Comparison with Clausal Temporal Resolution

In this section we show that operationally there is a close connection between LPSup and the Clausal Temporal Resolution (CTR) of Fisher et al. (2001). From this perspective, our formalism of labeled clauses can be seen as a new way to derive completeness of CTR that justifies the use of ordering restrictions and redundancy elimination in a transparent way. This has not been achieved yet in full by previous work: Hustadt et al. (2005) provide a proof theoretic argument, but only for the use of ordering restrictions, Konev et al. (2005) sketch the idea how to justify tautology removal and subsumption, but do not consider the abstract redundancy notion in the style of Bachmair and Ganzinger (2001), which we provide.

Moreover, there is also a correspondence between our layer-by-layer saturation followed by the application of the Leap inference and the BFS-Loop search of CTR as described by Gago et al. (2002); see also Ludwig and Hustadt (2009a). Apart from being interesting in its own right, this view sheds new light on explaining BFS-Loop search, as it gives meaning to the intermediate clauses generated in the process, and we thus do not need to take the detour through the DNF representation of Dixon (1996, 1998). Even here, the idea of labels clearly separates logical content of the clauses from the meta-logical one (c.f. the ad hoc marker literal of Gago et al., 2002).

Despite these similarities between LPSup and CTR, the calculi are by no means iden- tical. As discussed before, a temporal model can be extracted in a straightforward way from a satisfiable set of labeled clauses saturated by LPSup. This does not hold for CTR, where a more complex procedure, which simulates the model construction of Bachmair and Ganzinger (2001) only locally, needs to be applied (Ludwig and Hustadt, 2009b). In particular, because saturation by CTR does not give the model building procedure any guidance towards satisfying the goal clauses, the procedure needs to try out at every time point in a fair way all the possible orderings on the signature (in the worse case) to make sure the goal is eventually reached. As each change of the ordering calls for a subsequent re-saturation of the clause set in question (so that the local model construc- tion still works), it obviously diminishes the positive effect orderings in general have on reducing the search space.

Finally note that since we eventually rely on propositional superposition, we can also take into account the explicit use of partial models to further guide the search for a proof

Table 2.1: Clause alignment between CTR and LPSup. la, lb, and lcdenote literals. name CTR LPSup initial start_→W ala (0,∗) || Wala step V blb → Wclc (∗, ∗) ||Wblb∨Wclc0

or saturation. The idea is to build a partial model based on the ordering on propositional literals. Then it can be shown that resolution can be restricted to premises where one is false and the other true in the partial model (Bachmair and Ganzinger, 1990). This superposition approach on propositional clauses is closely related to the state-of-the-art CDCL algorithm for propositional logic (Marques-Silva et al., 2009). The missing bit is to “lift” this setting to our labeled clauses. We explore this idea in Chapter 3.

Aligning the syntax

The calculus CTR operates on temporal clauses of the Separated Normal Form (SNF) (see Section 2.2.2). The classical exposition (Fisher et al., 2001) adopts implicative notation for the temporal clauses and introduces a special temporal constant start interpreted to indicate the initial time point.

Recall that there are three kinds temporal clauses in SNF: initial, step, and eventuality clauses. The initial and step clauses used by CTR correspond to labeled counterparts of LPSup in a straightforward way (see Figure 2.1). While CTR in general works with several eventuality clauses of the form

 ^

b∈B

kb → ♦l

!

, (2.8)

LPSup uses the ideas of Degtyarev et al. (2002) to obtain a problem with only a single eventuality that is unconditional (with the set of the antecedent literals B being empty). On the other hand, LPSup relaxes the requirement that the eventuality be represented by a single literal l. Instead, the eventuality is described by a whole set of the (∗, 0)-clauses, understood conjunctively: ♦   ^ (∗,0) || C C  .

