scholarly journals Cuts for circular proofs

10.29007/54ps ◽  
2018 ◽  
Author(s):  
Jérôme Fortier ◽  
Luigi Santocanale
Keyword(s):  
Computer Science ◽  
Proof System ◽  
Cut Elimination ◽  
Cut Rule ◽  
Model Of Computation ◽  
Categorical Models ◽  
Parity Games ◽  
Finite Products ◽  
Proof Tree ◽  
Categorical Semantics

One of the authors introduced in [1] a calculus ofcircular proofs for studying the computability arising from thefollowing categorical operations: finite products and coproducts,initial algebras, final coalgebras. The calculus of[1] is cut-free; yet, even if sound and complete forprovability, it lacks an important property for the semantics ofproofs, namely fullness w.r.t. the class of natural categorical modelscalled μ-bicomplete category in [2].We fix, with this work, this problem by adding the cut rule to thecalculus. To this goal, we need to modifying the syntacticalconstraints on the cycles of proofs so to ensure soundness of thecalculus and at same time local termination of cut-elimination. Theenhanced proof system fully represents arrows of the intended model, afree μ-bicomplete category. We also describe a cut-eliminationprocedure as a model of computation arising from the above mentionedcategorical operations. The procedure constructs a cut-freeproof-tree with infinite branches out of a finite circular proof withcuts.[1] Luigi Santocanale. A calculus of circular proofs and its categorical semantics. In Mogens Nielsen and Uffe Engberg, editors, FoSSaCS, volume 2303 of Lecture Notes in Computer Science, pages 357–371. Springer, 2002.[2] Luigi Santocanale. μ-bicomplete categories and parity games. Theoretical Informatics and Applications, 36:195–227, September 2002.

scholarly journals A Calculus of Circular Proofs and its Categorical Semantics

2001 ◽  
Vol 8 (15) ◽  
Author(s):  
Luigi Santocanale
Keyword(s):  
Fixed Point ◽  
Cut Elimination ◽  
System Of Equations ◽  
Underlying Graph ◽  
Interactive Computation ◽  
Finite Products ◽  
Categorical Semantics

<p>We present a calculus of proofs, the intended models of which<br />are categories with finite products and coproducts, initial algebras<br />and final coalgebras of functors that are recursively constructible<br />out of these operations, that is, mu-bicomplete categories. The <br />calculus satisfies the cut elimination and its main characteristic is<br />that the underlying graph of a proof is allowed to contain a <br />certain amount of cycles. To each proof of the calculus we associate<br />a system of equations which has a meaning in every mu-bicomplete<br />category. We prove that this system admits always a unique <br />solution, and by means of this theorem we define the semantics of<br />the calculus.</p><p>Keywords: Initial algebras, final coalgebras. Fixed point calculi, mu-calculi.<br />Bicompletion of categories. Models of interactive computation.</p>

Proof Theory of the Cut Rule

2018 ◽  
Author(s):  
J. R. B. Cockett ◽  
R. A. G. Seely
Keyword(s):  
Proof Theory ◽  
Graphical Representation ◽  
Basic Component ◽  
Graphical Representations ◽  
Mathematical Language ◽  
Basic Logic ◽  
Cut Rule ◽  
Structural Rules ◽  
The One ◽  
Categorical Semantics

This chapter describes the categorical proof theory of the cut rule, a very basic component of any sequent-style presentation of a logic, assuming a minimum of structural rules and connectives, in fact, starting with none. It is shown how logical features can be added to this basic logic in a modular fashion, at each stage showing the appropriate corresponding categorical semantics of the proof theory, starting with multicategories, and moving to linearly distributive categories and *-autonomous categories. A key tool is the use of graphical representations of proofs (“proof circuits”) to represent formal derivations in these logics. This is a powerful symbolism, which on the one hand is a formal mathematical language, but crucially, at the same time, has an intuitive graphical representation.

scholarly journals Generalized Bounded Linear Logic and its Categorical Semantics

2021 ◽  
pp. 226-246
Author(s):  
Yōji Fukihara ◽  
Shin-ya Katsumata
Keyword(s):  
Linear Logic ◽  
Cut Elimination ◽  
Grading System ◽  
Bounded Linear ◽  
The Core ◽  
Categorical Structure ◽  
Categorical Semantics

AbstractWe introduce a generalization of Girard et al.’s called (and its affine variant ). It is designed to capture the core mechanism of dependency in , while it is also able to separate complexity aspects of . The main feature of is to adopt a multi-object pseudo-semiring as a grading system of the !-modality. We analyze the complexity of cut-elimination in , and give a translation from with constraints to with positivity axiom. We then introduce indexed linear exponential comonads (ILEC for short) as a categorical structure for interpreting the $${!}$$ ! -modality of . We give an elementary example of ILEC using folding product, and a technique to modify ILECs with symmetric monoidal comonads. We then consider a semantics of using the folding product on the category of assemblies of a BCI-algebra, and relate the semantics with the realizability category studied by Hofmann, Scott and Dal Lago.

