Jonathan Sterling

Jonathan Sterling
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Jonathan Sterling
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Computer Science - Logic in Computer Science (5)
 
Computer Science - Programming Languages (3)
 
Mathematics - Logic (2)
 
Computer Science - Computation and Language (1)

Publications Authored By Jonathan Sterling

We contribute a general apparatus for dependent tactic-based proof refinement in the LCF tradition, in which the statements of subgoals may express a dependency on the proofs of other subgoals; this form of dependency is extremely useful and can serve as an algorithmic alternative to extensions of LCF based on non-local instantiation of schematic variables. Additionally, we introduce a novel behavioral distinction between refinement rules and tactics based on naturality. Our framework, called Dependent LCF, is already deployed in the nascent RedPRL proof assistant for computational cubical type theory. Read More

Using Martin Escardo's "effectful forcing" technique, we demonstrate the constructive validity of Brouwer's monotone Bar Theorem for any System T-definable bar. We have not assumed any non-constructive (Classical or Brouwerian) principles in this proof, and have carried out the entire development formally in the Agda proof assistant for Martin-Loef's Constructive Type Theory. Read More

The syntax and semantics of user-supplied hypothesis names in tactic languages is a thorny problem, because the binding structure of a proof is a function of the goal at which a tactic script is executed. We contribute a new language to deal with the dynamic and interactive character of names in tactic scripts called Nominal LCF, and endow it with a denotational semantics in dI-domains. A large fragment of Nominal LCF has already been implemented and used to great effect in the new RedPRL proof assistant. Read More

The contribution of this paper is the development of the syntax and semantics of multi-sorted nominal abstract binding trees (abts), an extension of second order universal algebra to support symbol-indexed families of operators. Nominal abts are essential for correctly treating the syntax of languages with generative phenomena, including exceptions and mutable state. Additionally we have developed the categorical semantics for abstract binding trees formally in Constructive Type Theory using the Agda proof assistant. Read More

At the heart of intuitionistic type theory lies an intuitive semantics called the "meaning explanations"; crucially, when meaning explanations are taken as definitive for type theory, the core notion is no longer "proof" but "verification". We'll explore how type theories of this sort arise naturally as enrichments of logical theories with further judgements, and contrast this with modern proof-theoretic type theories which interpret the judgements and proofs of logics, not their propositions and verifications. Read More

What is the proper explanation of intuitionistic hypothetical judgment, and thence propositional implication? The answer is unclear from the writings of Brouwer and Heyting, who in their lifetimes propounded multiple (sometimes conflicting) explanations of the hypothetical judgment. To my mind, the determination of an acceptable explanation must take into account its adequacy for the expression of the bar theorem and, more generally, the development of an open-ended framework for transcendental arguments in mathematics. Read More

This paper proposes the use of dependent types for pragmatic phenomena such as pronoun binding and presupposition resolution as a type-theoretic alternative to formalisms such as Discourse Representation Theory and Dynamic Semantics. Read More