    A two-dimensional metric temporal logic

We introduce a two-dimensional metric (interval) temporal logic whose internal and external time flows are dense linear orderings. We provide a suitable semantics and a sequent calculus with axioms for equality and extralogical axioms. Then we prove completeness and a semantic partial cut-elimination theorem down to formulas of a certain type.

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1 Introduction

In recent years, metric temporal logics gained popularity because of their applicative aspects, for instance those relative to the formalization of time-critical systems. At the same time, under the paradigm of time granularity, there appeared in the literature a number of proposals on how to combine temporal logics into -dimensional systems (very often with ). See, for instance, [4, 5, 11]. To the best of our knowledge, proof-theoretic properties of those multi-dimensional systems have not been investigated yet.

In computer science, multi-dimensional temporal structures and the corresponding logics are often used. For instance, there are temporal structures modelling computations, with internal and external temporal quantifications ranging on states and computations respectively. There are also the structures for time granularity, which come equipped with an internal time flow (for each of the so called time grains) and an external one (for the whole structure of time grains). In this paper we took inspiration from the latter structures to develop a corresponding two-dimensional metric temporal logic, where both time flows are dense linear orderings. First of all we establish completeness of the proposed system with respect to a suitable class of structures. Secondly we get a semantic proof of cut-elimination as a byproduct of the technique used to establish completeness. More specifically, regarding cut-elimination, we aim at a semantic partial cut-elimination result in the vein of [7, Theorem 2.7.1]. We get our result by using semantic techniques, as done in [7, 3.1.9]. We show that every provable sequent has a proof whose cut formulas are among those occurring in some extralogical axiom.

We recall that, in presence of extralogical axioms, a full cut-elimination result that preserves the subformula property does not hold in general. See [7, Remark 2.7.7 ] for an example.

For simplicity, in the following we will simply refer to our result as to cut-elimination.

In  we introduced an infinitary extension of a fragment of the Metric Interval Temporal Logic over dense time domains, called We recall that Metric Interval Temporal Logic was introduced in  as a fragment of Metric Temporal Logic in which temporal operators may “quantify” over nonsingular intervals only. More precisely, the logic is a modification and an extension of a system proposed in . Same as with the latter system, only quantifications over nonsingular intervals are allowed in . Differently from the system in , an induction schema is provable in thanks to the presence of some infinitary rules and axioms. Moreover, applications of the cut rule can be restricted to very simple formulas (see [2, Theorem 3.4]). In there are relational formulas that describe properties of the underlying time flow and labelled formulas that express temporal statements. Regarding labelled deductive systems, we refer the reader to [6, 12].

In binary propositional connectives only apply to formulas of the same kind and the deduction rules deal separately (as much as possible) with the relational and the labelled component of the deductive system.

Apart from recovering an induction schema and retaining a suitable cut-elimination property, a motivation for studying was the remark made in  that, differently from the fully developed model checking techniques, proof-theoretic investigations of Metric Temporal Logic had been only partly attempted until that time.

Although familiarity with the content of  might be helpful, we will keep this paper as much self-contained as possible. We will refer to  to avoid unnecessary repetitions or for comments and further motivations for introducing .

In this paper we aim at temporalizing More precisely, but still vaguely, we want to put a copy of on top of itself and investigate the properties of the resulting two-dimensional system, to be called The semantic intuition is that there are two kinds of time flows, an internal one and an external one, each being a dense linear ordering with least but no greatest element. At each point of the external flow it corresponds a copy of the internal time flow. Roughly speaking, each copy of the internal time flow and the external one will be governed by the logic of . Temporal operators do quantify over (possibly unbounded) intervals in the same way as in

It is worth mentioning that the temporalization of certain systems has been studied in , where the authors investigate the so called external way of temporalizing a logic system. In the external approach it is not necessary to have detailed knowledge about the components of the system. We anticipate that, differently from , we will need a detailed knowledge of the components in order to establish cut elimination for the resulting system.

