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 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.

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.

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.

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.

scholarly journals Proof-search of propositional intuitionistic logic sequents by means of classical logic calculus

2008 ◽  
Vol 48 ◽  
Author(s):  
Romas Alonderis
Keyword(s):  
Intuitionistic Logic ◽  
Classical Logic ◽  
Sequent Calculus ◽  
Proof Search

In the paper, we define some classes of sequents of the propositional intuitionistic logic. These are classes of primarily and α-primarily reducible sequents. Then we show how derivability of these sequents in a propositional intuitionistic logic sequent calculus LJ0 can be checked by means of a propositional classical logic sequent calculus LK0.

A tutorial on computational classical logic and the sequent calculus

2018 ◽  
Vol 28 ◽  
Author(s):  
PAUL DOWNEN ◽  
ZENA M. ARIOLA
Keyword(s):  
Programming Languages ◽  
Classical Logic ◽  
Sequent Calculus ◽  
Control Flow ◽  
Model Of Computation ◽  
Type Safety ◽  
Proof Search ◽  
A Value ◽  
Evaluation Strategies ◽  
Compare And Contrast

AbstractWe present a model of computation that heavily emphasizes the concept of duality and the interaction between opposites–production interacts with consumption. The symmetry of this framework naturally explains more complicated features of programming languages through relatively familiar concepts. For example, binding a value to a variable is dual to manipulating the flow of control in a program. By looking at the computational interpretation of the sequent calculus, we find a language that lets us speak about duality, control flow, and evaluation order in programs as first-class concepts.We begin by reviewing Gentzen's LK sequent calculus and show how the Curry–Howard isomorphism still applies to give us a different basis for expressing computation. We then illustrate how the fundamental dilemma of computation in the sequent calculus gives rise to a duality between evaluation strategies: strict languages are dual to lazy languages. Finally, we discuss how the concept of focusing, developed in the setting of proof search, is related to the idea of type safety for computation expressed in the sequent calculus. In this regard, we compare and contrast two different methods of focusing that have appeared in the literature, static and dynamic focusing, and illustrate how they are two means to the same end.

The finite model property for various fragments of linear logic

1997 ◽  
Vol 62 (4) ◽  
pp. 1202-1208 ◽  
Author(s):  
Yves Lafont
Keyword(s):  
Classical Logic ◽  
Linear Logic ◽  
Boolean Model ◽  
Finite Model ◽  
Model Property ◽  
Boolean Models ◽  
Finite Model Property ◽  
First Order ◽  
Proof Search ◽  
Model Search

To show that a formula A is not provable in propositional classical logic, it suffices to exhibit a finite boolean model which does not satisfy A. A similar property holds in the intuitionistic case, with Kripke models instead of boolean models (see for instance [11]). One says that the propositional classical logic and the propositional intuitionistic logic satisfy a finite model property. In particular, they are decidable: there is a semi-algorithm for provability (proof search) and a semi-algorithm for non provability (model search). For that reason, a logic which is undecidable, such as first order logic, cannot satisfy a finite model property.The case of linear logic is more complicated. The full propositional fragment LL has a complete semantics in terms of phase spaces [2, 3], but it is undecidable [9]. The multiplicative additive fragment MALL is decidable, in fact PSPACE-complete [9], but the decidability of the multiplicative exponential fragment MELL is still an open problem. For affine logic, that is, linear logic with weakening, the situation is somewhat better: the full propositional fragment LLW is decidable [5].Here, we show that the finite phase semantics is complete for MALL and for LLW, but not for MELL. In particular, this gives a new proof of the decidability of LLW. The noncommutative case is mentioned, but not handled in detail.

scholarly journals The geometry of non-distributive logics

2005 ◽  
Vol 70 (4) ◽  
pp. 1108-1126 ◽  
Author(s):  
Greg Restall ◽  
Francesco Paoli
Keyword(s):  
Classical Logic ◽  
Natural Deduction ◽  
Sequent Calculus ◽  
Linear Logic ◽  
Cut Elimination ◽  
Deduction System ◽  
Inductive Definition ◽  
Theoretical Criterion ◽  
Natural Deduction System

AbstractIn this paper we introduce a new natural deduction system for the logic of lattices, and a number of extensions of lattice logic with different negation connectives. We provide the class of natural deduction proofs with both a standard inductive definition and a global graph-theoretical criterion for correctness, and we show how normalisation in this system corresponds to cut elimination in the sequent calculus for lattice logic. This natural deduction system is inspired both by Shoesmith and Smiley's multiple conclusion systems for classical logic and Girard's proofnets for linear logic.

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