12 research outputs found
HoCHC: A Refutationally Complete and Semantically Invariant System of Higher-order Logic Modulo Theories
We present a simple resolution proof system for higher-order constrained Horn
clauses (HoCHC) - a system of higher-order logic modulo theories - and prove
its soundness and refutational completeness w.r.t. the standard semantics. As
corollaries, we obtain the compactness theorem and semi-decidability of HoCHC
for semi-decidable background theories, and we prove that HoCHC satisfies a
canonical model property. Moreover a variant of the well-known translation from
higher-order to 1st-order logic is shown to be sound and complete for HoCHC in
standard semantics. We illustrate how to transfer decidability results for
(fragments of) 1st-order logic modulo theories to our higher-order setting,
using as example the Bernays-Schonfinkel-Ramsey fragment of HoCHC modulo a
restricted form of Linear Integer Arithmetic
Higher-Order SMT Solving (Work in Progress)
International audienceSatisfiability modulo theories (SMT) solvers have throughout the years been able to cope with increasingly expressive formulas, from ground logics to full first-order logic modulo theories. Nevertheless, higher-order logic within SMT (HOSMT) is still little explored. In this preliminary report we discuss how to extend SMT solvers to natively support higherorder reasoning without compromising their performances on FO problems. We present a pragmatic extension of the cvc4 solver in which we generalize existing data structures and algorithms to HOSMT, thus leveraging the extensive research and implementation efforts dedicated to efficient FO solving. Our evaluation shows that the initial implementation does not add significant overhead to FO problems and its performance is on par with the encoding-based approach for HOSMT. We also discuss an alternative extension being implemented in veriT, in which new data structures and algorithms are being developed from scratch to best support HOSMT, thus avoiding the inherent difficulties of generalizing in a graceful way existing infrastructure not indented to higher-order reasoning
Superposition for Lambda-Free Higher-Order Logic
International audienceWe introduce refutationally complete superposition calculi for intentional and extensional λ-free higher-order logic, two formalisms that allow partial application and applied variables. The calculi are parameterized by a term order that need not be fully monotonic, making it possible to employ the λ-free higher-order lexicographic path and Knuth-Bendix orders. We implemented the calculi in the Zipperposition prover and evaluated them on TPTP benchmarks. They appear promising as a stepping stone towards complete, efficient automatic theorem provers for full higher-order logic
Superposition for Lambda-Free Higher-Order Logic
We introduce refutationally complete superposition calculi for intentional and extensional clausal -free higher-order logic, two formalisms that allow partial application and applied variables. The calculi are parameterized by a term order that need not be fully monotonic, making it possible to employ the -free higher-order lexicographic path and Knuth-Bendix orders. We implemented the calculi in the Zipperposition prover and evaluated them on Isabelle/HOL and TPTP benchmarks. They appear promising as a stepping stone towards complete, highly efficient automatic theorem provers for full higher-order logic
How to Prove Higher Order Theorems in First Order Logic
In this paper we are interested in using a firstorder theorem prover to prove theorems thatare formulated in some higher order logic. Tothis end we present translations of higher or-der logics into first order logic with flat sortsand equality and give a sufficient criterion forthe soundness of these translations. In addi-tion translations are introduced that are soundand complete with respect to L. Henkin's gen-eral model semantics. Our higher order logicsare based on a restricted type structure in thesense of A. Church, they have typed functionsymbols and predicate symbols, but no sorts
How to Prove Higher Order Theorems in First Order Logic
In this paper we are interested in using a first order theorem prover to prove theorems that are formulated in some higher order logic. To this end we present translations of higher order logics into first order logic with flat sorts and equality and give a sufficient criterion for the soundness of these translations. In addition translations are introduced that are sound and complete with respect to L. Henkin's general model semantics. Our higher order logics are based on a restricted type structure in the sense of A. Church, they have typed function symbols and predicate symbols, but no sorts
How to Prove Higher Order Theorems in First Order Logic
In this paper we are interested in using a firstorder theorem prover to prove theorems thatare formulated in some higher order logic. Tothis end we present translations of higher or-der logics into first order logic with flat sortsand equality and give a sufficient criterion forthe soundness of these translations. In addi-tion translations are introduced that are soundand complete with respect to L. Henkin's gen-eral model semantics. Our higher order logicsare based on a restricted type structure in thesense of A. Church, they have typed functionsymbols and predicate symbols, but no sorts
Superposition for Lambda-Free Higher-Order Logic
We introduce refutationally complete superposition calculi for intentional
and extensional clausal -free higher-order logic, two formalisms that
allow partial application and applied variables. The calculi are parameterized
by a term order that need not be fully monotonic, making it possible to employ
the -free higher-order lexicographic path and Knuth-Bendix orders. We
implemented the calculi in the Zipperposition prover and evaluated them on
Isabelle/HOL and TPTP benchmarks. They appear promising as a stepping stone
towards complete, highly efficient automatic theorem provers for full
higher-order logic
Searching the space of representations: reasoning through transformations for mathematical problem solving
The role of representation in reasoning has been long and widely regarded as crucial.
It has remained one of the fundamental considerations in the design of information-processing
systems and, in particular, for computer systems that reason. However, the
process of change and choice of representation has struggled to achieve a status as a
task for the systems themselves. Instead, it has mostly remained a responsibility for
the human designers and programmers.
Many mathematical problems have the characteristic of being easy to solve only
after a unique choice of representation has been made. In this thesis we examine two
classes of problems in discrete mathematics which follow this pattern, in the light of
automated and interactive mechanical theorem provers. We present a general notion of
structural transformation, which accounts for the changes of representation seen in such
problems, and link this notion to the existing Transfer mechanism in the interactive
theorem prover Isabelle/HOL.
We present our mechanisation in Isabelle/HOL of some specific transformations identified as key in the solutions of the aforementioned mathematical problems. Furthermore,
we present some tools that we developed to extend the functionalities of the
Transfer mechanism, designed with the specific purpose of searching efficiently the
space of representations using our set of transformations. We describe some experiments
that we carried out using these tools, and analyse these results in terms of how
close the tools lead us to a solution, and how desirable these solutions are.
The thorough qualitative analysis we present in this thesis reveals some promise as
well as some challenges for the far-reaching problem of representation in reasoning, and
the automation of the processes of change and choice of representation