Cut elimination in multifocused linear logic

We study cut elimination for a multifocused variant of full linear logic in the sequent calculus. The multifocused normal form of proofs yields problems that do not appear in a standard focused system, related to the constraints in grouping rule instances in focusing phases. We show that cut elimination can be performed in a sensible way even though the proof requires some specific lemmas to deal with multifocusing phases, and discuss the difficulties arising with cut elimination when considering normal forms of proofs in linear logic.Comment: In Proceedings LINEARITY 2014, arXiv:1502.0441

Інноваціїність філософії педагогіки Пауло Фрейре

In the work of P. Freire, the recognized Brazilian thinker of the XX century, a philosopher of pedagogy, education appears in ontoanthropological, historical, social and political implications. The philosophy of pedagogy in this context is not an absolutization of knowledge as a monoparadigm, but a polyparadigmatic complex phenomenon. The questions posed in Freire's work concern the directions and projects of development of pedagogy of education as pedagogy of self-development of mankind. The critical analysis of the pedagogy of "banking education" as authoritarian is presented. It is proved that overcoming such a model is possible only under the conditions of liberation pedagogy, based on the unity of duality of subjects of educational reality: teacher-student and student-teacher, where there is a dialogue of partnership, not confrontation and paternalism.В роботі П. Фрейре, визнаного бразильського мислителя XX століття, філософа та педагога, освіта постає в онтоантропологічних, історичних, соціальних та політичних імплікаціях. Філософія педагогіки в такому контексті є не абсолютизацією знання як монопарадигми, а поліпарадигмальним складним феноменом. Питання, поставлені в роботі Фрейре, стосуються напрямків і проектів розвитку педагогіки освіти як педагогіки саморозвитку людства. Подано критичний аналіз педагогіки «банківського навчання», як авторитарної. Доведено, що подолання такої моделі можливо лише за умов здійсненням визвольної педагогіки, що базується на єдності двоїстості суб'єктів освітньої реальності: вчитель-учень і учень-вчитель, де існує діалог партнерства, а не протистояння і патерналізму

Інноваціїність філософії педагогіки Пауло Фрейре

In the work of P. Freire, the recognized Brazilian thinker of the XX century, a philosopher of pedagogy, education appears in ontoanthropological, historical, social and political implications. The philosophy of pedagogy in this context is not an absolutization of knowledge as a monoparadigm, but a polyparadigmatic complex phenomenon. The questions posed in Freire's work concern the directions and projects of development of pedagogy of education as pedagogy of self-development of mankind. The critical analysis of the pedagogy of "banking education" as authoritarian is presented. It is proved that overcoming such a model is possible only under the conditions of liberation pedagogy, based on the unity of duality of subjects of educational reality: teacher-student and student-teacher, where there is a dialogue of partnership, not confrontation and paternalism.В роботі П. Фрейре, визнаного бразильського мислителя XX століття, філософа та педагога, освіта постає в онтоантропологічних, історичних, соціальних та політичних імплікаціях. Філософія педагогіки в такому контексті є не абсолютизацією знання як монопарадигми, а поліпарадигмальним складним феноменом. Питання, поставлені в роботі Фрейре, стосуються напрямків і проектів розвитку педагогіки освіти як педагогіки саморозвитку людства. Подано критичний аналіз педагогіки «банківського навчання», як авторитарної. Доведено, що подолання такої моделі можливо лише за умов здійсненням визвольної педагогіки, що базується на єдності двоїстості суб'єктів освітньої реальності: вчитель-учень і учень-вчитель, де існує діалог партнерства, а не протистояння і патерналізму

Search for Program Structure

The community of programming language research loves the Curry-Howard correspondence between proofs and programs. Cut-elimination as computation, theorems for free, \u27call/cc\u27 as excluded middle, dependently typed languages as proof assistants, etc. Yet we have, for all these years, missed an obvious observation: "the structure of programs corresponds to the structure of proof search". For pure programs and intuitionistic logic, more is known about the latter than the former. We think we know what programs are, but logicians know better! To motivate the study of proof search for program structure, we retrace recent research on applying focusing to study the canonical structure of simply-typed lambda-terms. We then motivate the open problem of extending canonical forms to support richer type systems, such as polymorphism, by discussing a few enticing applications of more canonical program representations

