# Linear logic

TL;DR: This column presents an intuitive overview of linear logic, some recent theoretical results, and summarizes several applications oflinear logic to computer science.

Abstract: Linear logic was introduced by Girard in 1987 [11] . Since then many results have supported Girard' s statement, \"Linear logic is a resource conscious logic,\" and related slogans . Increasingly, computer scientists have come to recognize linear logic as an expressive and powerful logic with connection s to a variety of topics in computer science . This column presents a.n intuitive overview of linear logic, some recent theoretical results, an d summarizes several applications of linear logic to computer science . Other introductions to linear logic may be found in [12, 361 .

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15 Jan 1995

TL;DR: In this paper, the authors define abstract bases as the bases of compact elements of algebraic domains and define the notion of ideal completion as the relation with which a basis can be equipped.

Abstract: bases were introduced in [Smy77] where they are called “R-structures”. Examples of abstract bases are concrete bases of continuous domains, of course, where the relation≺ is the restriction of the order of approximation. Axiom (INT) is satisfied because of Lemma 2.2.15 and because we have required bases in domains to have directed sets of approximants for each element. Other examples are partially ordered sets, where (INT) is satisfied because of reflexivity. We will shortly identify posets as being exactly the bases of compact elements of algebraic domains. In what follows we will use the terminology developed at the beginning of this chapter, even though the relation ≺ on an abstract basis need neither be reflexive nor antisymmetric. This is convenient but in some instances looks more innocent than it is. An idealA in a basis, for example, has the property (following from directedness) that for everyx ∈ A there is another element y ∈ A with x ≺ y. In posets this doesn’t mean anything but here it becomes an important feature. Sometimes this is stressed by using the expression ‘ A is a round ideal’. Note that a set of the form↓x is always an ideal because of (INT) but that it need not contain x itself. We will refrain from calling ↓x ‘principal’ in these circumstances. Definition 2.2.21. For a basis〈B,≺〉 let Idl(B) be the set of all ideals ordered by inclusion. It is called theideal completionof B. Furthermore, leti : B → Idl(B) denote the function which maps x ∈ B to ↓x. If we want to stress the relation with whichB is equipped then we write Idl(B,≺) for the ideal completion. Proposition 2.2.22.Let 〈B,≺〉 be an abstract basis.

1,210 citations

### Cites methods from "Linear logic"

...Another proposal by Gordon Plotkin is to use Linear Types (in the sense of Linear Logic [Girard, 1987]) as a metalanguage for Domain Theory [Plotkin, 1993]....

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01 Jan 1993TL;DR: The π-calculus is a model of concurrent computation based upon the notion of naming that is generalized from monadic to polyadic form and semantics is done in terms of both a reduction system and a version of labelled transitions called commitment.

Abstract: The π-calculus is a model of concurrent computation based upon the notion of naming. It is first presented in its simplest and original form, with the help of several illustrative applications. Then it is generalized from monadic to polyadic form. Semantics is done in terms of both a reduction system and a version of labelled transitions called commitment; the known algebraic axiomatization of strong bisimilarity is given in the new setting, and so also is a characterization in modal logic. Some theorems about the replication operator are proved.

1,016 citations

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01 Oct 1995TL;DR: Sketches for Endofunctors: Catesian Closed Categories, Diagrams, and Toposes.

Abstract: Preliminaries. Categories. Functors. Diagrams. Naturality and Sketches. Products and Sums. Catesian Closed Categories. Finite Discrete Sketches. Limits and Colimits. More About Sketches. Fibrations. Adjoints. Algebras for Endofunctors. Toposes.

1,006 citations

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Intel

^{1}TL;DR: This paper investigates a weaker logic, MTL, which is intended to cope with the tautologies of left-continuous t-norms and their residua, and completeness of MTL with respect to linearly ordered MTL-algebras is proved.

Abstract: Hajek's BL logic is the fuzzy logic capturing the tautologies of continuous t-norms and their residua. In this paper we investigate a weaker logic, MTL, which is intended to cope with the tautologies of left-continuous t-norms and their residua. The corresponding algebraic structures, MTL-algebras, are defined and completeness of MTL with respect to linearly ordered MTL-algebras is proved. Besides, several schematic extensions of MTL are also considered as well as their corresponding predicate calculi.

900 citations

### Cites background from "Linear logic"

...In the picture we have also added the (propositional) AKne Multiplicative fragment of Girard’s Linear logic [9,17], a-MLL, which results to be equivalent to the extension of Monoidal logic with the axiom requiring the negation to be involutive (see [2])....

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TL;DR: It is shown that the syntactic restriction induced by LinLog is not performed at the cost of any expressive power: a mapping from full linear logic to LinLog, preserving focusing proofs, and analogous to the normalization to clausal form for classical logic, is presented.