Various techniques for combining temporal logic systems have been developed also in . In that approach, constraints need to be specified in order to ensure that properties of the combining logics are retained by the combined system. More precisely, in  the authors investigate the transfer of soundness, completeness and decidability from the components to the system they generate. We point out that they do not establish any proof-theoretic property of the resulting systems. In this paper, in addition to proving soundness and completeness of with respect to adequate semantics and sequent calculus, we will also get a semantic proof of a partial cut-elimination theorem.

We also point out that a two-dimensional system can be used to describe a temporal logic for time granularity. For such a logic can be regarded as a combination of simpler temporal logics in a way that properties of the resulting system can be derived from those of its components. For a survey on temporal logics for time granularity, we refer the reader to .

Eventually we recall that, in the literature, one can find general methods for obtaining a system that admits full cut-elimination, starting from a system with axioms. They are based on the replacement of axioms with rules. See the method introduced in  and subsequently developed by the same authors in other works. In our opinion, the technique proposed in  is a clever way of hiding the cuts within the added rules. The reader should bear in mind that, differently from our work,  originates from purely proof-theoretical motivations. A stronger motivation for dealing with a system which is not completely cut-free is that application of the technique in  to a concrete deduction system with axioms like ours results in the replacement of naturally formulated axioms (whose intuitive meaning is clearly understood) with rules which are far less intuitive. In our opinion, the latter phenomenon is not a point in favor of a potential usability of the resulting system.

The rest of the paper is organized as follows.

In Section 2, we introduce syntax and semantics of

In Section 3 we introduce the sequent calculus. We also point out that two induction schemas (one for each kind of time flow) are provable in

In Section 5 we establish completeness and we get cut-elimination, by suitably modifying the reduction tree construction technique of predicate logic. Differently from , we do not make use of any first order translation of the temporal formulas, but we directly construct the reduction tree of a sequent by means of the deduction rules.

Eventually, by applying a result in , we get completeness of with respect to the family of structures where the underlying time flows are given by the nonnegative rational numbers and the time intervals are determined by a very simple function.

2 Syntax and semantics

We recall that the logic is an extension of first order logic where countable conjunctions and disjunctions are allowed (see ).

As is usual we identify a language with its extra logical symbols. We write as an abbreviation for “ is an -formula”. Similarly with -terms.

We denote the set of natural numbers by

2.1 Syntax

The language of is formed by an internal language and an external language Each of these two languages comprises a relational component and a labelled component where We begin by introducing the two components of and the corresponding formulas.

• is a first order language with equality and , , are a unary function, a binary relation and a constant symbol respectively. We have one additional connective for (countable) infinitary disjunction. We use lower case letters (possibly indexed) for the (meta)variables of We denote by the set of -variables.

We denote by the set of -terms. We use letters (possibly indexed) for the -terms. The set of -formulas (to be called internal relational formulas) is the least set containing:

1. the finitary -formulas;

2. all the subformulas (in the sense of the logic) of the infinitary formulas of the form Hence the set of subformulas of contains the formula itself, and for all relational terms and all (Here and in the following we let for all terms )

• is a propositional language with a set of propositional letters; the propositional connectives and, for all in the temporal operators

 □[m,n] □]m,n] □[m,n[□]m,n[ □[m,∞[ □]m,∞[ .

The semantics of the temporal operators will be introduced in Section 2.2. We write to denote any of the above operators, where and

The set of -formulas (to be called internal temporal formulas) is the least set that contains the proposition symbols and is closed under application of the propositional connectives and under the following formation rules: if then is in where is any of the temporal operators.

The set of internal labelled formulas is the set of all expressions of the form where and

What we have introduced so far is basically the syntax of Logic has two kinds of formulas: the relational and the labelled ones. To sum up: the formulas in are those of the propositional metric temporal logic underlying A formula can be decorated by an internal label (an element of ) to form the internal labelled formula Once has been formed, it cannot be used to construct a more complicated formula. By this we mean that, for instance, the conjunction of two internal labelled formulas is not an internal labelled formula, but it will be a formula at the second level of the temporalization, as we explain below.

As we said already, is obtained by putting a copy of on top itself. Therefore we define (“e” for “external”) as above, by using the same symbols, this time in upper case, for the extralogical symbols and the variables. There will be no ambiguity in using for both the internal and the external predicate symbols.

