A verified abstract machine for functional coroutines

A normalization procedure is presented for a classical natural deduction (ND) proof system. This proof system, called N-Graphs, has a multiple conclusion proof structure, where cycles are allowed. With this, we have developed a thorough treatment of cycles, including cycles normalization via an algorithm. We also demonstrate the usefulness of the graphical framework of N-Graphs, where derivations are seen as digraphs. We use geometric perspective techniques to establish the normalization mechanism, thus giving a direct normalization proof. Moreover, the subformula and separation properties are determined.

A normalization procedure is given for classical natural deduction with the standard rule of indirect proof applied to arbitrary formulas. For normal derivability and the subformula property, it is sufficient to permute down instances of indirect proof whenever they have been used for concluding a major premiss of an elimination rule. The result applies even to natural deduction for classical modal logic.

We study how to postpone the application of the reductio ad absurdum rule ( $$\mathsf {raa}$$ ) in classical natural deduction. This technique is connected with two normalization strategies for classical logic, due to Prawitz and Seldin, respectively. We introduce a variant of Seldin’s strategy for the postponement of $$\mathsf {raa}$$ , which induces a negative translation (a variant of Kuroda’s one) from classical to intuitionistic and minimal logic. Through this translation, Glivenko’s theorem from classical to intuitionistic and minimal logic is proven.

Full classical S5 in natural deduction with weak normalization

A piece of functional programming folklore due to Piponi provides Lob's theorem from modal provability logic with a computational interpretation as an unusual fixed point. Interpreting modal necessity as an arbitrary Functor in Haskell, the type of Lob's theorem is inhabited by a fixed point function allowing each part of a structure to refer to the whole. However, Functor's logical interpretation may be used to prove Lob's theorem only by relying on its implicit functorial strength, an axiom not available in the provability modality. As a result, the well known loeb fixed point cheats by using functorial strength to implement its recursion. Rather than Functor, a closer Curry analogue to modal logic's Howard inspiration is a closed (semi-)comonad, of which Haskell's ComonadApply typeclass provides analogous structure. Its computational interpretation permits the definition of a novel fixed point function allowing each part of a structure to refer to its own context within the whole. This construction further guarantees maximal sharing and asymptotic efficiency superior to loeb for locally contextual computations upon a large class of structures. With the addition of a distributive law, closed comonads may be composed into spaces of arbitrary dimensionality while preserving the performance guarantees of this new fixed point. From these elements, we construct a small embedded domain-specific language to elegantly express and evaluate multidimensional spreadsheet-like recurrences for a variety of cellular automata.

We present a foundational language for spatially distributed programming, called Lambda 5, that addresses both mobility of code and locality of resources. In order to construct our system, we appeal to the powerful propositions-as-types interpretation of logic. Specifically, we take the possible worlds of the intuitionistic modal logic IS5 to be nodes on a network, and the connectives /spl square/ and /spl diams/ to reflect mobility and locality, respectively. We formulate a novel system of natural deduction for IS5, decomposing the introduction and elimination rules for /spl square/ and /spl diams/, thereby allowing the corresponding programs to be more direct. We then give an operational semantics to our calculus that is type-safe, logically faithful, and computationally realistic.

In this paper, we develop and study the following perspective -- just as higher-order functions give exponentials, higher-order continuations give coexponentials. From this, we design a language that combines exponentials and coexponentials, producing a duality of lambda abstraction. We formalise this language by giving an extension of a call-by-value simply-typed lambda-calculus with covalues, coabstraction, and coapplication. We develop the semantics of this language using the axiomatic structure of continuations, which we use to produce an equational theory, that gives a complete axiomatisation of control effects. We give a computational interpretation to this language using speculative execution and backtracking, and use this to derive the classical control operators and computational interpretation of classical logic, and encode common patterns of control flow using continuations. By dualising functional completeness, we further develop duals of first-order arrow languages using coexponentials. Finally, we discuss the implementation of this duality as control operators in programming, and develop some applications.

We introduce λµ...., an extension of Parigot's λµ-calculus where disjunction is taken as a primitive. The associated reduction relation, which includes the permutative conversions related to disjunction, is Church-Rosser, strongly normalizing, and such that the normal deductions satisfy the subformula property. From a computer science point of view, λµ.... may be seen as the core of a typed CBN functional language featuring product, coproduct, and control operators.

Multiary sequent terms were originally introduced as a tool for proving termination of permutative conversions in cut-free sequent calculus. This work develops the language of multiary sequent terms into a term calculus for the computational (Curry-Howard) interpretation of a fragment of sequent calculus with cuts and cut-elimination rules. The system, called generalized multiary λ-calculus , is a rich extension of the λ-calculus where the computational content of the sequent calculus format is explained through an enlarged form of the application constructor. Such constructor exhibits the features of multiarity (the ability to form lists of arguments) and generality (the ability to prescribe a kind of continuation). The system integrates in a modular way the multiary λ-calculus and an isomorphic copy of the λ-calculus with generalized application, Λ J (in particular, natural deduction is captured internally up to isomorphism). In addition, the system: (i) comes with permutative conversion rules, whose role is to eliminate the new features of application; (ii) is equipped with reduction rules — either the μ-rule, typical of the multiary setting, or rules for cut-elimination, which enlarge the ordinary β-rule. This article establishes the metatheory of the system, with emphasis on the role of the μ-rule, and including a study of the interaction of reduction and permutative conversions.

This paper studies the relationship between Argumentation Logic (AL), a recently defined logic based on the study of argumentation in AI, and classical Propositional Logic (PL). In particular, it shows that AL and PL are logically equivalent in that they have the same entailment relation from any given classically consistent theory. This equivalence follows from a correspondence between the non-acceptability of (arguments for) sentences in AL and Natural Deduction (ND) proofs of the complement of these sentences. The proof of this equivalence uses a restricted form of ND proofs, where hypotheses in the application of the Reductio of Absurdum inference rule are required to be “relevant” to the absurdity derived in the rule. The paper also discusses how the argumentative re-interpretation of PL could help control the application of ex-falso quodlibet in the presence of inconsistencies.

Propositional Godel logic G extends intuitionistic logic with the non-constructive principle of linearity (A → B) ∨ (B → A). We introduce a Curry-Howard correspondence for G and show that a simple natural deduction calculus can be used as a typing system. The resulting functional language extends the simply typed λ-calculus via a synchronous communication mechanism between parallel processes, which increases its expressive power. The normalization proof employs original termination arguments and proof transformations implementing forms of code mobility. Our results provide a computational interpretation of G, thus proving A. Avron's 1991 thesis.

In the previous chapters we have given a framework based on labelled deduction that provides a systematic solution to the problem of finding uniform and modular presentations of propositional non-classical logics. Here we consider quantified modal logics [89, 104, 141] as a significant case study of the additional complexity introduced by quantifiers with respect to the range of possible logics and semantics for them. (Other quantified non-classical logics, e.g. quantified relevance logics, can be presented similarly.) In this case we must choose not only properties of the accessibility relation in the Kripke frame, as in the propositional case, but also how the domains of individuals change between worlds; for example, do the domains vary arbitrarily (varying domains), or do the same objects exist in every world (constant domains), or are objects possibly created (increasing domains) or destroyed (decreasing domains) when moving to accessible worlds?

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