A Modal Calculus of Partially Ordered Events in a Logic Programming Framework

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A Modal Calculus of Partially

Appeared in the Proceedings of the Twelfth International Conference on Logic Program- ming — ICLP’95, (L. Sterling editor), pp. 299–313, MIT Press, Kanagawa, Japan, 13–16 June 1995.

Ordered Events in a Logic Programming Framework

Iliano Cervesato¹

Dipartimento di Informatica, Universit`a di Torino Corso Svizzera 185, 10149 Torino - ITALY

[email protected]

Luca Chittaro and Angelo Montanari²

Dipartimento di Matematica e Informatica, Universit`a di Udine Via delle Scienze 206, 33100 Udine - ITALY

{chittaro|montana}@dimi.uniud.it

Abstract

This paper proposes a modal logic reconstruction of temporal reasoning about partially ordered events in a logic programming framework. It consid- ers two variants of Kowalski and Sergot’s Event Calculus (EC): the Skeptical EC (SKEC) and the Credulous EC (CREC). In the presence of partially ordered sequences of events, SKEC and CREC derive the maximal validity intervals over which the relevant properties are necessarily and possibly true, respectively. SKEC and CREC are proved to be the operational counterparts of the modal operators of necessity and possibility in an appropriate modal logic and their properties in relation to EC are studied.

1 Introduction

The problem of computing which facts must be or can possibly be true over certain time intervals when information about the ordering of events is incomplete has been already addressed in the literature, e.g. [2, 4, 5]. This is a key issue for many real-world applications that either receive ordering information asynchronously with respect to the recording of events or cannot acquire complete ordering information. Case studies in the domains of planning and diagnosis have been analyzed in [10] and [3], respectively. In this paper, we propose a unifying framework based on Kowalski and Sergot’s Event Calculus [8] (hereinafter EC) and its skeptical and credulous variants [3] that allows us to formally characterize the state of knowledge about event ordering and its updates as well as queries about the truth of facts over time.

1Currently visiting the Computer Science Department of Carnegie Mellon University, 5000 Forbes Avenue, Pittsburgh, PA 15213-3891.

2On leave at ILLC, Universiteit van Amsterdam, Plantage Muidergracht 24, 1018 TV Amsterdam, The Netherlands.

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EC is a formalism for reasoning about time and change in a logic programming framework. Given a set of events, EC allows to derive maximal validity intervals (MVIs hereinafter) over which the properties they initiate or terminate hold. However, when only partial knowledge about the ordering of events is given, EC is neither able to derive all possible MVIs nor to distinguish which of the derived intervals are defeasible and which are not.

In this paper, we consider two variants of EC, namely the skeptical and the credulous EC (SKEC and CREC, respectively) [3]. The first derives MVIs that will be true in whatever final completion of the ordering, while the second derives those that may be true in at least one possible refinement of the ordering. We provide a uniform interpretation of EC, SKEC and CREC in a modal logic framework. In particular, we show how the two latter calculi can be respectively viewed as the operational counterparts of the modal operators of necessity and possibility in an appropriate modal logic.

The paper is organized as follows. In Section 2 we briefly describe the basic features of EC. In Section 3 we introduce its skeptical and credulous variants. In Section 4 we provide a modal logic interpretation of the three calculi. In particular, we prove the soundness and completeness results that relate the three calculi to their modal counterparts.

2 The basic Event Calculus

EC proposes a general approach to representing and reasoning about events and their effects in a logic programming framework. It takes the notions of event, property, time-point and time-interval as primitives and defines a model of change in which events happen at time-points and initiate and/or terminate time-intervals over which some property holds. EC also embodies a notion of default persistence according to which properties are assumed to persist until an event that interrupts them occurs. Formally, we represent an event occurrence by means of the predicate happens/1.

Events are assumed to be ordered relatively to each other rather than with respect to an absolute time line. Factual knowledge about event ordering is expressed through the predicate beforeFact/2. EC exploits the transitive closure of this ordering information, defined by means of the predicate before/2:

before(E1,E2) :-

beforeFact(E1,E2). (1)

before(E1,E2) :-

beforeFact(E1,E3),before(E3,E2). (2)

The relations between events and properties are defined by means of the predicates initiates/2 and terminates/2:

initiates(e1,p1). terminates(e2,p2).

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The initiates (terminates) declaration states that event e1 (e2) initiates (terminates) a period of time during which property p1 (p2) holds.

The plain EC model of time and change is defined by means of a set of axioms. The axiom holds/1 allows us to state that a property P holds maximally between events Ei and Et if Ei initiates P and occurs before Et that terminates P, provided that there are no known interruptions in between:

holds(period(Ei,P,Et)) :-

happens(Ei),initiates(Ei,P),happens(Et),

terminates(Et,P),before(Ei,Et),not broken(Ei,P,Et).

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The negation involving broken/3 is interpreted as negation-as-failure:

properties are assumed to hold uninterrupted over an interval of time on the basis of failure to determine an interrupting event. The predicate broken is defined as follows:

broken(Ei,P,Et) :-

happens(E),before(Ei,E),before(E,Et),

(initiates(E,Q);terminates(E,Q)),(exclusive(P,Q);P=Q).