Abstract: The deep symmetry of linear logic [18] makes it suitable for providing abstract models of computation, free from implementation details which are, by nature, oriented and nonsymmetrical. I propose here one such model, in the area of logic programming, where the basic computational principle is Computation = Proof search Proofs considered here are those of the Gentzen style sequent calculus for linear logic. However, proofs in this system may be redundant, in that two proofs can be syntactically different although identical up to some irrelevant reordering or simplification of the applications of the inference rules. This leads to an untractable proof search where the search procedure is forced to make costly choices which turn out to be irrelevant. To overcome this problem, a subclass of proofs, called the 'focusing' proofs, which is both complete (any derivable formula in linear logic has a focusing proof) and tractable (many irrelevant choices in the search are eliminated when aimed at focusing proofs) is identified. The main constraint underlying the specification of focusing proofs has been to preserve the symmetry of linear logic, which is its most salient feature. In particular, dual connectives have dual properties with respect to focusing proofs. Then, a programming language, called LinLog, consisting of a fragment of linear logic, in which focusing proofs have a more compact form, is presented. Linlog deals with formulae which have a syntax similar to that of the definite clauses and goals of Horn logic, but the crucial difference here is that it allows clauses with multiple atoms in the head, connected by the 'par' (multiplicative disjunction). It is then shown that the syntactic restriction induced by LinLog is not performed at the cost of any expressive power: a mapping from full linear logic to LinLog, preserving focusing proofs, and analogous to the normalization to clausal form for classical logic, is presented.

734 citations

##### References

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TL;DR: An effective rule (or algorithm) for distinguishing sentences from nonsentences is obtained, which works not only for the formal languages of interest to the mathematical logician, but also for natural languages such as English, or at least for fragments of such languages.

Abstract: (1958). The Mathematics of Sentence Structure. The American Mathematical Monthly: Vol. 65, No. 3, pp. 154-170.

1,432 citations

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TL;DR: It is shown that the syntactic restriction induced by LinLog is not performed at the cost of any expressive power: a mapping from full linear logic to LinLog, preserving focusing proofs, and analogous to the normalization to clausal form for classical logic, is presented.

Abstract: The deep symmetry of linear logic [18] makes it suitable for providing abstract models of computation, free from implementation details which are, by nature, oriented and nonsymmetrical. I propose here one such model, in the area of logic programming, where the basic computational principle is Computation = Proof search Proofs considered here are those of the Gentzen style sequent calculus for linear logic. However, proofs in this system may be redundant, in that two proofs can be syntactically different although identical up to some irrelevant reordering or simplification of the applications of the inference rules. This leads to an untractable proof search where the search procedure is forced to make costly choices which turn out to be irrelevant. To overcome this problem, a subclass of proofs, called the 'focusing' proofs, which is both complete (any derivable formula in linear logic has a focusing proof) and tractable (many irrelevant choices in the search are eliminated when aimed at focusing proofs) is identified. The main constraint underlying the specification of focusing proofs has been to preserve the symmetry of linear logic, which is its most salient feature. In particular, dual connectives have dual properties with respect to focusing proofs. Then, a programming language, called LinLog, consisting of a fragment of linear logic, in which focusing proofs have a more compact form, is presented. Linlog deals with formulae which have a syntax similar to that of the definite clauses and goals of Horn logic, but the crucial difference here is that it allows clauses with multiple atoms in the head, connected by the 'par' (multiplicative disjunction). It is then shown that the syntactic restriction induced by LinLog is not performed at the cost of any expressive power: a mapping from full linear logic to LinLog, preserving focusing proofs, and analogous to the normalization to clausal form for classical logic, is presented.

734 citations

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TL;DR: The main results of this paper show that the same notions of computability can be realized within the highly restricted monogenic formal systems called by Post the "Tag" systems, and within a peculiarly restricted variant of Turing machine which has two tapes, but can neither write on nor erase these tapes.

Abstract: The equivalence of the notions of effective computability as based (1) on formal systems (e.g., those of Post), and (2) on computing machines (e.g., those of Turing) has been shown in a number of ways. The main results of this paper show that the same notions of computability can be realized within (1) the highly restricted monogenic formal systems called by Post the "Tag" systems, and (2) within a peculiarly restricted variant of Turing machine which has two tapes, but can neither write on nor erase these tapes. From these, or rather from the arithmetization device used in their construction, we obtain also an interesting basis for recursive function theory involving programs of only the simplest arithmetic operations. We show first how Turing machines can be regarded as programmed computers. Then by defining a hierarchy of programs which perform certain arithmetic transformations, we obtain the representation in terms of the restricted two-tape machines. These machines, in turn, can be represented in terms of Post normal canonical systems in such a way that each instruction for the machine corresponds to a set of productions in a system which has the monogenic property (for each string in the Post system just one production can operate). This settles the questions raised

721 citations

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TL;DR: In this paper, a simple decision procedure for the word problems for commutative semigroups and polynomial deals is presented, which requires computational storage space growing exponentially with the size of the problem instance to which the procedure is applied.

Abstract: Any decision procedure for the word problems for commutative semigroups and polynomial deals inherently requires computational storage space growing exponentially with the size of the problem instance to which the procedure is applied. This bound is achieved by a simple procedure for the semigroup problem.

592 citations