Let be the the set obtained by closing under application of (external) propositional connectives and temporal operators. The intuition is that the formulas in play the role of the atomic temporal formulas for the external logic and is thus the set of all the external temporal formulas. The set of external labelled formulas is the set of all expressions of the form where and The definition of agrees with the spirit of where the relational and the labelled formulas are not “mixed”. We will use upper case letters (possibly indexed) to denote formulas in We say that a formula in is atomic if the formula is atomic.

Here is a simple example: let be an external term and be internal labelled formulas. As we said above, the expression is not an internal labelled formula, but is an external labelled one. Even if we have not defined the semantics yet, to help the reader’s intuition we anticipate that the latter formula will be true in a structure at the external time instant “” if, in the internal time flow corresponding to “”, formulas and are true at time instants “” and “” respectively.

In the following we will deal with sequents. As is usual, a sequent is a formal expression of the form for some where and are lists of formulas in

2.2 Semantics

The notation introduced in section 2.1 is in force. In the following we use standard model-theoretic notation (see, for instance, ).

A pre-structure is a a tuple of the form such that:

• where a dense linear ordering with least element 0 but no greatest element and a strictly increasing function such that, for all and the sequence is cofinal in (namely, for all there exists such that );

• for each is a first-order structure that satisfies analogous properties to those of

For notational simplicity, we will write for

In order to assign a truth value to a formula (internal or external) we fix

1. an assignment where we denote by the set of -variables;

2. a family of assignments where where we denote by the set of -variables;

3. a family of subsets of for each The set is intended to be the set of points in where the propositional letter is true.

We write to denote the interpretation of in under Let We denote by the assignment that differs from at most on the variable on which it takes value

The truth value of an -formula in under is given by its first-order semantics.

Let and let We fix Let We recursively define as follows.

1.  if

2. the cases relative to the propositional connectives are treated as expected;

3. if, for all such that it holds that

4. if, for all such that it holds that

(The cases relative to the other temporal operators are similar.)

We let and, with a little abuse of the model-theoretic terminology, we call a structure. We say that is based on the pre-structure

As with the internal relational formulas, the truth value of an -formula in under is given by by its first-order semantics. Relative to such an we will often write instead of since there is no ambiguity.

Relative to we will use the same notation previously introduced for Let and let be the interpretation of in under

If then if

Next we say when for Basically we will repeat (with the necessary changes) the definition that we gave in the case of the internal formulas. We provide some details for sake of clarity. Recall that the “atomic” formulas in are those of the form where and

• if

• the propositional cases are treated as expected;

• if, for all such that it holds that

(The other cases are similar.)

We say that a sequent is true, or satisfied, in a structure (and we write ) if does not satisfy some of the formulas in or satisfies some of the formulas in A sequent is valid if it is true in all structures.

3 Sequent calculus

In this section we introduce the axioms and the rules of Most of the axioms are formulated as Post rules.

In the following we first introduce some axiom schemas, then we will consider instances of those schemas. We assume that the reader can easily figure out what we mean by an instance of some axiom schema. For instance, for all in the sequent is an instance of the axiom schema .

The reason for dealing with instances rather than with schemas is to obtain a suitable cut-elimination theorem. The reader who is not interested in proof-theoretic issues may opt for axiom schemas.

Axioms for equality

Among the axioms for equality are all instances of the following axiom schemas.

 ⊢X=X;X=Y,ρ[X/Z]⊢ρ[Y/Z](ρ\ an % atomic formula in\ Fe,r)
 ⊢T:x=x;T:x=y,T:ρ[x/z]⊢T:ρ[y/z]

In addition, we have the following equality axioms:

 T=S, T:η⊢S:η(η an atomic formula in ¯¯¯¯Fi,l∪Fi,r)

As it can be immediately realized, the axiom schemas on the second line are obtained from those on the first one by using internal variables/symbols and by “decorating” with a same external label all the formulas involved. Loosely speaking, we will say that the latter schemas are the internal formulations of the former. Later we will adopt the same terminology relative to the rules.