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This axiom states that a given property P ceases to hold if there is an event E that happens between Ei and Et and initiates or terminates a property Q that is incompatible with P (predicate exclusive(P,Q)). We must also constrain the interferences due to incomplete sequences of events relat- ing to the same property. By considering the condition P=Q, we guarantee that the axiom for broken succeeds also when an initiating or terminating event for property P is found between the pair of events Ei and Et starting and terminating P, respectively.

3 Managing the temporal ordering of events

Database updates in EC provide information about the occurrence of events and their occurrence times, and are of additive nature only [7]. Since EC computes MVIs by applying a default persistence rule, an upgrading of its knowledge may result in some no longer derivable MVIs.

In this paper, we investigate how the MVIs derivable within the current set of events can change in response to the arrival of new ordering information (the practical relevance of this case has been illustrated in [3]). More precisely, we formally characterize the set of MVIs that cannot be invali- dated no matter how the ordering information is updated (as far as it is consistent), and the set of MVIs that may eventually become deducible de- pending on which ordered pairs are acquired. The corresponding two calculi, respectively nicknamed the Skeptical Event Calculus (SKEC ) and the Cred- ulous Event Calculus (CREC ), permit to derive the set of MVIs that are necessarily and possibly valid, respectively, in the possible completions of

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the current state of knowledge.

SKEC implements a sort of absolute persistence in order to exclude the possibility of deriving information that could be later retracted, provided that the given set of event occurrences does not change. The idea is to transform the definition of holds/1 so that holds(period(ei,p,et)) succeeds if and only if it is possible to conclude that no event affecting p will ever occur after ei and before et. In such a way, the computed MVIs are undefeasible with respect to refinements of the ordering specification: new ordering pieces coming in may result in new MVIs being derived, but every old MVI is still valid. SKEC replaces the predicates holds and broken of EC with the predicates sHolds/1 and sBroken/3, respectively:

sHolds(period(Ei,P,Et)) :-

terminates(Et,P),before(Ei,Et),not sBroken(Ei,P,Et).

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sBroken(Ei,P,Et) :-

happens(E),E \== Ei,E \== Et, not before(E,Ei),not before(Et,E),

(initiates(E,Q);terminates(E,Q)),(exclusive(P,Q);P=Q).

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Unlike SKEC, whenever it is not possible to derive that a terminating event et precedes an initiating event ei, CREC assumes that et follows ei. Such an assumption allows CREC to compute all MVIs which are not incompatible with a given set of partially ordered events. More precisely, CREC computes every MVI that holds with respect to at least one possible completion of the given partial ordering of events. Further constraining the ordering of events may indeed invalidate previously computed MVIs, but it never forces CREC to compute new MVIs.

The axioms of CREC are the same of EC except for the replacement of before(Ei,Et) with the negation of before(Et,Ei) in the definition of holds. The resulting predicate cHolds/1 is defined as follows:

cHolds(period(Ei,P,Et)) :-

terminates(Et,P),not before(Et,Ei),not broken(Ei,P,Et).

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A collection of facts with happens, initiates, terminates or exclusive as their predicate symbols is called factual knowledge. It represents information specific to the problem at hand. A collection of beforeFact facts is instead referred to as ordering information. It represents temporal data that become available during the execution³.

3For the ease of the reader, we associate names to certain sets of clauses. Specifically, clauses (1)–(7) are generically called the axioms of the extended Event Calculus and are referred to as EC+; clauses (1)–(2), (3)–(4) constitute the axioms of EC and are denoted by EC; clauses (1)–(2), (4), (7) are the axioms of CREC (CREC); clauses (1)–(2), (5)–(6) are the axioms of SKEC (SKEC); axioms (1)–(2) are the ordering axioms and are denoted by O.

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4 A modal logic interpretation of the calculi

EC, SKEC and CREC have a modal logic counterpart where orderings are interpreted as possible worlds, each one denoting a different state of knowledge.

Moreover, an MVI derived by EC translates into a formula which is true in the current world, and its derivability in SKEC (resp. CREC) corresponds to the truth of the formula in every accessible world (resp. in at least one of them).

4.1 Formalization of the Event Calculus with relative timing In order to formalize EC with relative timing, we first give a precise description of the factual knowledge. A structure for the Event Calculus with relative timing (hereinafter EC-structure) is a quintuple H = (E, P, [·i, h·], ]·,·[) such that: E = {e¹, . . . , eⁿ} is a finite set of events; P = {p¹, . . . , p^m} is a finite set of properties; [·i : P → 2^E and h·] : P → 2^E are respectively the initiating and terminating map of H (for every property p ∈ P , [pi and hp] represent the set of events that initiate and terminate p, respectively; [pi and hp] are disjoint for every p ∈ P ); ]·,·[⊆ P ×P is an irreflexive and symmetric relation, called the exclusivity relation, that models exclusivity among properties.

In order to formally define the notion of ordering, let R⁺ denote the transitive closure of a relation R on a set A. Given a set of events E, we define o ⊆ E × E to be a knowledge state for E if o⁺ is a (possibly partial) strict ordering on E. Let O_E be the set of all knowledge states for E and W_E ⊂ O_E be the set of all strict orderings on E (the subscripts will be kept implicit when no confusion can arise). Two knowledge states o1 and o2 are equally informative if o⁺1 = o⁺2. This induces an equivalence relation ∼ on O.

It is easy to prove that O_/∼ and W are isomorphic. Thus, in the following, a knowledge state o will always be identified with the corresponding element o⁺ of W , unless explicitly stated otherwise.

The pair (W, ⊆), where ⊆ is the ordinary subset relation, has the structure of an ordered set⁴. Let w belong to W . Any w^′ ∈ W such that w ⊆ w^′ is called an extension of w. We denote the set of all extensions of w as Ext(w).

It is trivial to prove that if (e1, e2) ∈ w, then, for every w^′ ∈ Ext(w), (e1, e2) ∈ w^′.

Let us now define a family of representation functions · ^·that relate the mathematical entities we have been using in this section to the terms of the logic programming language we have described in Sections 2 and 3. In the following, we will refer to an EC-structure H = (E, P, [·i, h·], ]·,·[).

First, we define the functions · ^E and · ^P that consider the concrete syntax of single events and properties, respectively. We explicitly assume that these functions are injective, i.e. that every event e in E (property p in

4It can be easily proved that (W, ∩, Ø) forms a finite lower semilattice, and that, for every w1, w2 ∈ W, w1 ↑ w2 = (w1∪ w2)⁺ is the lub of w1 and w2 with respect to ⊆ whenever this element belongs to W (w1 and w2 may indeed contain incompatible pairs).

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P) has a representation that is different from that of all other events (resp.

properties). This enables us to utilize the respective inverse functions, · _E and · P, whenever they are defined. Moreover, we want · ^E and · ^P give distinct representations to events and properties. The next step consists in defining the translation maps for [·i, h·] and ]·,·[, that correspond to the predicates initiates, terminates and exclusive. We need also a function that relates the set of events E to the predicate happens to complete the mapping onto the factual knowledge:

[·i^I = {initiates( e^E, p^P) : e ∈ E, p ∈ P, and e ∈ [pi};

h·]^T = {terminates( e^E, p^P) : e ∈ E, p ∈ P, and e ∈ hp]};

]·,·[^X = {exclusive( p^P, q^P) : p, q ∈ P and ]p, q[};

E^H = {happens( e^E) : e ∈ E}.

We define the representation of the EC-structure H as the union of the representations of its constituent entities: H^S = E^H ∪ [·iÎ ∪ h·]^T ∪ ]·,·[^X.Information concerning a knowledge state o on a set E of events will be related to its concrete syntax by means of the function · Ô: oÔ = {beforeFact( e1E, e2 E) : (e1, e2) ∈ o}. In order to simplify the notation, we will write the previously defined translation maps as · whenever possible.

4.2 A uniform modal framework for EC, SKEC and CREC In this section, we investigate the properties of EC-structures and their re- lationships to the event calculi introduced in Sections 2 and 3.

First we define the query language. Let H = (E, P, [·i, h·], ]·,·[) be an EC- structure. The query language for H, denoted as L_H (L when no ambiguity arises) is the set of all EC-formulae, as defined by the following equality:

L_H = {p(e1, e2), 2p(e¹, e2), 3p(e¹, e2) : p ∈ P and e1, e2 ∈ E}.

We call L⁰_H the sublanguage of L_H consisting of atomic formulae p(e1, e2) only. Notice that LH is a propositional language (do not be misled by the structured notation we used for atomic formulae).

A modal interpretation for EC formulae can be defined as follows. Given an EC-structure H = (E, P, [·i, h·], ]·,·[), an EC-frame for H is defined as a pair (W, ⊆), where W is the set of all strict orderings on E and ⊆ plays the role of accessibility relation. EC-frames are reflexive, transitive and anti- symmetric. An EC-interpretation M based on a frame (W, ⊆) is the triplet (W, ⊆, υ), where υ is the valuation function υ : L⁰× W → {true, f alse} that assigns a truth value to each atomic formula in L⁰ in each world w ∈ W in such a way that υ(p(e1, e2), w) = true if and only if

i. e1 ∈ [pi; e2 ∈ hp]; (e1, e2) ∈ w;

ii. ¬∃e ∈ E. (e1, e) ∈ w ∧ (e, e2) ∈ w

∧ ∃q ∈ P. ((e ∈ [qi ∨ e ∈ hq]) ∧ (]p, q[ ∨ p = q))).

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Condition (ii) expresses the requirement that p holds uninterruptedly between e1 and e2. As a matter of convenience, let us denote such a condition as nb(p, e1, e2, w). Notice also that, since w is a strict ordering, (e1, e) ∈ w and (e, e2) ∈ w entail e1 6= e and e 6= e2. The notion of satisfiability of an L_H-formula in a knowledge state is defined as follows:

w|= p(e1, e2) iff υ(p(e1, e2) = true;

w|= 2p(e¹, e2) iff for every w^′ ∈ W such that w ⊆ w^′, w^′ |= p(e1, e2) w|= 3p(e¹, e2) iff there is w^′ ∈ W such that w ⊆ w^′ and w^′|= p(e1, e2).

