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

Lightweight verification of functional programs

We have built several tools to help with testing and verifying functional programs. All three tools are based on QuickCheck properties. Our goal is to allow programmers to do more with QuickCheck properties than just test them.The first tool is QuickSpec, which finds equational specifications, and can be used to help with writing a specification or for program understanding. On top of QuickSpec, we have built HipSpec, which proves properties about Haskell programs, and uses QuickSpec to prove the necessary lemmas. We also describe PULSE and eqc_par_statem, which together can be used to find race conditions in Erlang programs.We believe that testable properties are a good basis for reasoning and verification, and that they give many of the benefits of formal verification without the cost of proof. The chief reason is that they are formal specifications for which the programmer can always get a counterexample when they are false. Furthermore, using testable properties allows us to write better tools. None of our tools would be possible if our properties were not testable.We also present work on encoding types in first-order logic, an essential component when using first-order provers to reason about programs. Our encodings are simple but extremely efficient, as evidenced by benchmarks. We develop the theory behind sound type encodings, and have written tools that implement our ideas

Lightweight verification of functional programs

We have built several tools to help with testing and verifying functional programs. All three tools are based on QuickCheck properties. Our goal is to allow programmers to do more with QuickCheck properties than just test them.The first tool is QuickSpec, which finds equational specifications, and can be used to help with writing a specification or for program understanding. On top of QuickSpec, we have built HipSpec, which proves properties about Haskell programs, and uses QuickSpec to prove the necessary lemmas. We also describe PULSE and eqc_par_statem, which together can be used to find race conditions in Erlang programs.We believe that testable properties are a good basis for reasoning and verification, and that they give many of the benefits of formal verification without the cost of proof. The chief reason is that they are formal specifications for which the programmer can always get a counterexample when they are false. Furthermore, using testable properties allows us to write better tools. None of our tools would be possible if our properties were not testable.We also present work on encoding types in first-order logic, an essential component when using first-order provers to reason about programs. Our encodings are simple but extremely efficient, as evidenced by benchmarks. We develop the theory behind sound type encodings, and have written tools that implement our ideas

Lightweight verification of functional programs

We have built several tools to help with testing and verifying functional programs. All three tools are based on QuickCheck properties. Our goal is to allow programmers to do more with QuickCheck properties than just test them.The first tool is QuickSpec, which finds equational specifications, and can be used to help with writing a specification or for program understanding. On top of QuickSpec, we have built HipSpec, which proves properties about Haskell programs, and uses QuickSpec to prove the necessary lemmas. We also describe PULSE and eqc_par_statem, which together can be used to find race conditions in Erlang programs.We believe that testable properties are a good basis for reasoning and verification, and that they give many of the benefits of formal verification without the cost of proof. The chief reason is that they are formal specifications for which the programmer can always get a counterexample when they are false. Furthermore, using testable properties allows us to write better tools. None of our tools would be possible if our properties were not testable.We also present work on encoding types in first-order logic, an essential component when using first-order provers to reason about programs. Our encodings are simple but extremely efficient, as evidenced by benchmarks. We develop the theory behind sound type encodings, and have written tools that implement our ideas

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

Order-sorted algebra I: equational deduction for multiple inheritance, overloading, exceptions and partial operations

AbstractThis paper generalizes many-sorted algebra (MSA) to order-sorted algebra (OSA) by allowing a partial ordering relation on the set of sorts. This supports abstract data types with multiple inheritance (in roughly the sense of object-oriented programming), several forms of polymorphism and overloading, partial operations (as total on equationally defined subsorts), exception handling, and an operational semantics based on term rewriting. We give the basic algebraic constructions for OSA, including quotient, image, product and term algebra, and we prove their basic properties, including quotient, homomorphism, and initiality theorems. The paper's major mathematical results include a notion of OSA deduction, a completeness theorem for it, and an OSA Birkhoff variety theorem. We also develop conditional OSA, including initiality, completeness, and McKinsey-Malcev quasivariety theorems, and we reduce OSA to (conditional) MSA, which allows lifting many known MSA results to OSA. Retracts, which intuitively are left inverses to subsort inclusions, provide relatively inexpensive run-time error handling. We show that it is safe to add retracts to any OSA signature, in the sense that it gives rise to a conservative extension. A final section compares and contrasts many different approaches to OSA. This paper also includes several examples demonstrating the flexibility and applicability of OSA, including some standard benchmarks like stack and list, as well as a much more substantial example, the number hierarchy from the naturals up to the quaternions

More SPASS with Isabelle: superposition with hard sorts and configurable simplification

Sledgehammer for Isabelle/HOL integrates automatic theorem provers to discharge interactive proof obligations. This paper considers a tighter integration of the superposition prover SPASS to increase Sledgehammer’s success rate. The main enhancements are native support for hard sorts (simple types) in SPASS, simplification that honors the orientation of Isabelle simp rules, and a pair of clause-selection strategies targeted at large lemma libraries. The usefulness of this integration is confirmed by an evaluation on a vast benchmark suite and by a case study featuring a formalization of language-based security