Classical realizability in the CPS target language

AbstractMotivated by considerations about Krivine's classical realizability, we introduce a term calculus for an intuitionistic logic with record types, which we call the CPS target language. We give a reformulation of the constructions of classical realizability in this language, using the categorical techniques of realizability triposes and toposes.We argue that the presentation of classical realizability in the CPS target language simplifies calculations in realizability toposes, in particular it admits a nice presentation of conjunction as intersection type which is inspired by Girard's ludics

A Sequent Calculus for a Semi-Associative Law

We introduce a sequent calculus with a simple restriction of Lambek\u27s product rules that precisely captures the classical Tamari order, i.e., the partial order on fully-bracketed words (equivalently, binary trees) induced by a semi-associative law (equivalently, tree rotation). We establish a focusing property for this sequent calculus (a strengthening of cut-elimination), which yields the following coherence theorem: every valid entailment in the Tamari order has exactly one focused derivation. One combinatorial application of this coherence theorem is a new proof of the Tutte-Chapoton formula for the number of intervals in the Tamari lattice Y_n. Elsewhere, we have also used the sequent calculus and the coherence theorem to build a surprising bijection between intervals of the Tamari order and a natural fragment of lambda calculus, consisting of the beta-normal planar lambda terms with no closed proper subterms

The polarized λ-calculus

A natural deduction system isomorphic to the focused sequent calculus for polarized intuitionistic logic is proposed. The system comes with a language of proof-terms, named polarized λ-calculus, whose reduction rules express simultaneously a normalization procedure and the isomorphic copy of the cut-elimination procedure pertaining to the focused sequent calculus. Noteworthy features of this natural deduction system are: how the polarity of a connective determines the style of its elimination rule; the existence of a proof-search strategy which is equivalent to focusing in the sequent calculus; the highlydisciplined organization of the syntax - even atoms have introduction, elimination and normalization rules. The polarized λ-calculus is a programming formalism close to call-by-push-value, but justified by its proof-theoretical pedigree.This research was financed by Portuguese Funds through FCT Fundac¸ao para a Ci ˜ encia ˆ e a Tecnologia, within the Project UID/MAT/00013/2013.info:eu-repo/semantics/publishedVersio

Curry-Howard for sequent calculus at last!

This paper tries to remove what seems to be the remaining stumbling blocks in the way to a full understanding of the Curry-Howard isomorphism for sequent calculus, namely the questions: What do variables in proof terms stand for? What is co-control and a co-continuation? How to define the dual of Parigot's mu-operator so that it is a co-control operator? Answering these questions leads to the interpretation that sequent calculus is a formal vector notation with first-class co-control. But this is just the "internal" interpretation, which has to be developed simultaneously with, and is justified by, an "external" one, offered by natural deduction: the sequent calculus corresponds to a bi-directional, agnostic (w.r.t. the call strategy), computational lambda-calculus. Next, the duality between control and co-control is studied and proved in the context of classical logic, where one discovers that the classical sequent calculus has a distortion towards control, and that sequent calculus is the de Morgan dual of natural deduction.(undefined

Proofs and Refutations for Intuitionistic and Second-Order Logic

The ?^{PRK}-calculus is a typed ?-calculus that exploits the duality between the notions of proof and refutation to provide a computational interpretation for classical propositional logic. In this work, we extend ?^{PRK} to encompass classical second-order logic, by incorporating parametric polymorphism and existential types. The system is shown to enjoy good computational properties, such as type preservation, confluence, and strong normalization, which is established by means of a reducibility argument. We identify a syntactic restriction on proofs that characterizes exactly the intuitionistic fragment of second-order ?^{PRK}, and we study canonicity results