• Corpus ID: 54031216

Set of Support for Higher-Order Reasoning

  title={Set of Support for Higher-Order Reasoning},
  author={Ahmed Bhayat and Giles Reger},
Higher-order logic (HOL) is utilised in numerous domains from program verification to the formalisation of mathematics. However, automated reasoning in the higher-order domain lags behind first-order automation. Many higher-order automated provers translate portions of HOL problems to first-order logic (FOL) and pass them to FOL provers. However, FOL provers are not optimised for dealing with these translations. One of the reasons for this is that the axioms introduced during the translation (e… 

Figures and Tables from this paper

Extensional Higher-Order Paramodulation in Leo-III
Leo-III is an automated theorem prover for extensional type theory with Henkin semantics and choice that supports reasoning in polymorphic first-order and higher-order logic, in all normal quantified modal logics, as well as in different deontic logics.
Extending SMT solvers to Higher-Order Logic ( Technical Report )
This work proposes a pragmatic extension of SMT solvers to natively support higher-order reasoning without compromising their performance on FOL problems, thus leveraging the extensive research and implementation efforts dedicated to efficient FOL solving.
Extending SMT Solvers to Higher-Order Logic
This work proposes a pragmatic extension for SMT solvers to support HOL reasoning natively without compromising performance on FOL reasoning, thus leveraging the extensive research and implementation efforts dedicated to efficient SMT solving.
Extending a brainiac prover to lambda-free higher-order logic
This work proposes to start with the state-of-the-art superposition prover E and gradually enrich it with higher-order features, explaining how to extend the prover’s data structures, algorithms, and heuristics to higher- order logic, a formalism that supports partial application and applied variables.
Restricted Combinatory Unification
A restricted version of Dougherty's algorithm that is incomplete, terminating and does not require polymorphism is presented, including a novel use of a substitution tree as a filtering index for higher-order unification.
Superposition for Full Higher-Order Logic (Technical Report)
This work designs a sound and refutationally complete calculus for higher-order logic with polymorphism, extensionality, Hilbert choice, and Henkin semantics, and implements its implementation in Zipperposition on a par with an earlier, pragmatic prototype of Booleans.
Aiming for the Goal with SInE
This paper implemented biasing clause selection to postpone introduction of input clauses further from the goal and goal sensitive simplification ordering in the automatic theorem prover Vampire and presented their experimental evaluation on the TPTP benchmark.
Some Thoughts About FOL-Translations in Vampire
This paper looks at translation activity from the perspective of a first-order ATP (mostly Vampire) and finds that not all translations are equal.
Superposition for Higher-Order Logic
We recently designed two calculi as stepping stones towards superposition for full higher-order logic: Boolean-free λ-superposition and superposition for first-order logic with interpreted Booleans.
Superposition for Full Higher-order Logic
This work aims to reach a sound and refutationally complete calculus for higher-order logic with polymorphism, extensionality, Hilbert choice, and Henkin semantics, and its implementation in Zipperposition outperforms all other higher- order theorem provers.


Translating Higher-Order Clauses to First-Order Clauses
Experimental data is presented that compares the translations of function applications, types, and λ-abstractions in respect of their success rates for three automatic provers.
A First Class Boolean Sort in First-Order Theorem Proving and TPTP
This paper presents an extension FOOL of many-sorted first- order logic, in which the boolean sort is treated as a first-class sort and defines the syntax and semantics of FOOL and its model-preserving translation to first-order logic.
The Higher-Order Prover Leo-III
The automated theorem prover Leo-III for classical higher-order logic with Henkin semantics and choice is presented and natively supports almost every normal higher- order modal logic.
Set of Support for Theory Reasoning
This paper describes initial experiments using the set of support strategy to improve how a saturationbased theorem prover performs theory reasoning with explicit theory axioms, and explores the effect of allowing some limited reasoning within this set.
Selecting the Selection
The notion of lookahead selection is introduced that estimates looks ahead the effect of selecting a particular literal on the number of immediate children of the given clause and selects to minimize this value.
Otter 2.0
Otter is a clause-based theorem prover for rst-order logic with equality that operates mainly on two lists of clauses, sos and usable, for drawing inferences and searching for a refutation with the set of support strategy.
Introducing StarExec: a Cross-Community Infrastructure for Logic Solving
StarExec allows community organizers to store, manage and make available benchmark libraries; competition organizers to run logic solver competitions; and community members to do comparative evaluations of logic solvers on public or private benchmark problems.
The vampire and the FOOL
New features recently implemented in the theorem prover Vampire are presented, namely support for first-order logic with a first class boolean sort (FOOL) and polymorphic arrays and presented extensions facilitate reasoning-based program analysis.
Satallax: An Automatic Higher-Order Prover
Satallax is an automatic higher-order theorem prover that generates propositional clauses encoding (ground) tableau rules and uses MiniSat to test for unsatisfiability. We describe the
Efficiency and Completeness of the Set of Support Strategy in Theorem Proving
Evidence of the efficiency of the set of support strategy is presented, and a theorem giving sufficient conditions for its logical completeness is proved.