to each formula a formula with variables and which occur free in . Intuitively, in the translation, and refer to the left and right endpoints of the current interval in , while the current position corresponds to the left endpoint of the current interval. Formally, the mapping is homomorphic w.r.t. the Boolean connectives and is inductively defined as follows: