arXiv subjects
Publications and source records attributed to Ruy J. G. B. de Queiroz.
In order to get $λ$-models with a rich structure of $\infty$-groupoid, which we call "homotopy $λ$-models", a general technique is described for solving domain equations on any cartesian closed $\infty$-category (c.c.i.) with enough points. Finally, the technique is applied in a particular c.c.i., where some examples of homotopy $λ$-models are given.
The use of the open and searchable Wittgenstein's Nachlass (The Wittgenstein Archives at the University of Bergen (WAB)) has proved instrumental in the quest for a common thread of Wittgenstein's view on the connections between meaning, use and consequences, going from the Notebooks to later writings (including the Philosophical Investigations) and beyond. Here we take this as the basis for a proposal for a formal counterpart of a 'meaning-as-use' (dialogical/game-theoretical) semantics for the language of predicate logic. In order to further consolidate this perspective, we shall need to bring in key excerpts from Wittgenstein oeuvre (including the Nachlass) and from those formal semanticists who advocate a different perspective on the connections between proofs and meaning. With this in mind we consider several passages from Wittgenstein's published as well as unpublished writings to build a whole picture of a formal counterpart to 'meaning is use' based on the idea that explanations of consequences via 'movements within language' ought to be taken as a central aspect to Wittgenstein's shift from 'interpretation of symbols in a state of affairs' to 'use of symbols' which underpins his 'meaning is use' paradigm. As in the Investigations "every interpretation hangs in the air together with what it interprets, and cannot give it any support. Interpretations by themselves do not determine meaning", as well as in a remark from his transitional period (1929-30): "Perhaps one should say that the expression 'interpretation of symbols' is misleading and one should instead say 'the use of symbols'." Significantly, we wish the present examination of the searchable Nachlass can make a relevant step towards a formal counterpart to the 'meaning is use' dictum, while highlighting an important common thread from Wittgenstein's very early to very late writings.
We use a labelled deduction system ( LND$_{ED-}$TRS ) based on the concept of computational paths (sequences of rewrites) as equalities between two terms of the same type, which allowed us to carry out in homotopic theory an approach using the concept of computational paths. From this, we show that the computational paths can be used to perform the proofs of the $LND_{EQ}-TRS_{2}$ rewriting system.
One takes advantage of some basic properties of every homotopic $λ$-model (e.g.\ extensional Kan complex) to explore the higher $βη$-conversions, which would correspond to proofs of equality between terms of a theory of equality of any extensional Kan complex. Besides, Identity types based on computational paths are adapted to a type-free theory with higher $λ$-terms, whose equality rules would be contained in the theory of any $λ$-homotopic model.
As a celebration of the \emph{Tractatus} 100th anniversary it might be worth revisiting its relation to the later writings. From the former to the latter, David Pears recalls that ``everyone is aware of the holistic character of Wittgenstein's later philosophy, but it is not so well known that it was already beginning to establish itself in the \emph{Tractatus}" (\emph{The False Prison}, 1987). From the latter to the former, Stephen Hilmy's (\emph{The Later Wittgenstein}, 1987) extensive study of the \emph{Nachlass} has helped removing classical misconceptions such as Hintikka's claim that ``Wittgenstein in the \emph{Philosophical Investigations} almost completely gave up the calculus analogy." Hilmy points out that even in the \emph{Investigations} one finds the use of the calculus/game paradigm to the understanding of language, such as ``in operating with the word" (Part I, §559) and ``it plays a different part in the calculus". Hilmy also quotes from a late (1946) unpublished manuscript (MS 130) ``this sentence has use in the calculus of language"), which seems to be compatible with ``asking whether and how a proposition can be verified is only a particular way of asking `How do you mean?'." Central in this back and forth there is an aspect which seems to deserve attention in the discussion of a semantics for the language of mathematics which might be based on (normalisation of) proofs and/or Hintikka/Lorenzen game-dialogue: the explication of consequences. Such a discussion is substantially supported by the use of the open and searchable \emph{The Wittgenstein Archives at the University of Bergen}. These findings are framed within the discussion of the meaning of logical constants in the context of natural deduction style rules of inference.