Extralogical axioms

With the extralogical axioms we axiomatize the class of structures under consideration (see Section 2.2).

We start with external extralogical axiom schemas stating that is a dense linear ordering with least element :

 X
 ⊢C

We do not include an axiom stating that there is no greatest element, because the latter property can be derived from the leftmost axiom of the following list.

The external extralogical axioms also express the properties of the function symbol and of the sequence :

 ⊢X

The external extralogical axioms are all instances of the above schemas.

As with the axioms for equality, we have internal versions of all the external extralogical axioms. We leave to the reader the easy task of formulating them.

In the following we refer to [7, Chapter 2] for a formulation of the rules of first-order sequent calculus.

Identity axiom and cut rule

They are:

 A⊢Aidtensy\vboxspread0.0pt\penalty1Γ⊢A,Δ   Γ,A⊢Δ% \hbox{\kern 0.0pt\vrule height 0.25pt depth 0.25pt width 89.998856pt% \hbox{\kern 3.0ptcut}}\kern 33.749599pt\hbox{Γ⊢Δ}

respectively. They apply to any formula

Structural rules

The structural rules are those of Weakening, Exchange and Contraction from sequent calculus. They apply to any formula in

Rules for relational formulas

The rules for the external relational formulas are those of the first-order sequent calculus for the finitary connectives and the quantifiers. In addition, we have the following rules for the infinitary disjunction:

We also have internal versions of all the rules above for the labelled external formulas of the form where and For sake of clarity, we formulate the rules concerning the infinitary disjunction:

Propositional rules

The propositional rules for the external labelled formulas closely follow the standard propositional rules. As an example, we give the the rules for implication:

Notice the constraint on the application of the two rules above, which is a consequence of the constraint that we put on the formation of the set of external labelled formulas. As we already said, we do not want to “mix up” the relational and temporal components of the system.

We also have internal versions of all the propositional rules for the external labelled formulas. For instance, the internal versions of the last two rules are the following:

Temporal rules

The rules for temporal operators are essentially the same as in . As above, we have a “duplication” of the rules to take into account the internal and the external level of the deductive system. For instance, the rules governing are:

The rules for the other temporal operators are similar. They can be easily derived from the rules above.

Proofs are standardly defined. Notice that, although each branch in a proof is finite, the proof itself may have infinite height because of the infinitary rules.

It may help the reader’s intuition to keep in mind that, from the point of view of the internal logic, the atomic temporal formulas are those in the set of propositional letters. The latter set gets closed under (internal) boolean connectives and (internal) modal operators to form the set of internal temporal formulas.

From the point of view of the external logic, the formulas in the set of internal labelled formulas play the role of atomic temporal formulas. By taking the closure of under (external) boolean connectives and (external) modal operators we get the external temporal formulas.

Forgetting about the labels, the deduction rules governing are those of propositional sequent calculus and those for the (internal) temporal operators. Similarly, the deduction rules governing are those of propositional sequent calculus and those for the (external) temporal operators. Basically, we have two copies of the same sequent calculus, one for each level.

Each relational component is governed by the deduction rules of the first-order sequent calculus, plus the corresponding infinitary rule for disjunction.

As is usual, we denote by the provability relation in

We end this section by remarking that two induction schemas are provable in More precisely, let and Then

 ⊢T:(α∧□[0,∞[(α→□[0,1]α))→□[0,∞[α(1)

is provable in Up to typographic changes, the proof is the same as the induction schema in . In a similar way, for every and we have

 ⊢T:t:(η∧□[0,∞[(η→□[0,1]η))→□[0,∞[η(2)

We refer the reader to  for a remark on the unprovability of (1) when the infinitary axiom gets removed. The example given therein can be easily adapted to show the unprovability of (2) without the axiom

4 Underlying unlabelled systems

Some labelled logics are built on top of unlabelled systems. Even if we have deliberately chosen not to describe in full detail a possible unlabelled system underlying in this subsection we provide some insight about the semantics of the latter. We are reluctant to call the resulting object a logic. For it should be clear that, for the purposes of this paper, a logic is made of two components: a semantic and a deductive one. Indeed the reader can easily convince himself that deduction rules for the temporal operators corresponding to those introduced in Section 3 cannot be formulated in absence of labels, even in case of the one-dimensional system introduced in .

