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تسجيل مستخدم جديدAutomating Induction by Reflection
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تأليف
Johannes Schoisswohl
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Despite recent advances in automating theorem proving in full first-order theories, inductive reasoning still poses a serious challenge to state-of-the-art theorem provers. The reason for that is that in first-order logic induction requires an infinite number of axioms, which is not a feasible input to a computer-aided theorem prover requiring a finite input. Mathematical practice is to specify these infinite sets of axioms as axiom schemes. Unfortunately these schematic definitions cannot be formalized in first-order logic, and therefore not supported as inputs for first-order theorem provers. In this work we introduce a new method, inspired by the field of axiomatic theories of truth, that allows to express schematic inductive definitions, in the standard syntax of multi-sorted first-order logic. Further we test the practical feasibility of the method with state-of-the-art theorem provers, comparing it to solvers native techniques for handling induction.
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Despite recent advances in automating theorem proving in full first-order theories, inductive reasoning still poses a serious challenge to state-of-the-art theorem provers. The reason for that is that in first-order logic induction requires an infini
te number of axioms, which is not a feasible input to a computer-aided theorem prover requiring a finite input. Mathematical practice is to specify these infinite sets of axioms as axiom schemes. Unfortunately these schematic definitions cannot be formalized in first-order logic, and therefore not supported as inputs for first-order theorem provers. In this work we introduce a new method, inspired by the field of axiomatic theories of truth, that allows to express schematic inductive definitions, in the standard syntax of multi-sorted first-order logic. Further we test the practical feasibility of the method with state-of-the-art theorem provers, comparing it to solvers native techniques for handling induction. This paper is an extended version of the LFMTP 21 submission with the same title.
The principle of strong induction, also known as k-induction is one of the first techniques for unbounded SAT-based Model Checking (SMC). While elegant and simple to apply, properties as such are rarely k-inductive and when they can be strengthened,
there is no effective strategy to guess the depth of induction. It has been mostly displaced by techniques that compute inductive strengthenings based on interpolation and property directed reachability (Pdr). In this paper, we present kAvy, an SMC algorithm that effectively uses k-induction to guide interpolation and Pdr-style inductive generalization. Unlike pure k-induction, kAvy uses Pdr-style generalization to compute and strengthen an inductive trace. Unlike pure Pdr, kAvy uses relative k-induction to construct an inductive invariant. The depth of induction is adjusted dynamically by minimizing a proof of unsatisfiability. We have implemented kAvy within the Avy Model Checker and evaluated it on HWMCC instances. Our results show that kAvy is more effective than both Avy and Pdr, and that using k-induction leads to faster running time and solving more instances. Further, on a class of benchmarks, called shift, kAvy is orders of magnitude faster than Avy, Pdr and k-induction.
In functional programming, datatypes a la carte provide a convenient modular representation of recursive datatypes, based on their initial algebra semantics. Unfortunately it is highly challenging to implement this technique in proof assistants that
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In control theory, to solve a finite-horizon sequential decision problem (SDP) commonly means to find a list of decision rules that result in an optimal expected total reward (or cost) when taking a given number of decision steps. SDPs are routinely
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