Fragment Of Linear Logic Research Articles

Using Z3 to Verify Inferences in Fragments of Linear Logic

Using Z3 to Verify Inferences in Fragments of Linear Logic

Glue Semantics

Glue Semantics (Glue) is a general framework for semantic composition and the syntax–semantics interface. The framework grew out of an interdisciplinary collaboration at the intersection of formal linguistics, formal logic, and computer science. Glue assumes a separate level of syntax; this can be any syntactic framework in which syntactic structures have heads. Glue uses a fragment of linear logic for semantic composition. General linear logic terms in Glue meaning constructors are instantiated relative to a syntactic parse. The separation of the logic of composition from structural syntax distinguishes Glue from other theories of semantic composition and the syntax–semantics interface. It allows Glue to capture semantic ambiguity, such as quantifier scope ambiguity, without necessarily positing an ambiguity in the syntactic structure. Glue is introduced here in relation to four of its key properties, which are used as organizing themes: resource-sensitive composition, flexible composition, autonomy of syntax, and syntax/semantics non-isomorphism.

Making Logical Form type-logical: Glue semantics for Minimalist syntax

Glue semantics is a theory of the syntax–semantics interface according to which the syntactic structure of a sentence produces premises in a fragment of linear logic, and the semantic interpretation(s) of the sentence correspond to the proof(s) derivable from those premises. This paper describes how Glue can be connected to a Minimalist syntactic theory and compares the result with the more mainstream approach to the syntax–semantics interface in Minimalism, according to which the input to semantic interpretation is a syntactic structure (Logical Form) derived by covert movement operations. I argue that the Glue approach has advantages that make it worth exploring.

Rich Coalitional Resource Games

We propose a simple model of interaction for resource-conscious agents. The resources involved are expressed in fragments of Linear Logic. We investigate a few problems relevant to cooperative games, such as deciding whether a group of agents can form a coalition and act together in a way that satisfies all of them. In terms of solution concepts, we study the computational aspects of the core of a game. The main contributions are a formal link with the existing literature, and complexity results for several classes of models.

A Geometrical Representation of the Basic Laws of Categorial Grammar

We present a geometrical analysis of the principles that lay at the basis of Categorial Grammar and of the Lambek Calculus. In Abrusci (On residuation, 2014) it is shown that the basic properties known as Residuation laws can be characterized in the framework of Cyclic Multiplicative Linear Logic, a purely non-commutative fragment of Linear Logic. We present a summary of this result and, pursuing this line of investigation, we analyze a well-known set of categorial grammar laws: Monotonicity, Application, Expansion, Type-raising, Composition, Geach laws and Switching laws.

A correspondence between maximal abelian sub-algebras and linear logic fragments

We show a correspondence between a classification of maximal abelian sub-algebras (MASAs) proposed by Jacques Dixmier (Dixmier 1954.Annals of Mathematics59(2) 279–286) and fragments of linear logic. We expose for this purpose a modified construction of Girard's hyperfinite geometry of interaction (Girard 2011.Theoretical Computer Science412(20) 1860–1883). The expressivity of the logic soundly interpreted in this model is dependent on properties of a MASA which is a parameter of the interpretation. We also unveil the essential role played by MASAs in previous geometry of interaction constructions.

The undecidability theorem for the Horn-like fragment of linear logic (Revisited)

There has been an increased interest in the decision problems for linear logic and its fragments. Here, we give a fully self-contained, easy-to-follow, but fully detailed, direct and constructive proof of the undecidability of a very simple Horn-like fragment of linear logic, which is accessible to a wide range of people. Namely, we show that there is a direct correspondence between terminated computations of a Minsky machine M and cut-free linear logic derivations for a Horn-like sequent of the form \begin{equation*} \bang{\Phi_M},\ l_1 \vdash l_0, \end{equation*} where ΦM consists only of Horn-like implications of the following simple forms \begin{equation*} (l \llto l'),\ \ ((l\otimes r) \llto l'),\ \ (l\llto (r\otimes l')),\ \ and \ \ (l\llto (l'\oplus l'')), \end{equation*} where l1, l0, l, l′, l″ and r stand for literals.Neither negation, nor &, nor constants, nor embedded implications/bangs are used here.Furthermore, our particular correspondence constructed above provides decidability for each of the Horn-like fragments whenever we confine ourselves to any two forms of the above Horn-like implications, along with the complexity bounds that come from the proof.

Infinitary affine proofs

Even though the multiplicative–additive fragment of linear logic forbids structural rules in general, is does admit a bounded form of exponential modalities enjoying a bounded form of structural rules. The approximation theorem, originally proved by Girard, states that if full linear logic proves a propositional formula, then the multiplicative–additive fragment proves every bounded approximation of it. This may be understood as the fact that multiplicative–additive linear logic is somehow dense in full linear logic. Our goal is to give a technical formulation of this informal remark. We introduce a Cauchy-complete space of infinitary affine term-proofs and we show that it yields a fully complete model of multiplicative exponential polarised linear logic, in the style of Girard's ludics. Moreover, the subspace of finite term-proofs, which is a model of multiplicative polarised linear logic, is dense in the space of all term-proofs.

The decidability of the intensional fragment of classical linear logic

The intensional fragment of classical propositional linear logic combines modalities with contraction-free relevance logic — adding modalized versions of the thinning and contraction rules. This paper provides a proof of the decidability of this logic based on a sequent calculus formulation. Some related logics and some other fragments of linear logic are also shown decidable.

