Intuitionistic Podelski-Rybalchenko Theorem and Equivalence Between Inductive Definitions and Cyclic Proofs

Berardi, Stefano; Tatsuta, Makoto

doi:10.1007/978-3-030-00389-0_3

“…For example, the type t = 1 t is empty because we may assume that t is empty while testing 1 t. Instead, we express this and similar kinds of arguments using valid circular reasoning. If one were to formalize it, it would be in CLKID ω [14], although the succedent of any sequent is either empty or a singleton (as in CLJID ω [12]).…”

Section: Empty and Full Typesmentioning

confidence: 99%

“…Again, we normalize the signature before running the algorithm. From a circular derivation we now construct a valid circular proof in an intuitionistic metalogic [12]. For example, t ≤ u is interpreted as t ⊆ u, that is, every value in t is also a value in u.…”

Section: Syntactic Subtypingmentioning

confidence: 99%

Polarized Subtyping

Lakhani¹,

Das²,

DeYoung³

et al. 2022

Preprint

0

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Polarization of types in call-by-push-value naturally leads to the separation of inductively defined observable values (classified by positive types), and coinductively defined computations (classified by negative types), with adjoint modalities mediating between them. Taking this separation as a starting point, we develop a semantic characterization of typing with step indexing to capture observation depth of recursive computations. This semantics justifies a rich set of subtyping rules for an equirecursive variant of call-by-push-value, including variant and lazy records. We further present a bidirectional syntactic typing system for both values and computations that elegantly and pragmatically circumvents difficulties of type inference in the presence of width and depth subtyping for variant and lazy records. We demonstrate the flexibility of our system by systematically deriving related systems of subtyping for (a) isorecursive types, (b) call-by-name, and (c) call-by-value, all using a structural rather than a nominal interpretation of types.

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“…In the future, we plan to adapt our approach to make more effective other soundness criteria based on minimal cycles, e.g., those involving cyclic formula-based Noetherian induction reasoning [10,12], and other systems where the soundness can be checked by the global trace condition, as CLJID ω [3].…”

Section: Defining the Ordering And Derivability Conditionsmentioning

confidence: 99%

Validating Back-links of FOLID Cyclic Pre-proofs

Stratulat

¹

2018

Electron. Proc. Theor. Comput. Sci.

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Cyclic pre-proofs can be represented as sets of finite tree derivations with back-links. In the frame of the first-order logic with inductive definitions (FOL ID ), the nodes of the tree derivations are labelled by sequents and the back-links connect particular terminal nodes, referred to as buds, to other nodes labelled by a same sequent. However, only some back-links can constitute sound pre-proofs. Previously, it has been shown that special ordering and derivability conditions, defined along the minimal cycles of the digraph representing a particular normal form of the cyclic pre-proof, are sufficient for validating the back-links. In that approach, a same constraint could be checked several times when processing different minimal cycles, hence one may require additional recording mechanisms to avoid redundant computation in order to downgrade the time complexity to polynomial.We present a new approach that does not need to process minimal cycles. It based on a normal form that allows to define the validation conditions by taking into account only the root-bud paths from the non-singleton strongly connected components of its digraph.

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Polarized Subtyping

Lakhani

¹

,

Das

²

,

DeYoung

³

et al. 2022

Lecture Notes in Computer Science

3

0

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Polarization of types in call-by-push-value naturally leads to the separation of inductively defined observable values (classified by positive types), and coinductively defined computations (classified by negative types), with adjoint modalities mediating between them. Taking this separation as a starting point, we develop a semantic characterization of typing with step indexing to capture observation depth of recursive computations. This semantics justifies a rich set of subtyping rules for an equirecursive variant of call-by-push-value, including variant and lazy records. We further present a bidirectional syntactic typing system for both values and computations that elegantly and pragmatically circumvents difficulties of type inference in the presence of width and depth subtyping for variant and lazy records. We demonstrate the flexibility of our system by systematically deriving related systems of subtyping for (a) isorecursive types, (b) call-by-name, and (c) call-by-value, all using a structural rather than a nominal interpretation of types.

show abstract

Intuitionistic Podelski-Rybalchenko Theorem and Equivalence Between Inductive Definitions and Cyclic Proofs

Cited by 3 publications

References 18 publications

Polarized Subtyping

Polarized Subtyping

Validating Back-links of FOLID Cyclic Pre-proofs

Polarized Subtyping

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