An appropriate framework is put forward for the construction of $λ$-models with $\infty$-groupoid structure, which we call \textit{homotopic $λ$-models}, through the use of an $\infty$-category with cartesian closure and enough points. With this, we establish the start of a project of generalization of Domain Theory and $λ$-calculus, in the sense that the concept of proof (path) of equality of $λ$-terms is raised to \textit{higher proof} (homotopy).
We use a labelled deduction system based on the concept of computational paths (sequences of rewrites) as equalities between two terms of the same type. We also define a term rewriting system that is used to make computations between these computational paths, establishing equalities between equalities. We then proceed to show the main result here: using this system to obtain the calculation of the fundamental group of the circle, of the torus and the real projective plane.
The lambda calculus is a universal programming language. It can represent the computable functions, and such offers a formal counterpart to the point of view of functions as rules. Terms represent functions and this allows for the application of a term/function to any other term/function, including itself. The calculus can be seen as a formal theory with certain pre-established axioms and inference rules, which can be interpreted by models. Dana Scott proposed the first non-trivial model of the extensional lambda calculus, known as $ D_\infty$, to represent the $λ$-terms as the typical functions of set theory, where it is not allowed to apply a function to itself. Here we propose a construction of an $\infty$-groupoid from any lambda model endowed with a topology. We apply this construction for the particular case $D_\infty$, and we see that the Scott topology does not provide enough information about the relationship between higher homotopies. This motivates a new line of research focused on the exploration of $λ$-models with the structure of a non-trivial $\infty$-groupoid to generalize the proofs of term conversion (e.g., $β$-equality, $η$-equality) to higher-proofs in $λ$-calculus.
In this work, we use a labelled deduction system based on the concept of computational paths (sequence of rewrites) as equalities between two terms of the same type. We also define a term rewriting system that is used to make computations between these computational paths, establishing equalities between equalities. We use a labelled deduction system based on the concept of computational paths (sequence of rewrites) to obtain some results of algebraic topology and with support of the Seifet-Van Kampen Theorem we will calculate, in a way less complex than the one made in mathematics \cite{Munkres} and the technique of homotopy type theory \cite{hott}, the fundamental group of Klein Blottle $\mathbb{K}^2$, of the Torus $\mathbb{T}^2$ and Two holed Torus $\mathbb{M}_2=\mathbb{T}^2\# \mathbb{T}^2$ (the connected sum two torus).
One of the most interesting entities of homotopy type theory is the identity type. It gives rise to an interesting interpretation of the equality, since one can semantically interpret the equality between two terms of the same type as a collection of homotopical paths between points of the same space. Since this is only a semantical interpretation, the addition of paths to the syntax of homotopy type theory has been recently proposed by De Queiroz, Ramos and De Oliveira . In these works, the authors propose an entity known as `computational path', proposed by De Queiroz and Gabbay in 1994, and show that it can be used to formalize the identity type. We have found that it is possible to use these computational paths as a tool to achieve one central result of algebraic topology and homotopy type theory: the calculation of fundamental groups of surfaces. We review the concept of computational paths and the $LND_{EQ}-TRS$, which is a term rewriting system proposed by De Oliveira in 1994 to map redundancies between computational paths. We then proceed to calculate the fundamental group of the circle, cylinder, M{ö}bius band, torus and the real projective plane. Moreover, we show that the use of computational paths make these calculations simple and straightforward, whereas the same result is much harder to obtain using the traditional code-encode-decode approach of homotopy type theory.
The treatment of equality as a type in type theory gives rise to an interesting type-theoretic structure known as `identity type'. The idea is that, given terms $a,b$ of a type $A$, one may form the type $Id_{A}(a,b)$, whose elements are proofs that $a$ and $b$ are equal elements of type $A$. A term of this type, $p : Id_{A}(a,b)$, makes up for the grounds (or proof) that establishes that $a$ is indeed equal to $b$. Based on that, a proof of equality can be seen as a sequence of substitutions and rewrites, also known as a `computational path'. One interesting fact is that it is possible to rewrite computational paths using a set of reduction rules arising from an analysis of redundancies in paths. These rules were mapped by De Oliveira in 1994 in a term rewrite system known as $LND_{EQ}-TRS$. Here we use computational paths and this term rewrite system to develop the main foundations of homotopy type theory, i.e., we develop the lemmas and theorems connected to the main types of this theory, types such as products, coproducts, identity type, transport and many others. We also show that it is possible to directly construct path spaces through computational paths. To show this, we construct the natural numbers and the fundamental group of the circle, showing results connected to these structures.
