SN and CR for free-style LKtq: linear decorations and simulation of normalization

2002 ◽  
Vol 67 (1) ◽  
pp. 162-196 ◽  
Author(s):  
Jean-Baptiste Joinet ◽  
Harold Schellinx ◽  
Lorenzo Tortora De Falco
Keyword(s):  
Classical Logic ◽  
Sequent Calculus ◽  
Linear Logic ◽  
Present Report ◽  
Point Of View ◽  
Strong Normalization ◽  
Structural Rules ◽  
Specific Combination ◽  
Proof Nets ◽  
Classical Proof

AbstractThe present report is a, somewhat lengthy, addendum to [4], where the elimination of cuts from derivations in sequent calculus for classical logic was studied ‘from the point of view of linear logic’. To that purpose a formulation of classical logic was used, that - as in linear logic - distinguishes between multiplicative and additive versions of the binary connectives.The main novelty here is the observation that this type-distinction is not essential: we can allow classical sequent derivations to use any combination of additive and multiplicative introduction rules for each of the connectives, and still have strong normalization and confluence of tq-reductions.We give a detailed description of the simulation of tq-reductions by means of reductions of the interpretation of any given classical proof as a proof net of PN2 (the system of second order proof nets for linear logic), in which moreover all connectives can be taken to be of one type, e.g., multiplicative.We finally observe that dynamically the different logical cuts, as determined by the four possible combinations of introduction rules, are independent: it is not possible to simulate them internally, i.e.. by only one specific combination, and structural rules.

A new deconstructive logic: linear logic

1997 ◽  
Vol 62 (3) ◽  
pp. 755-807 ◽  
Author(s):  
Vincent Danos ◽  
Jean-Baptiste Joinet ◽  
Harold Schellinx
Keyword(s):  
Classical Logic ◽  
Sequent Calculus ◽  
Predicate Logic ◽  
Linear Logic ◽  
Second Order ◽  
Main Concern ◽  
Simple Pattern ◽  
Strong Normalization ◽  
Viable Systems ◽  
Proof Nets

AbstractThe main concern of this paper is the design of a noetherian and confluent normalization for LK2 (that is, classical second order predicate logic presented as a sequent calculus).The method we present is powerful: since it allows us to recover as fragments formalisms as seemingly different as Girard's LC and Parigot's λμ, FD ([10, 12, 32, 36]), delineates other viable systems as well, and gives means to extend the Krivine/Leivant paradigm of ‘programming-with-proofs’ ([26, 27]) to classical logic; it is painless: since we reduce strong normalization and confluence to the same properties for linear logic (for non-additive proof nets, to be precise) using appropriate embeddings (so-called decorations); it is unifying: it organizes known solutions in a simple pattern that makes apparent the how and why of their making.A comparison of our method to that of embedding LK into LJ (intuitionistic sequent calculus) brings to the fore the latter's defects for these ‘deconstructive purposes’.

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.

scholarly journals Jump from parallel to sequential proofs: exponentials

2016 ◽  
Vol 28 (7) ◽  
pp. 1204-1252
Author(s):  
PAOLO DI GIAMBERARDINO
Keyword(s):  
Distinctive Feature ◽  
Linear Logic ◽  
The Other ◽  
Cut Elimination ◽  
Strong Normalization ◽  
Structural Rules ◽  
New Class ◽  
Game Semantics ◽  
Proof Nets

In previous works, by importing ideas from game semantics (notably Faggian–Maurel–Curien'sludics nets), we defined a new class of multiplicative/additive polarized proof nets, calledJ-proof nets. The distinctive feature of J-proof nets with respect to other proof net syntaxes, is the possibility of representing proof nets which are partially sequentialized, by usingjumps(that is, untyped extra edges) as sequentiality constraints. Starting from this result, in the present work, we extend J-proof nets to the multiplicative/exponential fragment, in order to take into account structural rules: More precisely, we replace the familiar linear logic notion of exponential box with a less restricting one (calledcone) defined by means of jumps. As a consequence, we get a syntax for polarized nets where, instead of a structure of boxes nested one into the other, we have one of cones which can bepartially overlapping. Moreover, we define cut-elimination for exponential J-proof nets, proving, by a variant of Gandy's method, that even in case of ‘superposed’ cones, reduction enjoys confluence and strong normalization.

scholarly journals Deep Proof Search in MELL

10.29007/p1fd ◽  
2018 ◽  
Author(s):  
Ozan Kahramanogullari
Keyword(s):  
Classical Logic ◽  
Sequent Calculus ◽  
Linear Logic ◽  
Proof Search ◽  
Proof Construction ◽  
Deep Inference ◽  
Speed Up