The following two lemmas relate satisfiability and local properties of EC- structures.

Lemma 4.1 (Pointwise condition for necessity)

Let H = (E, P, [·i, h·], ]·,·[) be an EC structure. Then for any e1, e2 ∈ E, p∈ P and w ∈ W , w |= 2p(e¹, e2) iff the following conditions are satisfied:

e1 ∈ [pi, e2 ∈ hp], (e1, e2) ∈ w, and nsb(p, e1, e2, w), where nsb(p, e1, e2, w) stands for the expression

∀e ∈ E. ∀q ∈ P. e = e1 ∨ e = e2 ∨ (e, e1) ∈ w ∨ (e2, e) ∈ w

∨ (e ∈ [qi ∨ e ∈ hq] ⇒ ¬]p, q[ ∧ p 6= q).

Proof: (⇐) Let us proceed by contradiction. Assume that e1 ∈ [pi, e2 ∈ hp], (e1, e2) ∈ w and nsb(p, e1, e2, w), but there exists an extension w^′ of w such that w^′ |= p(e1, e2) does not hold, i.e. such that nb(e1, e2, w^′) is false. After some logical manipulations, the latter statement rewrites to

∃e ∈ E. q ∈ P.((e1, e) ∈ w^′∧ (e, e2) ∈ w^′∧ (e ∈ [qi ∨ e ∈ hq]) ∧ (]p, q[ ∨ p = q).

Let e^′and q^′witness the validity of this formula. By instantiation, we obtain:

(e1, e^′) ∈ w^′ ∧ (e^′, e2) ∈ w^′ ∧ (e^′ ∈ [q^′i ∨ e^′ ∈ hq^′]) ∧ (]p, q^′[ ∨ p = q^′) (1) We can instantiate the expression for nsb(p, e1, e2, w) with these values too.

The resulting formula is:

e^′= e1 ∨ e^′= e2 ∨ (e^′, e1) ∈ w ∨ (e2, e^′) ∈ w ∨

∨ (e^′ ∈ [q^′i ∧ e^′ ∈ hq^′] ⇒ ¬]p, q^′[ ∧ p 6= q^′). (2) We must show that none of the alternatives in formula (2) applies. Since w is a strict order, the validity of (1) implies that e^′ can be neither e1 nor e2. Analogously, by monotonicity of extensions, either (e^′, e1) ∈ w or (e2, e^′) ∈ w would violate the antisymmetry of w^′. Finally, the choice of q^′ contradicts the last alternative, i.e. that (e^′ ∈ [q^′i ∨ e^′ ∈ hq^′] ⇒ ¬]p, q^′[ ∧ p 6= q^′). This concludes this direction of the proof.

(⇒) We will again proceed by contradiction. Clearly, if e1 6∈ [pi or e2 6∈ hp], then we cannot obtain w^′ |= p(e1, e2) in any state of knowledge w^′.

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If (e1, e2) 6∈ w, then there exist extensions of w containing (e2, e1). Because of antisymmetry, these extensions cannot contain (e1, e2), thus, p(e1, e2) cannot be valid in them. Assume now that nsb(p, e1, e2, w) does not hold. Therefore, there are an event e^′ and a property q^′ such that:

e^′6= e1 ∧ e^′6= e2 ∧ (e^′, e1) 6∈ w ∧ (e2, e^′) 6∈ w ∧

∧ (e^′ ∈ [q^′i ∨ e^′ ∈ hq^′]) ∧ (]p, q^′[ ∨ p = q^′).

Since (e1, e2) ∈ w, there exists at least one extension w^′ of w such that (e1, e^′) ∈ w^′ and (e^′, e2) ∈ w^′. Therefore,

(e1, e^′) ∈ w^′ ∧ (e^′, e2) ∈ w^′ ∧ (e^′ ∈ [q^′i ∨ e^′ ∈ hq^′]) ∧ (]p, q^′[ ∨ p = q^′) hence nb(p, e1, e2, w^′) does not hold. This contradicts the hypothesis that w|= 2p(e¹, e2). 2

Notice that, unlike nb(p, e1, e2, w), nsb(p, e1, e2, w) requires that there is no e ∈ E such that (e, e1) 6∈ w and not only that (e1, e) ∈ w; similarly for (e2, e).

Lemma 4.2 (Pointwise condition for possibility)

Let H = (E, P, [·i, h·], ]·,·[) be an EC structure. Then for any e1, e2 ∈ E, p∈ P and w ∈ W , w |= 3p(e¹, e2) iff the following conditions are satisfied:

e1 ∈ [pi, e2 ∈ hp], (e2, e1) 6∈ w, and nb(p, e1, e2, w).

Proof: (⇐) Let us construct an extension w^′of w such that w^′|= p(e1, e2).