The treatment of equality as a type in type theory gives rise to an interesting type-theoretic structure known as `identity type'. The idea is that, given terms $a,b$ of a type $A$, one may form the type $Id_{A}(a,b)$, whose elements are proofs that $a$ and $b$ are equal elements of type $A$. A term of this type, $p : Id_{A}(a,b)$, makes up for the grounds (or proof) that establishes that $a$ is indeed equal to $b$. Based on that, a proof of equality can be seen as a sequence of substitutions and rewrites, also known as a `computational path'. One interesting fact is that it is possible to rewrite computational paths using a set of reduction rules arising from an analysis of redundancies in paths. These rules were mapped by De Oliveira in 1994 in a term rewrite system known as $LND_{EQ}-TRS$. Here we use computational paths and this term rewrite system to work with path spaces. In homotopy type theory, the main technique used to define path spaces is the code-encode-decode approach. Our objective is to propose an alternative approach based on the theory of computational paths. We believe this new approach is simpler and more straightforward than the code-encode-decode one. We then use our approach to obtain two important results of homotopy type theory: the construction of the path space of the naturals and the calculation of the fundamental group of the circle.
Since proof-nets for MLL- were introduced by Girard (1987), several studies have appeared dealing with its soundness proof. Bellin & Van de Wiele (1995) produced an elegant proof based on properties of subnets (empires and kingdoms) and Robinson (2003) proposed a straightforward generalization of this presentation for proof-nets from sequent calculus for classical logic. In 2014 it was presented an extension of these studies to obtain a proof of the sequentialization theorem for the fragment of N-Graphs with conjunction, disjunction and negation connectives, via the notion of sub-N-Graphs. N-Graphs is a symmetric natural deduction calculus with multiple conclusions that adopts Danos-Regnier's criterion and has defocussing switchable links. In this paper, we present a sequentization for full propositional classical N-Graphs, showing how to find a split node in the middle of the proof even with a global rule for discharging hypothesis.
The main objective of this work is to study mathematical properties of computational paths. Originally proposed by de Queiroz \& Gabbay (1994) as `sequences or rewrites', computational paths are taken to be terms of the identity type of Martin Löf's Intensional Type Theory, since these paths can be seen as the grounds on which the propositional equality between two computational objects stand. From this perspective, this work aims to show that one of the properties of the identity type is present on computational paths. We are referring to the fact that that the identity type induces a groupoid structure, as proposed by Hofmann \& Streicher (1994). Using categorical semantics, we show that computational paths induce a groupoid structure. We also show that computational paths are capable of inducing higher categorical structures.
Context-free language theory is a subject of high importance in computer language processing technology as well as in formal language theory. This paper presents a formalization, using the Coq proof assistant, of fundamental results related to context-free grammars and languages. These include closure properties (union, concatenation and Kleene star), grammar simplification (elimination of useless symbols inaccessible symbols, empty rules and unit rules) and the existence of a Chomsky Normal Form for context-free grammars.
Context-free grammar simplification is a subject of high importance in computer language processing technology as well as in formal language theory. This paper presents a formalization, using the Coq proof assistant, of the fact that general context-free grammars generate languages that can be also generated by simpler and equivalent context-free grammars. Namely, useless symbol elimination, inaccessible symbol elimination, unit rules elimination and empty rules elimination operations were described and proven correct with respect to the preservation of the language generated by the original grammar.
Context-free languages (CFLs) are highly important in computer language processing technology as well as in formal language theory. The Pumping Lemma is a property that is valid for all context-free languages, and is used to show the existence of non context-free languages. This paper presents a formalization, using the Coq proof assistant, of the Pumping Lemma for context-free languages.
The main objective of this work is to study mathematical properties of computational paths. Originally proposed by de Queiroz \& Gabbay (1994) as `sequences of rewrites', computational paths can be seen as the grounds on which the propositional equality between two computational objects stand. Using computational paths and categorical semantics, we take any type $A$ of type theory and construct a groupoid for this type. We call this groupoid the fundamental groupoid of a type $A$, since it is similar to the one obtained using the homotopical interpretation of the identity type. The main difference is that instead of being just a semantical interpretation, computational paths are entities of the syntax of type theory. We also expand our results, using computational paths to construct fundamental groupoids of higher levels.