Dynamic Spaces in Concurrent Constraint Programming

Concurrent constraint programming (CCP) is a declarative model for concurrency where agents interact with each other by posting (telling) and asking constraints (formulas in logic) in a shared store of partial information. With the advent of emergent applications as security protocols, social networks and cloud computing, the CCP model has been extended in different directions to faithfully model such systems as follows: (1) It has been shown that a name-passing discipline, where agents can communicate local names, can be described through the interplay of local (∃) processes along with universally (∀) quantified asks. This strategy has been used, for instance, to model the generation and communication of fresh values (nonces) in mobile reactive systems as security protocols; and (2) the underlying constraint system in CCP has been enhanced with local stores for the specification of distributed spaces. Then, agents are allowed to share some information with others but keep some facts for themselves. Recently, we have shown that local stores can be neatly represented in CCP by considering a constraint system where constraints are built from a fragment of linear logic with subexponentials (SELL⋒). In this paper, we explore the use of existential (⋓) and universal (⋒) quantification over subexponentials in SELL⋒ in order to endow CCP with the ability to communicate location (space) names. The resulting CCP language that we obtain is a model of distributed computation where it is possible to dynamically establish new shared spaces for communication. We thus extend the sort of mobility achieved in (1) –for variables – to dynamically change the shared spaces among agents – (2) above. Finally, we argue that the new CCP language can be used in the specification of service oriented computing systems.

What is relevance logic?

We suggest two precise abstract definitions of the notion of ‘relevance logic’ which are both independent of any proof system or semantics. We show that according to the simpler one, R→¬ (the intensional fragment of R) is the minimal relevance logic, but R itself is not. In contrast, R and many other logics are relevance logics according to the second (more complicated) definition, while all fragments of linear logic are not.

RASP and ASP as a fragment of linear logic

RASP is a recent extension to Answer Set Programming (ASP) that permits declarative specification and reasoning on the consumption and production of resources. ASP can be seen as a particular case of RASP. In this paper, we study the relationship between linear logic and RASP problem specification. We prove that RASP programs can be translated into (a fragment of) linear logic, and vice versa. In doing so, we introduce a linear logic representation of default negation as understood in ASP. We are also able to establish a link between linear logic and here-and-there (HT) logic.

Interpolation via translations

AbstractA new technique is presented for proving that a consequence system enjoys Craig interpolation or Maehara interpolation based on the fact that these properties hold in another consequence system. This technique is based on the existence of a back and forth translation satisfying some properties between the consequence systems. Some examples of translations satisfying those properties are described. Namely a translation between the global/local consequence systems induced by fragments of linear logic, a Kolmogorov‐Gentzen‐Gödel style translation, and a new translation between the global consequence systems induced by full Lambek calculus and linear logic, mixing features of a Kiriyama‐Ono style translation with features of a Kolmogorov‐Gentzen‐Gödel style translation. These translations establish a strong relationship between the logics involved and are used to obtain new results about whether Craig interpolation and Maehara interpolation hold in that logics (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Implicational Paradoxes and the Meaning of Logical Constants

I discuss paradoxes of implication in the setting of a proof-conditional theory of meaning for logical constants. I argue that a proper logic of implication should be not only relevant, but also constructive and nonmonotonic. This leads me to select as a plausible candidate LL, a fragment of linear logic that differs from R in that it rejects both contraction and distribution.

Model checking linear logic specifications

The overall goal of this paper is to investigate the theoretical foundations of algorithmic verification techniques for first order linear logic specifications. The fragment of linear logic we consider in this paper is based on the linear logic programming language called LO (Andreoli and Pareschi, 1990) enriched with universally quantified goal formulas. Although LO was originally introduced as a theoretical foundation for extensions of logic programming languages, it can also be viewed as a very general language to specify a wide range of infinite-state concurrent systems (Andreoli, 1992; Cervesato, 1995). Our approach is based on the relation between backward reachability and provability highlighted in our previous work on propositional LO programs (Bozzano et al., 2002). Following this line of research, we define here a general framework for the bottom-up. evaluation of first order linear logic specifications. The evaluation procedure is based on an effective fixpoint operator working on a symbolic representation of infinite collections of first order linear logic formulas. The theory of well quasi-orderings Abdulla et al., 1996; Finkel and Schnoebelen, 2001) can be used to provide sufficient conditions for the termination of the evaluation of non trivial fragments of first order linear logic.

Obsessional experiments for linear logic proof-nets

We address the question of injectivity of coherent semantics of linear logic proof-nets. Starting from Girard's definition of experiment, we introduce the key-notion of ‘injective obsessional experiment’, which allows us to give a positive answer to our question for certain fragments of linear logic, and to build counter-examples to the injectivity of coherent semantics in the general case.

The Logic of Multisets Continued: The Case of Disjunction

We continue our work [5] on the logic of multisets (or on the multiset semantics of linear logic), by interpreting further the additive disjunction ⊔. To this purpose we employ a more general class of processes, called free, the axiomatization of which requires a new rule (not compatible with the full LL), the cancellation rule. Disjunctive multisets are modeled as finite sets of multisets. The ⊔-Horn fragment of linear logic, with the cut rule slightly restricted, is sound with respect to this semantics. Another rule, which is a slight modification of cancellation, added to HF⊔ makes the system sound and complete.

Linguistic Applications of First Order Intuitionistic Linear Logic

In this paper we will discuss the first order multiplicative intuitionistic fragment of linear logic, MILL1, and its applications to linguistics. We give an embedding translation from formulas in t...

Concurrent programming as proof net construction

We propose a concurrent process calculus, called Calcul Parallèle Logique (CPL), based on the paradigm of computation as proof net construction in linear logic. CPL uses a fragment of first-order intuitionistic linear logic where formulas represent processes and proof nets represent successful computations. In these computations, communication is expressed in an asynchronous way by means of axiom links. We define testing equivalences for processes, which are based on a concept of interface, and use the power of proof theory in linear logic.

The finite model property for various fragments of linear logic

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.