Let w^′ be (w ∪ {(e1, e2)})⁺. First notice that w^′ is consistent since w is consistent and (e2, e1) 6∈ w. Then observe that nb(p, e1, e2, w^′) holds by the definition of w^′ and the monotonicity of extensions. Otherwise, we should be able to conclude that there is an event e ∈ E such that (e1, e) ∈ w^′, (e, e2) ∈ w^′ and e ∈ [qi or e ∈ hq] for some property q ∈ P with ]p, q[

and p 6= q, but in that case, (e1, e) ∈ w and (e, e2) ∈ w contradicting the assumption that nb(p, e1, e2, w). Therefore, conditions (i-ii) are satisfied w.r.t. w^′, hence w^′ |= p(e1, e2), thus w |= 3p(e¹, e2).

(⇒) We proceed by contradiction. Clearly, if e1 6∈ [pi or e2 6∈ hp], then we cannot obtain w^′ |= p(e1, e2) in any state of knowledge w^′. Analogously, if (e2, e1) ∈ w, then (e2, e1) belongs to every extension of w, forbidding in this way condition (i) of the definition of valuation to be satisfied. Finally, if nb(p, e1, e2, w) does not hold, then, by monotonicity of extensions, the same condition would apply to every extension w^′ as well, thus nb(p, e1, e2, w^′) would not hold in any extension w^′ of w. 2

4.3 Soundness and Completeness results

In this section we prove the soundness and completeness results that relate the clausal representation of EC, SKEC and CREC introduced in Sections 2 and 3 to the formal system described above. For this purpose, we need to

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refer to the provability relation built in the inference engine of Prolog. Given a program P and a goal formula g, we will denote this relation as P ⊢_sldnf g, meaning that the goal g is provable by SLDNF-resolution from program P.

In the following, we will need a couple of basic results about negation-as- failure. See [9] for more detail. Specifically, we have that, if P is a definite program and g is a goal, the following statements hold:

P ⊢_sldnf not g⇒ P 6⊢_sldnf g; (NAF–A)

P 6⊢_sldnf gbut P ⊢_sldnf g terminates ⇒ P ⊢_sldnf not g. (NAF–B) We begin with two lemmas referring to the properties of before.

Lemma 4.3 (Soundness/completeness of before w.r.t. transitive closure) Let H = (E, P, [·i, h·], ]·,·[) be an EC structure and o a state of knowledge, then for any e1, e1 ∈ E

O, H , o ⊢_sldnf before( e1, e2) iff (e1, e2) ∈ o⁺.

Proof: (⇒) We will prove the statement by induction on the height of a resolution tree for O, H , o ⊢_sldnf before( e1 , e2). If the height is 1, then clause (1) must have been used. Thus O, H , o ⊢_sldnf beforeFact( e1 , e2) and therefore, (e1, e2) ∈ o ⊆ o⁺. Otherwise, let us assume that this tree has height h + 1. The first rule applied must be clause (2). By unfolding, we obtain that O, H , o ⊢_sldnf beforeFact( e1, e) and O, H , o ⊢_sldnf before( e , e2) for some event e, where the latter has a derivation tree of height h. Thus, (e1, e) ∈ o and, by induction, (e, e2) ∈ o⁺. Now, by the definition of transitive closure, (e1, e2) ∈ o⁺.

(⇐) Let σ = e^′₁, . . . , e^′_l, with e^′₁= e1 and e^′_l= e2 be a sequence of events such that for i = 1 . . . l − 1 (e^′_i, e^′_i+1) ∈ o, proving in this way that (e1, e2) ∈ o⁺. We conduce the proof by induction on the length l of this sequence. If l = 1, then (e1, e2) ∈ o, thus O, H , o ⊢_sldnf beforeFact( e1 , e2) and, by clause (1), O, H , o ⊢sldnf before( e1, e2 ). Otherwise, σ = e1, e^′, . . . , e2 with (e1, e^′) ∈ o and (e^′, e2) ∈ o⁺. Thus O, H , o ⊢_sldnf beforeFact( e1 , e^′) and, by induction, O, H , o ⊢_sldnf before( e^′, e2). Finally, by applying clause (2), we obtain O, H , o ⊢_sldnf before( e1, e2). 2

Lemma 4.4 (Soundness/completeness of not before)

Let H = (E, P, [·i, h·], ]·,·[) be an EC structure and o a state of knowledge, then, for any e1, e1 ∈ E,

O, H , o ⊢_sldnf not before( e1 , e2) iff (e1, e2) 6∈ o⁺. Proof: By lemma 4.3, we reduce this statement to

O, H , o ⊢_sldnf not before( e1, e2 ) iff O, H , o 6⊢_sldnf before( e1, e2).

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In order to prove this relation, we need to rely on the properties of negation- as-failure stated at the beginning of the section. The proof proceeds as follows:

(⇒) Immediate from (NAF–A).

