A focused linear logical framework and its application to metatheory of object logics

2021 ◽  
pp. 1-29
Author(s):  
Amy Felty ◽  
Carlos Olarte ◽  
Bruno Xavier
Keyword(s):  
Programming Languages ◽  
Sequent Calculus ◽  
Necessary Conditions ◽  
Linear Logic ◽  
Cut Elimination ◽  
Logical Framework ◽  
Minimal Logic ◽  
First Order ◽  
Fundamental Properties ◽  
Logical Frameworks

Abstract Linear logic (LL) has been used as a foundation (and inspiration) for the development of programming languages, logical frameworks, and models for concurrency. LL’s cut-elimination and the completeness of focusing are two of its fundamental properties that have been exploited in such applications. This paper formalizes the proof of cut-elimination for focused LL. For that, we propose a set of five cut-rules that allows us to prove cut-elimination directly on the focused system. We also encode the inference rules of other logics as LL theories and formalize the necessary conditions for those logics to have cut-elimination. We then obtain, for free, cut-elimination for first-order classical, intuitionistic, and variants of LL. We also use the LL metatheory to formalize the relative completeness of natural deduction and sequent calculus in first-order minimal logic. Hence, we propose a framework that can be used to formalize fundamental properties of logical systems specified as LL theories.

scholarly journals Subexponentials in non-commutative linear logic

2018 ◽  
Vol 29 (8) ◽  
pp. 1217-1249 ◽  
Author(s):  
MAX KANOVICH ◽  
STEPAN KUZNETSOV ◽  
VIVEK NIGAM ◽  
ANDRE SCEDROV
Keyword(s):  
Programming Languages ◽  
Proof Theory ◽  
Linear Logic ◽  
Cut Elimination ◽  
Proof Systems ◽  
Logical Frameworks ◽  
Local Contraction ◽  
Concurrent Programming Languages ◽  
Exchange Rule ◽  
Non Local

Linear logical frameworks with subexponentials have been used for the specification of, among other systems, proof systems, concurrent programming languages and linear authorisation logics. In these frameworks, subexponentials can be configured to allow or not for the application of the contraction and weakening rules while the exchange rule can always be applied. This means that formulae in such frameworks can only be organised as sets and multisets of formulae not being possible to organise formulae as lists of formulae. This paper investigates the proof theory of linear logic proof systems in the non-commutative variant. These systems can disallow the application of exchange rule on some subexponentials. We investigate conditions for when cut elimination is admissible in the presence of non-commutative subexponentials, investigating the interaction of the exchange rule with the local and non-local contraction rules. We also obtain some new undecidability and decidability results on non-commutative linear logic with subexponentials.

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.

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.

Basic logic: reflection, symmetry, visibility

2000 ◽  
Vol 65 (3) ◽  
pp. 979-1013 ◽  
Author(s):  
Giovanni Sambin ◽  
Giulia Battilotti ◽  
Claudia Faggian
Keyword(s):  
Sequent Calculus ◽  
Linear Logic ◽  
Logical Constant ◽  
Cut Elimination ◽  
Inference Rules ◽  
Reflection Symmetry ◽  
Basic Logic ◽  
Strict Control ◽  
Geometric Symmetry

AbstractWe introduce a sequent calculus B for a new logic, named basic logic. The aim of basic logic is to find a structure in the space of logics. Classical, intuitionistic. quantum and non-modal linear logics, are all obtained as extensions in a uniform way and in a single framework. We isolate three properties, which characterize B positively: reflection, symmetry and visibility.A logical constant obeys to the principle of reflection if it is characterized semantically by an equation binding it with a metalinguistic link between assertions, and if its syntactic inference rules are obtained by solving that equation. All connectives of basic logic satisfy reflection.To the control of weakening and contraction of linear logic, basic logic adds a strict control of contexts, by requiring that all active formulae in all rules are isolated, that is visible. From visibility, cut-elimination follows. The full, geometric symmetry of basic logic induces known symmetries of its extensions, and adds a symmetry among them, producing the structure of a cube.

Non-commutative logic II: sequent calculus and phase semantics

2000 ◽  
Vol 10 (2) ◽  
pp. 277-312 ◽  
Author(s):  
PAUL RUET
Keyword(s):  
Special Class ◽  
Sequent Calculus ◽  
Linear Logic ◽  
Partial Orders ◽  
Cut Elimination ◽  
Phase Semantics ◽  
Entropy Relation

Non-commutative logic, which is a unification of commutative linear logic and cyclic linear logic, is extended to all linear connectives: additives, exponentials and constants. We give two equivalent versions of the sequent calculus (directly with the structure of order varieties, and with their presentations as partial orders), phase semantics and a cut-elimination theorem. This involves, in particular, the study of the entropy relation between partial orders, and the introduction of a special class of order varieties: the series–parallel order varieties.

On the unification of classical, intuitionistic and affine logics

2018 ◽  
Vol 29 (8) ◽  
pp. 1177-1216
Author(s):  
CHUCK LIANG
Keyword(s):  
Intuitionistic Logic ◽  
Classical Logic ◽  
Sequent Calculus ◽  
Linear Logic ◽  
Kripke Semantics ◽  
Cut Elimination ◽  
Structural Rules ◽  
Phase Semantics ◽  
Intuitionistic Logics

This article presents a unified logic that combines classical logic, intuitionistic logic and affine linear logic (restricting contraction but not weakening). We show that this unification can be achieved semantically, syntactically and in the computational interpretation of proofs. It extends our previous work in combining classical and intuitionistic logics. Compared to linear logic, classical fragments of proofs are better isolated from non-classical fragments. We define a phase semantics for this logic that naturally extends the Kripke semantics of intuitionistic logic. We present a sequent calculus with novel structural rules, which entail a more elaborate procedure for cut elimination. Computationally, this system allows affine-linear interpretations of proofs to be combined with classical interpretations, such as the λμ calculus. We show how cut elimination must respect the boundaries between classical and non-classical modes of proof that correspond to delimited control effects.