For sake of simplicity, let us consider The formation rules for the underlying unlabelled formulas are those of propositional first-order logic expanded with those for the temporal operators with and More precisely: if is a formula then so is for all possible choices of the operator

The pre-structures which are adequate to interpret those unlabelled formulas are of the form where is a dense linear ordering with least element 0 but no greatest element and a strictly increasing function such that, for all and the sequence is cofinal in In order to get a structure we must provide a family of subsets of . We let

 M=(M,(pM)p∈P) (1)

The semantics is Kripke-like. Let be as in (1) and The definition of is by induction on We skip the the details (see Section 2 or  for insights).

We stipulate that an unlabelled formula is valid if, for all as in (1) and all

We make the trivial observation that, for as (1) and by letting and by denoting with the same names the restrictions of by and to respectively, the tuple together with the family forms a structure as in (1). It follows that validity of is equivalent to the following:

 for all M as in (???),M,0⊨α.

Eventually, we notice that the latter is equivalent to the validity of the labelled formula with respect to the semantics. By soundness and completeness of (see [2, Theorem 3.4]), validity of the unlabelled formula is equivalent to the provability in of the labelled sequent

Therefore we may use the calculus to investigate validity in the underlying unlabelled system. Admittedly, this is of limited interest because of the above mentioned fact that the unlabelled system lacks deduction rules for its temporal operators. As such, it is not a logic whose validity or provability relation we may want to investigate within another logic.

In light of soundness and completeness (see Corollary 5.2), we may proceed as above in the case of In this latter case, the underlying system should have two families of temporal operators and with and and with the obvious intuitive meaning of the superscripts i and e. We should also impose, among others, syntactic restrictions on the formation rules of formulas. For instance, no can occur on the right-hand side of a operator. The structures which are adequate to interpret the resulting class of formulas can be easily defined, by closely following Section 2.2.

As in the case of it is possibile to relate validity in the unlabelled system to validity with respect to the semantics. We skip the details and we provide a simple example: validity of the unlabelled formula turns out to be equivalent to the validity of the sequent

The same considerations as in the one-dimensional case apply. Unfortunately, even in this case the unlabelled system lacks deduction rules for the temporal operators.

5 Reduction tree construction and completeness

At the end of Section 2.2 we already said when a sequent is valid. It is a lengthy but easy task to verify that all the axioms are valid and that every provable sequence is valid. To prove the latter, proceed by induction on a derivation of a provable sequent.

We aim at proving that is complete with respect to the class of structures that we introduced in Section 2.2.

5.1 The reduction tree

In this section we suitably modify the reduction tree construction technique of first order logic (see, for instance, Theorem 3.1.9, Remarks 3.1.11 and 3.1.12 in [7, §3.1]). Our aim is to get either a proof in of a sequent or a tree with an infinite branch whose root is labelled In Section 5.2 we will derive the unprovability of from the existence of such an infinite branch.

For simplicity we identify a node in a tree with its associated label.

We fix a sequent We also fix

1. an ordering of all the formulas in (recall that all the ’s belong to );

2. an enumeration of all axioms different from id, in which each axiom is repeated infinitely many times;

3. an enumeration of the set of -terms;

4. an enumeration of the set of -terms.

Next we recursively define a sequence of well founded trees. We say that a branch in is closed if its leaf is an axiom (of any kind), otherwise it is called open. It is intended that the following construction only applies to the open branches in a tree. Therefore, if for some all branches in are closed, then for all

basis

is .

recursion

Assume that the trees have been already defined in a way that, for all is an upwards extension of If all the branches in are closed, we stop. If not, we define by extending each of the open branches in upwards. With each open branch we proceed as follows.

Let be the leaf of an open branch in  The formulas in come equipped with an ordering that has been determined by the previous steps of the construction (see cases 1–3 below).