(⇐) We must prove that O, H , o ⊢_sldnf before( e1 , e2) has only (possibly failed) finite computations (by property (NAF–A)). Assume by absur- dum that the goal before( e1 , e2) leads to an infinite computation w.r.t.

the program O, H , o . Hence, clause (2) has been applied infinitely many times. In particular, we have that O, H , o ⊢sldnf beforeFact( ê_i,eˆ_i+1) for an infinite sequence of events {ê_i}_i∈ω (with ê0= e1). Thus (ê_i,eˆ_i+1) ∈ o for all i ∈ ω. At this point, we must remember that E is finite. Therefore, there must be two distinct indices i, j with i < j such that ê_i = ê_j. Then, by transitivity, we have that (ê_i,eˆ_j) ∈ o⁺, but this violates the irreflexivity of o⁺. 2

Lemma 4.5 (Correspondence between broken (sBroken) and nb (nsb)) Let H = (E, P, [·i, h·], ]·,·[) be an EC structure and o a state of knowledge, then

a. EC+, H , o ⊢_sldnf broken( e1 , p , e2 ) iff ¬nb(p, e1, e2, o⁺) holds in H;

b. EC+, H , o ⊢_sldnf sBroken( e1, p , e2)iff ¬nsb(p, e1, e2, o⁺) holds in H.

Proof: (a ⇒) Assume that EC+, H , o ⊢_sldnf broken( e1, p , e2). After unfolding clause (4), we obtain the following relations. The left column shows the single subgoals. The central column gives a formal counterpart (in the sense of logical equivalence) of the corresponding clausal relation in the left column. Finally, the right column justifies the deduction step on that row.

EC+, H , o ⊢_sldnf happens( e ) e∈ E by def. of E EC+, H , o ⊢_sldnf before( e1, e) (e1, e) ∈ o⁺ by lemma 4.3 EC+, H , o ⊢_sldnf before( e , e2) (e, e2) ∈ o⁺ by lemma 4.3 EC+, H , o ⊢_sldnf initiates( e , q ); (e ∈ [qi ∨ by def. of [·i

terminates( e , q ) e∈ hq]) and of h·]

EC+, H , o ⊢_sldnf exclusive( p , q ); (e ∈]p, q[ ∨ by def. of ]·,·[

p= q p= q)

We can put the single pieces together in order to obtain a statement equivalent to EC+, H , o ⊢_sldnf broken( e1, p , e2):

(e1, e) ∈ o⁺ ∧ (e, e2) ∈ o⁺ ∧ (e ∈ [qi ∨ e ∈ hq]) ∧ (]p, q[ ∨ p = q).

We now abstract over the event e and the property q and obtain the formula

∃e ∈ E. q ∈ P. ((e1, e) ∈ o⁺ ∧ (e, e2) ∈ o⁺

∧ (e ∈ [qi ∨ e ∈ hq]) ∧ (]p, q[ ∨ p = q))

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that is equivalent, after some logical manipulations, to nb(p, e1, e2, o⁺).

(a ⇐) Assume now that ¬nb(p, e1, e2, o⁺) is valid in H, i.e. that

∃e ∈ E. ((e1, e) ∈ o⁺ ∧ (e, e2) ∈ o⁺

∧ ∃q ∈ P. ((e ∈ [qi ∨ e ∈ hq]) ∧ (]p, q[ ∨ p = q))).

Let e^′and q^′ be such e and q respectively. Then, by instantiation, we obtain:

(e1, e^′) ∈ o⁺ ∧ (e^′, e2) ∈ o⁺ ∧ (e^′ ∈ [q^′i ∨ e^′ ∈ hq^′]) ∧ (]p, q^′[ ∨ p = q^′).

By the definition of E and since e^′ ∈ E, happens( e^′ ) ∈ H . This implies that EC+, H , o ⊢_sldnf happens( e^′ ) Consider the first conjunct of the previous formula. By lemma 4.3, we have that EC+, H , o ⊢_sldnf before( e1 , e^′ )

Similarly, we obtain that

EC+, H , o ⊢_sldnf before( e , e2 )

EC+, H , o ⊢_sldnf initiates( e , q ); terminates( e , q ) EC+, H , o ⊢_sldnf exclusive( p , q ); p = q

We have proved in this way every goal in the body of clause (4). Thus the head of this clause is valid, i.e. EC+, H , o ⊢_sldnf broken( e1, p , e2).

(b ⇒) We proceed as in (a ⇐), with the only complication that we have to deal with negated goals.

Assume that EC+, H , o ⊢_sldnf sBroken( e1, p , e2). By unfolding clause (4), we obtain the following relations:

EC+, H , o ⊢_sldnf happens( e ) e∈ E by def. of E EC+, H , o ⊢_sldnf e \== e1 e6= e1 by def. of · ^E EC+, H , o ⊢_sldnf e \== e1 e6= e2 by def. of · ^E EC+, H , o ⊢_sldnf not before( e , e1) (e, e1) 6∈ o⁺ by lemma 4.4 EC+, H , o ⊢_sldnf not before( e2 , e) (e2, e) 6∈ o⁺ by lemma 4.4 EC+, H , o ⊢_sldnf initiates( e , q ); (e ∈ [qi ∨ by def. of [·i

terminates( e , q ) e∈ hq]) and of h·]

EC+, H , o ⊢_sldnf exclusive( p , q ); (e ∈]p, q[ ∨ by def. of ]·,·[

p= q p= q)

As in (a ⇒), we now consider the conjunction of the formulae displayed on the right-hand side:

e6= e1 ∧ e 6= e2 ∧ (e, e1) 6∈ o⁺∧

∧ (e2, e) 6∈ o⁺ ∧ (e ∈ [qi ∨ e ∈ hq]) ∧ (]p, q[ ∨ p = q) By abstracting over e and q, obtain

∃e ∈ E. ∃q ∈ P. e 6= e1 ∧ e 6= e2 ∧ (e, e1) 6∈ o⁺

∧ (e2, e) 6∈ o⁺ ∧ (e ∈ [qi ∨ e ∈ hq]) ∧ (]p, q[ ∨ p = q)) that is equivalent, after some logical manipulations, to nsb(p, e1, e2, o⁺).