The consistency of arithmetic

2021 ◽  
pp. 268-311
Author(s):  
Paolo Mancosu ◽  
Sergio Galvan ◽  
Richard Zach
Keyword(s):  
Simple Proof ◽  
Sequent Calculus ◽  
Cut Elimination ◽  
Successor Function ◽  
Cut Rule ◽  
Original Proof ◽  
First Order ◽  
Successive Elimination ◽  
Pure Logic ◽  
Classical Arithmetic

This chapter opens the part of the book that deals with ordinal proof theory. Here the systems of interest are not purely logical ones, but rather formalized versions of mathematical theories, and in particular the first-order version of classical arithmetic built on top of the sequent calculus. Classical arithmetic goes beyond pure logic in that it contains a number of specific axioms for, among other symbols, 0 and the successor function. In particular, it contains the rule of induction, which is the essential rule characterizing the natural numbers. Proving a cut-elimination theorem for this system is hopeless, but something analogous to the cut-elimination theorem can be obtained. Indeed, one can show that every proof of a sequent containing only atomic formulas can be transformed into a proof that only applies the cut rule to atomic formulas. Such proofs, which do not make use of the induction rule and which only concern sequents consisting of atomic formulas, are called simple. It is shown that simple proofs cannot be proofs of the empty sequent, i.e., of a contradiction. The process of transforming the original proof into a simple proof is quite involved and requires the successive elimination, among other things, of “complex” cuts and applications of the rules of induction. The chapter describes in some detail how this transformation works, working through a number of illustrative examples. However, the transformation on its own does not guarantee that the process will eventually terminate in a simple proof.

An untyped higher order logic with Y combinator

2007 ◽  
Vol 72 (4) ◽  
pp. 1385-1404
Author(s):  
James H. Andrews
Keyword(s):  
Model Theory ◽  
Lambda Calculus ◽  
Higher Order ◽  
Order Logic ◽  
Proof System ◽  
Cut Elimination ◽  
Higher Order Logic ◽  
Consistent Model

AbstractWe define a higher order logic which has only a notion of sort rather than a notion of type, and which permits all terms of the untyped lambda calculus and allows the use of the Y combinator in writing recursive predicates. The consistency of the logic is maintained by a distinction between use and mention, as in Gilmore's logics. We give a consistent model theory, a proof system which is sound with respect to the model theory, and a cut-elimination proof for the proof system. We also give examples showing what formulas can and cannot be used in the logic.

scholarly journals On geometry of interaction for polarized linear logic

2017 ◽  
Vol 28 (10) ◽  
pp. 1639-1694
Author(s):  
MASAHIRO HAMANO ◽  
PHILIP SCOTT
Keyword(s):  
Proof Theory ◽  
Monoidal Category ◽  
Linear Logic ◽  
Cut Elimination ◽  
List Type ◽  
Monoidal Categories ◽  
Categorical Structure ◽  
Categorical Models ◽  
Focusing Property ◽  
Special Case

We present Geometry of Interaction (GoI) models for Multiplicative Polarized Linear Logic, MLLP, which is the multiplicative fragment of Olivier Laurent's Polarized Linear Logic. This is done by uniformly adding multi-points to various categorical models of GoI. Multi-points are shown to play an essential role in semantically characterizing the dynamics of proof networks in polarized proof theory. For example, they permit us to characterize the key feature of polarization, focusing, as well as being fundamental to our construction of concrete polarized GoI models.Our approach to polarized GoI involves following two independent studies, based on different categorical perspectives of GoI: (i)Inspired by the work of Abramsky, Haghverdi and Scott, a polarized GoI situation is defined in which multi-points are added to a traced monoidal category equipped with a reflexive object U. Using this framework, categorical versions of Girard's execution formula are defined, as well as the GoI interpretation of MLLP proofs. Running the execution formula is shown to characterize the focusing property (and thus polarities) as well as the dynamics of cut elimination.(ii)The Int construction of Joyal–Street–Verity is another fundamental categorical structure for modelling GoI. Here, we investigate it in a multi-pointed setting. Our presentation yields a compact version of Hamano–Scott's polarized categories, and thus denotational models of MLLP. These arise from a contravariant duality between monoidal categories of positive and negative objects, along with an appropriate bimodule structure (representing ‘non-focused proofs’) between them.Finally, as a special case of (ii) above, a compact model of MLLP is also presented based on Rel (the category of sets and relations) equipped with multi-points.

