Notes of the ECAI-10 Workshop on Automated Reasoning about Context and Ontology Evolution

Reasoning with embedded formulas is relevant for the SUMO ontology but there is limited automation support so far. We investigate whether higher-order automated theorem provers are applicable for the task. Moreover, we point to a challenge that we have revealed as part of our experiments: modal operators in SUMO are in conflict with Boolean extensionality. A solution is proposed. 1 EMBEDDED FORMULAS IN SUMO The open source Suggested Upper Merged Ontology (SUMO) [9] (and similarly, proprietary Cyc [13]) contains a small but significant amount of higher-order representations. The approach taken in these systems to address higher-order challenges has been to employ specific translation ’tricks’, possibly in combination or in addition to some pre-processing techniques. Examples of such means are the quoting techniques for embedded formulas as employed in SUMO [11] and the heuristic-level modules in CYC [13]. Unfortunately, however, these solutions are strongly limited. The effect is that many desirable inferences are currently not supported, so that many relevant queries cannot be answered. This includes statements in which formulas are embedded as arguments of terms, for example, statements that employ epistemic operators such as believes or knows, temporal operators such as holdsDuring, and further operators such as disapproves or hasPurpose. While first-order automated theorem proving (FOATP) for SUMO has strongly improved recently [12], there is still only very limited support for reasoning with non-trivial embedded formulas; we give an example (free variables in premises are universal and those in the query are existential): Ex. 1 (Reasoning in temporal contexts.) What holds that holds at all times. Mary likes Bill. During 2009 Sue liked whoever Mary liked. Is there a year in which Sue has liked somebody? A: (=> ? P (holdsDuring ? Y ? P)) B: (lk Mary Bill) C: (holdsDuring (YearFn 2009) (forall (? X) (=> (lk Mary ? X) (lk Sue ? X)))) Q: (holdsDuring (YearFn ? Y) (lk Sue ? X)) This example, which is a challenge for FO-ATP (note the embedded first-order formula), is actually trivial for higher-order automated theorem provers (HO-ATP): the prover LEO-II [5] can solve it in 0.16 1 This work is funded by the German Research Foundation under grant BE 2501/6-1. 2 Articulate Software, email: cbenzmueller|apease@articulatesoftware.com 3 SUMO is available at http://www.ontologyportal. org 4 To save space ’likes’ is written as ’lk’. sec. on a standard MacBook. A slight modification of Ex.1, which LEO-II proves in 0.08 sec. , is: Ex. 2 (Ex.1 modified; A is replaced by ’True always holds’.) A’: (holdsDuring ? Y True) B: (lk Mary Bill) C: (holdsDuring (YearFn 2009) (forall (? X) (=> (lk Mary ? X) (lk Sue ? X)))) Q: (holdsDuring (YearFn ? Y) (lk Sue ? X)) Further examples are studied in [6]; there we also outline the translation from SUMO’s SUO-KIF representation language [10, 7] as used above to the new higher-order TPTP THF syntax [14] as supported by several HO-ATPs including LEO-II. 2 THE PROBLEM WITH MODAL OPERATORS Validity of Ex.1 and Ex.2 is easily shown provided that Boolean extensionality is assumed (this ensures that the denotation of each formula, also the embedded ones, is either true of false). This assumption has actually never been questioned for SUMO, neither in [7] nor in [10]. However, this assumption also leads to problematic effects as the following slight modification of Ex.2 illustrates: Ex. 3 (Ex.2 modified; now formulated for an epistimec context) A”: (knows ? Y True) B: (lk Mary Bill) C’: (knows Chris (forall (? X) (=> (lk Mary ? X) (lk Sue ? X))) Q’: (knows Chris (lk Sue Bill)) Using Boolean extensionality the query is easily shown valid and LEO-II can prove it in 0.04 sec. However, now this inference is disturbing since we have not explicitly required that (knows Chris (lk Mary Bill)) holds which intuitively seems mandatory. Hence, we here (re-)discover an issue that some logicians possibly claim as widely known: modalities have to be treated with great care in classical, extensional higher-order logic. Our ongoing work therefore studies how we can suitably adapt the modeling of affected modalities in SUMO in order to appropriately address this issue. A respective proposal is sketched next. 5 It is important to note that True in A’ can actually be replaced by other tautologies, e.g. by (equal Mary Mary); this may appear more natural and the example can still be proved by LEO-II in milliseconds. 6 For a detailed discussion of functional and Boolean extensionality in classical higher-order logic we refer to [2]. ARCOE-10 Workshop Notes