The deep inference presentation of multiplicative exponential linear logic (MELL) benefits from a rich combinatoric analysis with many more proofs in comparison to its sequent calculus presentation. In the deep inference setting, all the sequent calculus proofs are preserved. Moreover, many other proofs become available, and some of these proofs are much shorter. However, proof search in deep inference is subject to a greater nondeterminism, and this nondeterminism constitutes a bottleneck for applications. To this end, we address the problem of reducing nondeterminism in MELL by refining and extending our technique that has been previously applied to multiplicative linear logic and classical logic. We show that, besides the nondeterminism in commutative contexts, the nondeterminism in exponential contexts can be reduced in a proof theoretically clean manner. The method conserves the exponential speed-up in proof construction due to deep inference, exemplified by Statman tautologies. We validate the improvement in accessing the shorter proofs by experiments with our implementations.

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.

A modal view of linear logic

1994 ◽  
Vol 59 (3) ◽  
pp. 888-899 ◽  
Author(s):  
Simone Martini ◽  
Andrea Masini
Keyword(s):  
Modal Logic ◽  
Classical Logic ◽  
Decision Problem ◽  
Sequent Calculus ◽  
Linear Logic

AbstractWe present a sequent calculus for the modal logic S4, and building on some relevant features of this system (the absence of contraction rules and the confinement of weakenings into axioms and modal rules) we show how S4 can easily be translated into full prepositional linear logic, extending the Grishin-Ono translation of classical logic into linear logic. The translation introduces linear modalities (exponentials) only in correspondence with S4 modalities. We discuss the complexity of the decision problem for several classes of linear formulas naturally arising from the proposed translations.

Phase semantics and sequent calculus for pure noncommutative classical linear propositional logic

1991 ◽  
Vol 56 (4) ◽  
pp. 1403-1451 ◽  
Author(s):  
V. Michele Abrusci
Keyword(s):  
Propositional Logic ◽  
Sequent Calculus ◽  
Linear Logic ◽  
Commutative Monoid ◽  
Structural Rules ◽  
Exchange Rule ◽  
Phase Semantics ◽  
Correct Program ◽  
Image Position

The linear logic introduced in [3] by J.-Y. Girard keeps one of the so-called structural rules of the sequent calculus: the exchange rule. In a one-sided sequent calculus this rule can be formulated asThe exchange rule allows one to disregard the order of the assumptions and the order of the conclusions of a proof, and this means, when the proof corresponds to a logically correct program, to disregard the order in which the inputs and the outputs occur in a program.In the linear logic introduced in [3], the exchange rule allows one to prove the commutativity of the multiplicative connectives, conjunction (⊗) and disjunction (⅋). Due to the presence of the exchange rule in linear logic, in the phase semantics for linear logic one starts with a commutative monoid. So, the usual linear logic may be called commutative linear logic.The aim of the investigations underlying this paper was to see, first, what happens when we remove the exchange rule from the sequent calculus for the linear propositional logic at all, and then, how to recover the strength of the exchange rule by means of exponential connectives (in the same way as by means of the exponential connectives ! and ? we recover the strength of the weakening and contraction rules).

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.

scholarly journals A fine-grained computational interpretation of Girard’s intuitionistic proof-nets

2022 ◽  
Vol 6 (POPL) ◽  
pp. 1-28
Author(s):  
Delia Kesner
Keyword(s):  
Linear Logic ◽  
Operational Semantics ◽  
Functional Term ◽  
Strong Normalization ◽  
Fine Grained ◽  
Fundamental Properties ◽  
Structural Equivalence ◽  
Proof Nets ◽  
Intuitionistic Linear Logic ◽  
Single Exponential

This paper introduces a functional term calculus, called pn, that captures the essence of the operational semantics of Intuitionistic Linear Logic Proof-Nets with a faithful degree of granularity, both statically and dynamically. On the static side, we identify an equivalence relation on pn-terms which is sound and complete with respect to the classical notion of structural equivalence for proof-nets. On the dynamic side, we show that every single (exponential) step in the term calculus translates to a different single (exponential) step in the graphical formalism, thus capturing the original Girard’s granularity of proof-nets but on the level of terms. We also show some fundamental properties of the calculus such as confluence, strong normalization, preservation of β-strong normalization and the existence of a strong bisimulation that captures pairs of pn-terms having the same graph reduction.

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