scholarly journals Pretopologies and completeness proofs

1995 ◽  
Vol 60 (3) ◽  
pp. 861-878 ◽  
Author(s):  
Giovanni Sambin
Keyword(s):  
Classical Logic ◽  
Sequent Calculus ◽  
Predicate Logic ◽  
Linear Logic ◽  
Double Negation ◽  
Open Problems ◽  
Completeness Proof ◽  
Structural Rules ◽  
First Order ◽  
Definition Of

Pretopologies were introduced in [S], and there shown to give a complete semantics for a propositional sequent calculus BL, here called basic linear logic, as well as for its extensions by structural rules, ex falso quodlibet or double negation. Immediately after Logic Colloquium '88, a conversation with Per Martin-Löf helped me to see how the pretopology semantics should be extended to predicate logic; the result now is a simple and fully constructive completeness proof for first order BL and virtually all its extensions, including the usual, or structured, intuitionistic and classical logic. Such a proof clearly illustrates the fact that stronger set-theoretic principles and classical metalogic are necessary only when completeness is sought with respect to a special class of models, such as the usual two-valued models.To make the paper self-contained, I briefly review in §1 the definition of pretopologies; §2 deals with syntax and §3 with semantics. The completeness proof in §4, though similar in structure, is sensibly simpler than that in [S], and this is why it is given in detail. In §5 it is shown how little is needed to obtain completeness for extensions of BL in the same language. Finally, in §6 connections with proofs with respect to more traditional semantics are briefly investigated, and some open problems are put forward.

Multiple conclusion linear logic: cut elimination and more

2020 ◽  
Vol 30 (1) ◽  
pp. 157-174 ◽  
Author(s):  
Harley Eades III ◽  
Valeria de Paiva
Keyword(s):  
Sequent Calculus ◽  
Linear Logic ◽  
Direct Proof ◽  
Cut Elimination ◽  
Tensor Model ◽  
Double Negation ◽  
Proof Nets ◽  
Small Change ◽  
Definition Of ◽  
Intuitionistic Linear Logic

Abstract Full intuitionistic linear logic (FILL) was first introduced by Hyland and de Paiva, and went against current beliefs that it was not possible to incorporate all of the linear connectives, e.g. tensor, par and implication, into an intuitionistic linear logic. Bierman showed that their formalization of FILL did not enjoy cut elimination as such, but Bellin proposed a small change to the definition of FILL regaining cut elimination and using proof nets. In this note we adopt Bellin’s proposed change and give a direct proof of cut elimination for the sequent calculus. Then we show that a categorical model of FILL in the basic dialectica category is also a linear/non-linear model of Benton and a full tensor model of Melliès’ and Tabareau’s tensorial logic. We give a double-negation translation of linear logic into FILL that explicitly uses par in addition to tensor. Lastly, we introduce a new library to be used in the proof assistant Agda for proving properties of dialectica categories.

Quantifier elimination and parametric polymorphism in programming languages

1992 ◽  
Vol 2 (2) ◽  
pp. 213-226 ◽  
Author(s):  
Harry G. Mairson
Keyword(s):  
Programming Languages ◽  
Lambda Calculus ◽  
Quantifier Elimination ◽  
Type Inference ◽  
Cut Elimination ◽  
Standard Interpretation ◽  
Inference Problem ◽  
First Order ◽  
Typed Lambda Calculus ◽  
Parametric Polymorphism

AbstractWe present a simple and easy-to-understand explanation of ML type inference and parametric polymorphism within the framework of type monomorphism, as in the first order typed lambda calculus. We prove the equivalence of this system with the standard interpretation using type polymorphism, and extend the equivalence to include polymorphic fixpoints. The monomorphic interpretation gives a purely combinatorial understanding of the type inference problem, and is a classic instance of quantifier elimination, as well as an example of Gentzen-style cut elimination in the framework of the Curry-Howard propositions-as-types analogy.

PRESERVATION OF STRUCTURAL PROPERTIES IN INTUITIONISTIC EXTENSIONS OF AN INFERENCE RELATION

2018 ◽  
Vol 24 (3) ◽  
pp. 291-305 ◽  
Author(s):  
TOR SANDQVIST
Keyword(s):  
Structural Properties ◽  
Sequent Calculus ◽  
General Nature ◽  
Cut Elimination ◽  
Formal Proofs ◽  
Structural Rules ◽  
Logical Operators ◽  
First Order ◽  
Order Language ◽  
The Given

AbstractThe article approaches cut elimination from a new angle. On the basis of an arbitrary inference relation among logically atomic formulae, an inference relation on a language possessing logical operators is defined by means of inductive clauses similar to the operator-introducing rules of a cut-free intuitionistic sequent calculus. The logical terminology of the richer language is not uniquely specified, but assumed to satisfy certain conditions of a general nature, allowing for, but not requiring, the existence of infinite conjunctions and disjunctions. We investigate to what extent structural properties of the given atomic relation are preserved through the extension to the full language. While closure under the Cut rule narrowly construed is not in general thus preserved, two properties jointly amounting to closure under the ordinary structural rules, including Cut, are.Attention is then narrowed down to the special case of a standard first-order language, where a similar result is obtained also for closure under substitution of terms for individual parameters. Taken together, the three preservation results imply the familiar cut-elimination theorem for pure first-order intuitionistic sequent calculus.In the interest of conceptual economy, all deducibility relations are specified purely inductively, rather than in terms of the existence of formal proofs of any kind.

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