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(b ⇐) The proof follows the pattern of (a ⇐). If ¬nsb(p, e1, e2, o⁺), then there exist an event e^′ and a property p^′ such that

e^′ 6= e1 ∧ e^′ 6= e2 ∧ (e^′, e1) 6∈ o⁺ ∧ (e2, e^′) 6∈ o⁺∧

∧ (e^′ ∈ [q^′i ∨ e^′ ∈ hq^′]) ∧ (]p, q^′[ ∨ p = q^′).

By the definition of E , happens( e^′) ∈ H . Thus EC+, H , o ⊢_sldnf happens( e^′ )

By considering all the conjuncts in the last formula and applying the definition of · and lemma 4.4, we have that

EC+, H , o ⊢_sldnf e^′ \== e1

EC+, H , o ⊢_sldnf not before( e^′ , e1) EC+, H , o ⊢_sldnf not before( e2, e^′)

EC+, H , o ⊢_sldnf initiates( e , q ); terminates( e , q ) EC+, H , o ⊢_sldnf exclusive( p , q ); p = q

Therefore, the head of clause (6) is provable and we have that EC+, H , o ⊢_sldnf sBroken( e1, p , e2). 2

Lemma 4.6 (Soundness/completeness of not broken and not sBroken) Let H = (E, P, [·i, h·], ]·,·[) be an EC structure and o a state of knowledge, then

a. EC+, H , o ⊢_sldnf not broken( e1 , p , e2) iff nb(p, e1, e2, o⁺) holds in H;

b. EC+, H , o ⊢_sldnf not sBroken( e1, p , e2)iff nsb(p, e1, e2, o⁺) holds in H.

Proof: (a) Given lemma 4.5, this statement can be reduced to proving that EC+, H , o ⊢_sldnf not broken( e1, p , e2) if and only if EC+, H , o 6⊢_sldnf broken( e1 , p , e2 ). For this purpose, we can apply the technique used in the proof of lemma 4.4.

(⇒) Immediate from (NAF–A).

(⇐) We prove that EC+, H , o ⊢_sldnf broken( e1, p , e2) has only (possibly failed) finite computations (by property (NAF–B)). To this end, we need to examine the definition of broken in clause (4). This clause is non- recursive and moreover its body only contains lookups to the factual knowledge, which terminate since factual knowledge is represented by atomic facts, and to before, which, from lemmas 4.3 and 4.4, has finite derivations only.

(b) Similar. 2

We can now state the major result of this paper.

Theorem 4.7 (Soundness/completeness of EC, SKEC and CREC w.r.t.

the EC-frame semantics)

Let H = (E, P, [·i, h·], ]·,·[) be an EC structure and o a state of knowledge, then

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a. EC+, H , o ⊢_sldnf holds(period( e1, p , e2)) iff o⁺|= p(e1, e2);

b. EC+, H , o ⊢_sldnf sHolds(period( e1 , p , e2 )) iff o⁺|= 2p(e¹, e2);

c. EC+, H , o ⊢_sldnf cHolds(period( e1 , p , e2 )) iff o⁺|= 3p(e¹, e2).

Proof: (a ⇒) Let EC+, H , o ⊢sldnf holds(period( e1, p , e2)). We first prove that, with this hypothesis, υ(p(e1, e2), o⁺) = true; the thesis will follow by the definition of validity.

By unfolding clause (3) we obtain the following relations:

EC+, H , o ⊢_sldnf happens( e1) e1 ∈ E by def. of E EC+, H , o ⊢_sldnf initiates( e1, p) e1 ∈ [pi by def. of [·i EC+, H , o ⊢_sldnf happens( e2) e2 ∈ E by def. of E EC+, H , o ⊢_sldnf terminates( e2 , p) e2 ∈ hp] by def. of h·]

EC+, H , o ⊢_sldnf before( e1, e2) (e1, e2) ∈ o⁺ by lemma 4.3 EC+, H , o ⊢_sldnf not broken( e1 , p , e2 ) nb(p, e1, e2, o⁺) by lemma 4.6 Now, it suffices to notice that the first five relations on the right-hand side correspond to condition (i) of the definition of valuation and that the sixth corresponds to condition (ii). Therefore υ(p(e1, e2), o⁺) = true and thus o⁺|= p(e1, e2).

(a ⇐) Apply the technique used to prove the (⇐) direction in lemma 4.5.

(b ⇒) Assume that EC+, H , o ⊢_sldnf sHolds(period( e1, p , e2)). We will prove that, from this hypothesis, e1 ∈ [pi, e2 ∈ hp], (e1, e2) ∈ o⁺ and nsb(p, e1, e2, o⁺); the thesis will follow by lemma 4.1.