Note that the technique of Degtyarev et al. (2002) allows us to obtain a formulation of a problem that contains only a single unconditional eventuality in a form of a single literal. That is an “intersection” format directly accessible to both LPSup and CTR. Comparing deductions

There are two step resolution rules in CTR (Fisher et al., 2001) dealing with initial and step clauses, respectively. When equipped with ordering constraints (Hustadt et al.,

2005) these can be seen to be equivalent to the Ordered Resolution inference of LPSup acting on the corresponding labeled clauses. Similarly, the upper half of the following clause conversion rule of CTR

R φ→ false

true_{→ ¬φ}

start_{→ ¬φ}

can be matched by the Temporal Shift inference. The other half of the rule, which turns the step clause with unsatisfiable succedent into an initial clause is not needed in LPSup, where the assumption is kept instead and allowed to interact with initial clauses directly (by relying on merge of the respective labels).

Let us now compare how the two calculi deal with eventualities. The inference in CTR dedicated to this purpose is called temporal resolution. It combines several step clauses into groups (so called merged-SNF clauses) and resolves those against one eventuality clause. There is a nontrivial side condition to be verified that amounts to proving that the step clauses involved form a so called loop, meaning that they together conditionally imply that the eventuality may become false forever. As a last step, the inference’s conclusion, which is not a temporal clause in general, must be translated into SNF after it is derived.

Several methods have been proposed how to actually implement temporal resolution (Dixon, 1996). Here we focus on breadth first search for the loop as described by Gago et al. (2002). The idea is to perform the loop search by iteratively applying step resolu- tion inferences to certain clauses and to organize the individual iterations by enriching the participating clauses with a special marking literal (see also Ludwig and Hustadt, 2009a). The marking literals are numbered by the iteration index. This helps to sepa- rate the clauses of the individual iterations and allows for their reuse in subsequent loop searches for the same eventuality literal.

Interestingly, we can map this form of loop search to the layer-by-layer saturation process of LPSup. We identify the marker literal with the label (∗, k), i.e. the label of the clauses associated with the goal, and interpret k as the iteration index. There is, however, a small but important difference in how the clauses with a new value of the index k are created. In LPSup they arise as conclusions of the Temporal Shift inference

I₍ (∗, k) || C ∗, k + 1) || (C)0.

The corresponding inference of CTR, when adopted to our notation, becomes I (∗, k) || C

(∗, k + 1) || (C ∨ l)0, (2.9)

where l is the respective eventuality literal from (2.8). Weakening the derived clause by l0 _{is not sound with respect to the semantics based on (K, L)-models, but can be}

interpreted and justified with the help of semantic graphs.

As explained in Section 2.4.2, in LPSup the (_{∗, k)-clauses of a given clause set can} be seen to represent a set of those vertexes of the semantic graph that can reach a goal

vertex in exactly k steps. As an effect of the added eventuality literal in (2.9) above, the corresponding (∗, k)-clauses in CTR represent the vertexes that can reach a goal vertex in at most k steps. Intuitively, the goal vertexes of order zero are effectively reinserted to the computed preimage after each iteration. As a side-effect, the sequence of the represented vertex sets grows monotonically with respect to the subset relation.

In LPSup, layer-by-layer saturation ends when a repetition is detected. We then invoke the Leap inference and potentially derive additional (∗, k)-clauses. Analogously, a successful repetition check concludes the loop search in CTR. There, the new clauses collected by an equivalent of Leap obtain the status of simple step clauses, i.e., they effectively become (∗, ∗)-clauses. These clauses stand for a fixpoint result of the iterative loop search and represent the set of those vertexes that can reach a goal vertex in any number of steps. It is therefore sound to assert these clauses to hold universally, because only the vertexes they represent can be part of any potential model path.

Remark 2.4. Because, unlike in LPSup, the sequence of vertexes represented during loop search by the respective clause sets grows monotonically, we can for CTR derive a better theoretical bound on the maximal number of iterations before repetition occurs. Indeed, under similar conditions on the saturation process as those of Section 2.4.3, there is at most|2Σ_{| iterations for CTR, because each iteration must include at least one new vertex}

unless it stabilizes. On the other hand, our current best bound for LPSup derives from 2|2Σ|– the number of all subsets of the set of vertexes 2Σ_.

On the other hand, the inconspicuous addition of the eventuality literal seems to have a negative effect on the performance of CTR in practice, because it means that more inferences typically need to be performed before the computation proceeds from one layer to the next. We explore this phenomenon empirically in the next section.