Logic programming in a fragment of intuitionistic linear logic

Hodas, Joshua S.; Miller, Dale

doi:10.1109/lics.1991.151628

Cited by 105 publications

(160 citation statements)

References 24 publications

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“…The focused inverse method therefore directly generalizes these two classical proof search strategies. We also demonstrate through an implementation and experimental results that this choice can be important in practical proof search situations and that the standard polarity assumed for atoms in intuitionistic [15] or classical [22] logic programming is often the less efficient one.…”

Section: Introductionmentioning

confidence: 99%

A Logical Characterization of Forward and Backward Chaining in the Inverse Method

2008

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The inverse method is a generalization of resolution that can be applied to non-classical logics. We have recently shown how Andreoli's focusing strategy can be adapted for the inverse method in linear logic. In this paper we introduce the notion of focusing bias for atoms and show that it gives rise to forward and backward chaining, generalizing both hyperresolution (forward) and SLD resolution (backward) on the Horn fragment. A key feature of our characterization is the structural, rather than purely operational, explanation for forward and backward chaining. A search procedure like the inverse method is thus able to perform both operations as appropriate, even simultaneously. We also present experimental results and an evaluation of the practical benefits of biased atoms for a number of examples from different problem domains.

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Section: Introductionmentioning

confidence: 99%

A Logical Characterization of Forward and Backward Chaining in the Inverse Method

2008

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“…It is worth contrasting this characteristic with the tenets of logic programming as uniform provability [46], which instead extracts the operational semantics of a logical operator from its right sequent rules. This approach has robustly been extended to linear logic programming [5,30,43]. In a partial departure from this short tradition, Kobayashi and Yonezawa's ACL [34] derives its semantics from specialized versions of left rules of linear logic (when examined through the lense of duality).…”

Section: Discussionmentioning

confidence: 99%

“…Similarly to Barber's DILL [8] and Hodas and Miller's L [30], LV isolates reusable assumptions in the unrestricted context Γ (subject to exchange, weakening and contraction), while assumptions to be used exactly once are contained in the linear context Δ (subject only to exchange). The combination corresponds to the single context (!Γ, Δ) of Girard [26].…”

Section: A Very Brief Review Of Linear Logicmentioning

confidence: 99%

Relating State-Based and Process-Based Concurrency through Linear Logic

Cervesato

Scedrov

2006

Electronic Notes in Theoretical Computer Science

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This paper has the purpose of reviewing some of the established relationships between logic and concurrency, and of exploring new ones. Concurrent and distributed systems are notoriously hard to get right. Therefore, following an approach that has proved highly beneficial for sequential programs, much effort has been invested in tracing the foundations of concurrency in logic. The starting points of such investigations have been various idealized languages of concurrent and distributed programming, in particular the well-established state-transformation model inspired to Petri nets and multiset rewriting, and the prolific process-based models such as the π-calculus and other process algebras. In nearly all cases, the target of these investigations has been linear logic, a formal language that supports a view of formulas as consumable resources. In the first part of this paper, we review some of these interpretations of concurrent languages into linear logic. In the second part of the paper, we propose a completely new approach to understanding concurrent and distributed programming as a manifestation of logic, which yields a language that merges those two main paradigms of concurrency. Specifically, we present a new semantics for multiset rewriting founded on an alternative view of linear logic. The resulting interpretation is extended with a majority of linear connectives into the language of ω-multisets. This interpretation drops the distinction between multiset elements and rewrite rules, and considerably enriches the expressive power of standard multiset rewriting with embedded rules, choice, replication, and more. Derivations are now primarily viewed as open objects, and are closed only to examine intermediate rewriting states. The resulting language can also be interpreted as a process algebra. For example, a simple translation maps process constructors of the asynchronous π-calculus to rewrite operators, while the structural equivalence corresponds directly to logically-motivated structural properties of ω-multisets (with one exception).

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“…Linear logic is then the natural choice, since it offers a notion of context where each assumption must be used exactly once; a declarative encoding of store update can be obtained via linear operations that, by accessing the context, consume the old assumption and insert the new one. This is one of the motivations for proposing frameworks based on linear logics (see [72] for an overview) such as Lolli [52], Forum [71], and LLF [18], a conservative extension of LF with multiplicative implication, additive conjunction, and unit. Yet, at the time of writing this article, work on the automation of reasoning in such frameworks is still in its infancy [65] and may take other directions, such as hybrid logics [100].…”

Section: Ordered Linear Logic As a Specification Logicmentioning

confidence: 99%

Hybrid

Felty

Momigliano

2010

J Autom Reasoning

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Combining higher-order abstract syntax and (co)-induction in a logical framework is well known to be problematic. We describe the theory and the practice of a tool called Hybrid, within Isabelle/HOL and Coq, which aims to address many of these difficulties. It allows object logics to be represented using higher-order abstract syntax, and reasoned about using tactical theorem proving and principles of (co)induction. Moreover, it is definitional, which guarantees consistency within a classical type theory. The idea is to have a de Bruijn representation of λ-terms providing a definitional layer that allows the user to represent object languages using higher-order abstract syntax, while offering tools for reasoning about them at the higher level. In this paper we describe how to use Hybrid in a multi-level reasoning fashion, similar in spirit to other systems such as Twelf and Abella. By explicitly referencing provability in a middle layer called a specification logic, we solve the problem of reasoning by (co)induction in the presence of non-stratifiable hypothetical judgments, which allow very elegant and succinct specifications of object logic inference rules. We first demonstrate the method on a simple example, formally proving type soundness (subject reduction) for a fragment of a pure functional language, using a minimal intuitionistic logic as the specification logic. We then prove an analogous result for a continuation-machine presentation of the operational semantics of the same language, encoded this time in an ordered linear logic that serves as the specification layer. This example demonstrates the ease with which we can incorporate new specification logics, and also illustrates a significantly more complex object logic whose encoding is elegantly expressed using features of the new specification logic.

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Logic programming in a fragment of intuitionistic linear logic

Cited by 105 publications

References 24 publications

A Logical Characterization of Forward and Backward Chaining in the Inverse Method

A Logical Characterization of Forward and Backward Chaining in the Inverse Method

Relating State-Based and Process-Based Concurrency through Linear Logic

Hybrid

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