On categorical models of classical logic and the Geometry of Interaction

2007 ◽  
Vol 17 (5) ◽  
pp. 957-1027 ◽  
Author(s):  
CARSTEN FÜHRMANN ◽  
DAVID PYM
Keyword(s):  
Classical Logic ◽  
Disjoint Union ◽  
Sequent Calculus ◽  
Linear Logic ◽  
Structural Theory ◽  
Cut Elimination ◽  
Categorical Models ◽  
Boolean Lattices ◽  
Deep Exploration ◽  
Straightforward Proof

It is well known that weakening and contraction cause naive categorical models of the classical sequent calculus to collapse to Boolean lattices. In previous work, summarised briefly herein, we have provided a class of models calledclassical categoriesthat is sound and complete and avoids this collapse by interpreting cut reduction by a poset enrichment. Examples of classical categories include boolean lattices and the category of sets and relations, where both conjunction and disjunction are modelled by the set-theoretic product. In this article, which is self-contained, we present an improved axiomatisation of classical categories, together with a deep exploration of their structural theory. Observing that the collapse already happens in the absence of negation, we start with negation-free models calledDummett categories. Examples of these include, besides the classical categories mentioned above, the category of sets and relations, where both conjunction and disjunction are modelled by the disjoint union. We prove that Dummett categories are MIX, and that the partial order can be derived from hom-semilattices, which have a straightforward proof-theoretic definition. Moreover, we show that the Geometry-of-Interaction construction can be extended from multiplicative linear logic to classical logic by applying it to obtain a classical category from a Dummett category.Along the way, we gain detailed insights into the changes that proofs undergo during cut elimination in the presence of weakening and contraction.

scholarly journals On semantics of a term calculus for classical logic

2012 ◽  
Vol 92 (106) ◽  
pp. 79-95
Author(s):  
Silvia Likavec ◽  
Pierre Lescanne
Keyword(s):  
Classical Logic ◽  
Sequent Calculus ◽  
Denotational Semantics ◽  
Proof System ◽  
Cut Elimination ◽  
Kleisli Category

The calculus of Curien and Herbelin was introduced to provide the Curry-Howard correspondence for classical logic. The terms of this calculus represent derivations in the sequent calculus proof system and reduction reflects the process of cut-elimination. We investigate some properties of two well-behaved subcalculi of untyped calculus of Curien and Herbelin, closed under the call-by-name and the call-by-value reduction, respectively. Continuation semantics is given using the category of negated domains and Moggi?s Kleisli category over predomains for the continuation monad. Soundness theorems are given for both versions thus relating operational and denotational semantics. A thorough overview of the work on continuation semantics is given.

Categorical models of the differential λ-calculus

2019 ◽  
Vol 29 (10) ◽  
pp. 1513-1555
Author(s):  
Robin Cockett ◽  
Jonathan Gallagher
Keyword(s):  
Conference Paper ◽  
Additive Structure ◽  
Full Version ◽  
Differential Structure ◽  
Categorical Models ◽  
Cartesian Closed ◽  
Categorical Semantics

AbstractThe paper shows how the Scott–Koymans theorem for the untyped λ-calculus can be extended to the differential λ-calculus. The main result is that every model of the untyped differential λ-calculus may be viewed as a differential reflexive object in a Cartesian-closed differential category. This extension of the Scott–Koymans theorem depends critically on unraveling the somewhat subtle issue of which idempotents can be split so that differential structure lifts to the idempotent splitting.The paper uses (total) Turing categories with “canonical codes” as the basic categorical semantics for the λ-calculus. It develops the main result in a modular fashion by showing how to add left-additive structure to a Turing category, and then – on top of that – differential structure. For both levels of structure, it is necessary to identify how “canonical codes” must behave with respect to the added structure and, furthermore, how “universal objects” must behave. The latter is closely tied to the question – which is the crux of the paper – of which idempotents can be split while preserving the differential structure of the setting.This paper is the full version of a conference paper and includes the proofs which were omitted from that version due to page-length restrictions.

Pure Proof Theory Aims, Methods and Results: Extended Version of Talks Given at Oberwolfach and Haifa

1996 ◽  
Vol 2 (2) ◽  
pp. 159-188 ◽  
Author(s):  
Wolfram Pohlers
Keyword(s):  
Mathematical Logic ◽  
Computer Science ◽  
Proof Theory ◽  
Axiom System ◽  
Theoretical Research ◽  
Cut Elimination ◽  
Extended Version ◽  
Consistency Proof ◽  
Exciting Field ◽  
Axiom Systems

Apologies. The purpose of the following talk is to give an overview of the present state of aims, methods and results in Pure Proof Theory. Shortage of time forces me to concentrate on my very personal views. This entails that I will emphasize the work which I know best, i.e., work that has been done in the triangle Stanford, Munich and Münster. I am of course well aware that there are as important results coming from outside this triangle and I apologize for not displaying these results as well.Moreover the audience should be aware that in some points I have to oversimplify matters. Those who complain about that are invited to consult the original papers.1.1. General. Proof theory startedwithHilbert's Programme which aimed at a finitistic consistency proof for mathematics.By Gödel's Theorems, however, we know that we can neither formalize all mathematics nor even prove the consistency of formalized fragments by finitistic means. Inspite of this fact I want to give some reasons why I consider proof theory in the style of Gentzen's work still as an important and exciting field of Mathematical Logic. I will not go into applications of Gentzen's cut-elimination technique to computer science problems—this may be considered as applied proof theory—but want to concentrate on metamathematical problems and results. In this sense I am talking about Pure Proof Theory.Mathematicians are interested in structures. There is only one way to find the theorems of a structure. Start with an axiom system for the structure and deduce the theorems logically. These axiom systems are the objects of proof-theoretical research. Studying axiom systems there is a series of more or less obvious questions.

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