By unfolding clause (5) we obtain the first five relations of case (a ⇒) plus the following one:

EC+, H , o ⊢_sldnf not sBroken( e1, p , e2) nsb(p, e1, e2, o⁺) by lemma 4.6 (b ⇐) The same as in (a ⇐).

(c ⇒) Assume that EC+, H , o ⊢_sldnf cHolds(period( e1, p , e2)). Let us prove that, with this hypothesis, e1 ∈ [pi, e2 ∈ hp], (e2, e1) 6∈ o⁺ and nb(p, e1, e2, o⁺); the thesis will follow by lemma 4.2.

By unfolding clause (7) we obtain the first four relations of case (a ⇒) plus the following ones:

EC+, H , o ⊢_sldnf not before( e2 , e1) (e2, e1) 6∈ o⁺ by lemma 4.4 EC+, H , o ⊢_sldnf not broken( e1 , p , e2 ) nb(p, e1, e2, o⁺) by lemma 4.6

(c ⇐) The same as in (a ⇐). 2

These results are the formal counterpart of the intuitive description of the behavior of holds, sHolds and cHolds presented in the previous sections.

The relative cardinalities of the sets of values satisfying these predicates are related in the following corollary, which, again, agrees with our intuition.

Corollary 4.8 (SKEC and CREC are a lower and an upper bound on EC)

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Let H = (E, P, [·i, h·], ]·,·[) be an EC structure and o a state of knowledge, then

a. if EC+, H , o ⊢_sldnf sHolds(period( e1, p , e2)) then EC+, H , o ⊢_sldnf holds(period( e1, p , e2));

b. if EC+, H , o ⊢_sldnf holds(period( e1, p , e2)) then EC+, H , o ⊢_sldnf cHolds(period( e1, p , e2)).

Proof: By the previous theorem, these relations can be rewritten as:

a. if o⁺|= 2p(e¹, e2), then o⁺|= p(e1, e2), and b. if o⁺|= p(e1, e2), then o⁺|= 3p(e¹, e2).

The validity of these expressions is a direct consequence of the reflexivity of the accessibility relation of EC-frames. Indeed, o⁺|= 2p(e¹, e2) iff p(e1, e2) is valid in every extension of o⁺, in particular in o⁺ itself. Analogously, if o⁺|= p(e1, e2), then o⁺|= 3p(e¹, e2). 2

We can restate this corollary in a form closer to the Event Calculus terminology. Let M V I_EC(o), M V I_SKEC(o) and M V I_CREC(o) denote the sets of MVIs derivable from EC, SKEC and CREC, respectively, in a given state of knowledge o. Then M V I_SKEC(o) ⊆ M V I_EC(o) ⊆ M V I_CREC(o).

Corollary 4.9 (Monotonicity of SKEC and CREC)

Let H = (E, P, [·i, h·], ]·,·[) be an EC structure and o and o^′ two states of knowledge such that o ⊆ o^′, then

a. if EC+, H , o ⊢_sldnf sHolds(period( e1, p , e2)), then EC+, H , o^′ ⊢_sldnf sHolds(period( e1, p , e2));

b. if EC+, H , o^′ ⊢_sldnf cHolds(period( e1, p , e2)), then EC+, H , o ⊢_sldnf cHolds(period( e1, p , e2)).

Proof: By theorem 4.7, we can rewrite this statement as follows: given o⊆ o^′, then

a. if o⁺|= 2p(e¹, e2), then o^′+|= 2p(e¹, e2), and b. if o^′+|= 3p(e¹, e2), then o⁺|= 3p(e¹, e2).

Given the definition of EC-frame, these relations hold trivially: if o⁺ |=

2p(e¹, e2), then p(e1, e2) is valid in all the extensions of o⁺, which comprise all the extensions of o^′+, and thus o^′+ |= 2p(e¹, e2); similarly, if o^′+ |=

3p(e¹, e2) then p(e1, e2) holds in an extension o^∗of o^′+, but since o ⊆ o^′ and

⊆ is transitive, o^∗ is an extension of o⁺ as well, and thus o⁺ |= 3p(e¹, e2).

2

Again, we can restate this corollary in terms of sets of MVIs. We obtain that

a. if o ⊆ o^′, then M V I_SKEC(o) ⊆ M V I_SKEC(o^′);

b. if o ⊆ o^′, then M V I_CREC(o^′) ⊆ M V I_CREC(o).

This corresponds to saying that SKEC is a monotonic version of EC while CREC is anti-monotonic.

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5 Conclusions

In this paper, we have studied SKEC and CREC, two extensions of Kowalski and Sergot’s Event Calculus proposed in [3]. A formal description of EC is presented in propositional logic and is proved sound and complete w.r.t. the clausal formulation of the calculus. It is then shown how a simple extension of this description in a modal setting models SKEC and CREC. The interpretation of necessity-moded formulae coincides with the set of intervals derivable in SKEC. Similarly, the semantics of the possibility operators is caught by the clausal representation of CREC. A soundness and completeness result for the extended system is presented. The axiomatizability of the proposed modal formalization is currently under investigation.

Acknowledgments

This work has been partially supported by the P.A.O.L.A. Consortium and by the Italian National Research Council.

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