Abstract
AbstractInduction and coinduction are both used extensively within mathematics and computer science. Algebraic formulations of these principles make the duality between them apparent, but do not account well for the way they are commonly used in deduction. Generally, the formalization of these reasoning methods employs inference rules that express a generalexplicit(co)induction scheme. Non-well-founded proof theory provides an alternative, more robust approach for formalizingimplicit(co)inductive reasoning. This approach has been extremely successful in recent years in supporting implicit inductive reasoning, but is not as well-developed in the context of coinductive reasoning. This paper reviews the general method of non-well-founded proofs, and puts forward a concrete natural framework for (co)inductive reasoning, based on (co)closure operators, that offers a concise framework in which inductive and coinductive reasoning are captured as we intuitively understand and use them. Through this framework we demonstrate the enormous potential of non-well-founded deduction, both in the foundational theoretical exploration of (co)inductive reasoning and in the provision of proof support for (co)inductive reasoning within (semi-)automated proof tools.
Highlights
The principle of induction is a key technique in mathematical reasoning that is widely used in computer science for reasoning about recursive data types and computations
One concrete example of an inductively defined set is that of finite lists, which can be constructed starting from the empty list and one constructor operator that adds an element to the head of the list
We present the general method of non-well-founded proof theory (Section 3.1), and provides a concrete example of a non-well-founded proof system for inductive reasoning in the setting of the transitive closure (Section 3.2), where the implicit form of inductive reasoning is compared against the explicit one
Summary
The principle of induction is a key technique in mathematical reasoning that is widely used in computer science for reasoning about recursive data types (such as numbers or lists) and computations. This logic offers a succinct and intuitive dual treatment to induction and coinduction, while still supporting their common practices in deduction, making it great for prototyping It has the benefits of (1) conciseness: no need for a separate language or interpretation for definitions, nor for fully general least/greatest-fixed-point operators; (2) intuitiveness: the concept of transitive closure is basic, and the dual closure is simple to grasp, resulting in a simpler metatheory; (3) illumination: similarities, dualities, and differences between induction and coinduction are clearly demonstrated; and (4) naturality: local reasoning is rudimentary, and the global structure of proofs directly reflects higher-level reasoning. Verification, and the task of determining the precise relationship between systems for cyclic reasoning and standard systems for explicit reasoning
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