We examine various cases, depending on the first formula in the ordering.

In the following, we write if the sequent can be proved from the sequent by application of finitely many structural rules.

1. is provable by means of structural rules only from some axiom (of any kind), say from . In this case we extend upwards with the corresponding bit of derivation.

Notice that the branch so obtained is closed. Therefore it will not be further extended.

2. Case 1 does not happen and is atomic.

In this case we examine the -th element in the enumeration of the axioms. Let be such element.

If all occur in we extend upwards as follows:

For each the ordering of the formulas in the leaf is

If some among does not occur in we do not add any node to the current branch and we re-order the formulas in as follows:

3. Case 1 does not happen and is not atomic. We consider several cases and subcases (one for each logical rule of the calculus). In the following we deal with few cases only. We invite the reader to work out the missing cases.

1. occurs in

1. is We extend upwards as follows:

The new orderings of the formulas in the left and the right leaf are and respectively.

2. is Let be the first term in the enumeration that has not been used in the current branch in a previous application of this case to the formula . We extend upwards as follows

The orderings of the formulas in the new leaves are (from left to right):

 A2,…,Aq,T:fi(t)≤s,A1A2,…,Aq,T:s≤fj(t),A1

and

3. is We extend upwards as follows

The ordering of the formulas in the leaf is for all

2. occurs in

1. if is We extend upwards as follows

The ordering of the formulas in the new leaf is:

2. is We pick an internal variable that does not occur free in and we extend upwards as follows

The ordering of the formulas in the new leaf is

3. is Let be the least natural number such that does not occur in We extend upwards as follows

The ordering of the formulas in the new leaf is

Remark 5.1.

By construction, is a tree, called the reduction tree of Notice that the presence of axioms with empty premiss and the fact that each axiom different from id is repeated infinitely many times in the enumeration imply that any open branch keeps on being extended upwards. Therefore if has no infinite branch there exists some such that all the branches in the tree are closed. It follows immediately from the construction that yields a proof Since all applications of the cut rule take place in case 2, we actually get a proof of where all cut-formulas occur in the righthand side of some axiom different from id.

If there is some infinite branch in in Section 5.2 we show how to get a countermodel of from such an infinite branch.

5.2 Completeness

Let us assume that the reduction tree of the sequent constructed in Section 5.1 has an infinite branch Let and

We define a structure by using and

First we define We define a relation on the set of external terms by letting

 S∼T ⇔ (S=T)∈¯¯¯¯Γ.

The tree construction and the first order axioms for equality imply that is an equivalence relation. Let be the -equivalence class of term We let

 M={T∼: T∈Te}; FM(T∼)=F(T)∼; CM=C∼; S∼

We let (Notice the small notational abuse.) The axioms for equality and the reduction tree construction imply that everything is well-defined. Furthermore, the extralogical axioms and the reduction tree construction imply that satisfies the required properties. Namely, is a dense linear ordering with least element but no greatest element; the function is strictly increasing; for all and the sequence is cofinal in

We define as follows: By induction on external terms it follows that the interpretation of term in under is just for each

Next we basically repeat the construction of to get for each For each we define an equivalence relation on as follows:

 t1∼St2 ⇔ (S:t1=t2)∈¯¯¯¯Γ.

Let be the -equivalence class of For each we let

 NS={tS: t∈Ti};fNS(tS)=f(t)S;cNS=cS;
 tS1

Moreover, for each we let (Recall that we denote by the set of points in where is true.)

As above, the reduction tree construction implies that everything is well-defined and that, for all satisfies the required properties.

We define as follows: It follows easily that, for each

The careful reader may have already noticed that the support of the external time flow is the but we defined above a family (instead of an indexed family on the -equivalence classes of as required by the definition of pre-structure). We explain why: from the axioms and the construction of the reduction tree it follows that whenever For instance, let us assume that and are both in and that is the -th element in the enumeration of the axioms (one such does exist because of the infinitely many repetitions). Then, extending upwards as prescribed by case 2 of the reduction tree construction, we get that is Hence is in Similarly, assuming we get and for all

Hence we simply let for some (for all) such that