Extending the Automated Reasoning Toolbox

Due to the semi-decidable nature of first-order logic, it can be desirable to address a wider range of problems than the standard ones of satisfiability and derivability. We extend the automated reasoning toolbox by introducing three new tools for analysing problems in first-order logic. Infinox aims to show finite unsatisfiability, i.e. the absence of models with finite domains, and is a useful complement to finite model-finding. Infinox can also be used to reason about the relative sizes of model domains in sorted first-order logic. Monotonox uses a novel analysis that can identify sorts with extendable domains, improving on well-known existing translations between sorted and unsorted logic. This enables reasoning tools for unsorted logic to tackle problems in sorted logic. Conversely, finite model finders benefit from sort information which Monotonox can add to unsorted problems. Equalox, the third tool in our toolbox, can improve the per- formance of first-order provers on problems involving transitive relations. The insight is that first-order provers are poor at applying the transitivity axiom effectively, but that the problem can always be transformed to safely remove the transitivity axiom. Finally, we explore the field of computational linguistics as an application of automated reasoning. The tool Morfar uses a constraint solver to analyse the morphology of an input language. The result is a novel automatic method for segmentation and labelling that works well even when there is very little training data available

Encoding monomorphic and polymorphic types

Most automatic theorem provers are restricted to untyped logics, and existing translations from typed logics are bulky or unsound. Recent research proposes monotonicity as a means to remove some clutter. Here we pursue this approach systematically, analysing formally a variety of encodings that further improve on efficiency while retaining soundness and completeness. We extend the approach to rank-1 polymorphism and present alternative schemes that lighten the translation of polymorphic symbols based on the novel notion of “cover”. The new encodings are implemented, and partly proved correct, in Isabelle/HOL. Our evaluation finds them vastly superior to previous schemes

Encoding monomorphic and polymorphic types

Most automatic theorem provers are restricted to untyped logics, and existing translations from typed logics are bulky or unsound. Recent research proposes monotonicity as a means to remove some clutter. Here we pursue this approach systematically, analysing formally a variety of encodings that further improve on efficiency while retaining soundness and completeness. We extend the approach to rank-1 polymorphism and present alternative schemes that lighten the translation of polymorphic symbols based on the novel notion of “cover”. The new encodings are implemented, and partly proved correct, in Isabelle/HOL. Our evaluation finds them vastly superior to previous schemes

Deciding First-Order Satisfiability when Universal and Existential Variables are Separated

We introduce a new decidable fragment of first-order logic with equality, which strictly generalizes two already well-known ones -- the Bernays-Sch\"onfinkel-Ramsey (BSR) Fragment and the Monadic Fragment. The defining principle is the syntactic separation of universally quantified variables from existentially quantified ones at the level of atoms. Thus, our classification neither rests on restrictions on quantifier prefixes (as in the BSR case) nor on restrictions on the arity of predicate symbols (as in the monadic case). We demonstrate that the new fragment exhibits the finite model property and derive a non-elementary upper bound on the computing time required for deciding satisfiability in the new fragment. For the subfragment of prenex sentences with the quantifier prefix $\exists^* \forall^* \exists^*$ the satisfiability problem is shown to be complete for NEXPTIME. Finally, we discuss how automated reasoning procedures can take advantage of our results.Comment: Extended version of our LICS 2016 conference paper, 23 page

Encoding monomorphic and polymorphic types

Many automatic theorem provers are restricted to untyped logics, and existing translations from typed logics are bulky or unsound. Recent research proposes monotonicity as a means to remove some clutter when translating monomorphic to un-typed first-order logic. Here we pursue this approach systematically, analysing formally a variety of encodings that further improve on efficiency while retaining soundness and completeness. We extend the approach to rank-1 polymorphism and present alternative schemes that lighten the translation of polymorphic symbols based on the novel notion of “cover”. The new encodings are implemented in Isabelle/HOL as part of the Sledgehammer tool. We include informal proofs of soundness and correctness, and have formalized the monomorphic part of this work in Isabelle/HOL. Our evaluation finds the new encodings vastly superior to previous schemes

Encoding monomorphic and polymorphic types

Many automatic theorem provers are restricted to untyped logics, and existing translations from typed logics are bulky or unsound. Recent research proposes monotonicity as a means to remove some clutter when translating monomorphic to un-typed first-order logic. Here we pursue this approach systematically, analysing formally a variety of encodings that further improve on efficiency while retaining soundness and completeness. We extend the approach to rank-1 polymorphism and present alternative schemes that lighten the translation of polymorphic symbols based on the novel notion of “cover”. The new encodings are implemented in Isabelle/HOL as part of the Sledgehammer tool. We include informal proofs of soundness and correctness, and have formalized the monomorphic part of this work in Isabelle/HOL. Our evaluation finds the new encodings vastly superior to previous schemes

Automated Inference of Finite Unsatisfiability

We present Infinox, an automated tool for analyzing first-order logic problems, aimed at showing finite unsatisfiability, i.e., the absence of models with finite domains. Finite satisfiability is not a decidable problem (only semi-decidable), which means that such a tool can never be complete. Nonetheless, our hope is that Infinox be a useful complement to finite model finders in practice. Infinox uses several different proof techniques for showing infinity of a set, each of which requires the identification of a function or a relation with particular properties. Infinox enumerates candidates to such functions and relations, and subsequently uses an automated theorem prover as a sub-procedure to try to prove the resulting proof obligations. We have evaluated Infinox on the relevant problems from the TPTP benchmark suite, and we are able to automatically show finite unsatisfiability for over 25% of these problems