Encoding transition systems in sequent calculus

McDowell, Raymond; Miller, Dale; Palamidessi, Catuscia

doi:10.1016/s0304-3975(01)00168-2

Cited by 32 publications

(27 citation statements)

References 18 publications

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“…Abramsky [2] and Bellin and Scott [9] rely on classical linear logic for this purpose. Miller et al have performed a similar investigation using intuitionistic linear logic [39], and more recently using a refinement of linear logic with a new quantifier that resembles name generation [45,57]. Abramsky has recently suggested extracting processes from proofs [3].…”

Section: Introductionmentioning

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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Section: Introductionmentioning

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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“…The encoding of the syntax of deductive systems inside formal logic can benefit from the use of higher-order abstract syntax (HOAS) [40], a high-level and declarative treatment of object-level bound variables and substitution. At the same time, we want to use such a logic in order to reason over the meta-theoretical properties of object languages, for example type preservation in operational semantics [26], soundness and completeness of compilation [32] or congruence of bisimulation in transition systems [27]. Typically this involves reasoning by (structural) induction and, when dealing with infinite behavior, co-induction [23].…”

Section: Introductionmentioning

confidence: 99%

Induction and Co-induction in Sequent Calculus

Momigliano

Tiu

2004

Lecture Notes in Computer Science

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Abstract. Proof search has been used to specify a wide range of computation systems. In order to build a framework for reasoning about such specifications, we make use of a sequent calculus involving induction and co-induction. These proof principles are based on a proof theoretic (rather than set-theoretic) notion of definition [13,20,25,51]. Definitions are akin to (stratified) logic programs, where the left and right rules for defined atoms allow one to view theories as "closed" or defining fixed points. The use of definitions makes it possible to reason intensionally about syntax, in particular enforcing free equality via unification. We add in a consistent way rules for pre and post fixed points, thus allowing the user to reason inductively and co-inductively about properties of computational system making full use of higher-order abstract syntax. Consistency is guaranteed via cut-elimination, where we give the first, to our knowledge, cut-elimination procedure in the presence of general inductive and co-inductive definitions.

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“…For example, defR is essentially the backchaining rule found in logic programming, while defL is essentially a case analysis on how an atom can be proved and can be used to establish finite failure. Together, these two rules can be used to encode simulation and bisimulation in certain abstract transition systems [11]. Other uses involve reasoning about computational system [10].…”

Section: Definition 2 Given a Formula B Its Level Lvl(b) Is Defined mentioning

confidence: 99%

“…or defined constants. Extending this work to infinite process expressions should be possible by adding induction (as in [11]) or co-induction to our proof system. We shall require three primitive syntactic categories: n for channels, p for processes, and a for actions.…”

Section: Figure 4 the Rules For The (Late) π-Calculusmentioning

confidence: 99%

“…Similarly, focusing on their extensional nature guaranteed by cut-elimination, enrichments to the sequent calculus have been proposed by [7,24,6,9] in which eigenvariables are intended as variables to be substituted. This enrichment to proof theory (discussed here in Section 4) holds promise for providing proof systems for the direct reasoning of logic specifications (see, for example, the above mentioned papers as well as [10,11]). These two approaches are, however, at odds with each other.…”

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confidence: 99%

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A proof theory for generic judgments: an extended abstract

Miller

Tiu

18th Annual IEEE Symposium of Logic in Computer Science, 2003. Proceedings.

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A powerful and declarative means of specifying computations containing abstractions involves meta-level, universally quantified generic judgments. We present a proof theory for such judgments in which signatures are associated to each sequent (used to account for eigenvariables of the sequent) and to each formula in the sequent (used to account for generic variables locally scoped over the formula). A new quantifier, ∇, is introduced to explicitly manipulate the local signature. Intuitionistic logic extended with ∇ satisfies cut-elimination even when the logic is additionally strengthened with a proof theoretic notion of definitions. The resulting logic can be used to encode naturally a number of examples involving name abstractions, and we illustrate using the π-calculus and the encoding of objectlevel provability.Keywords: proof search, reasoning about operational semantics, generic judgments, higher-order abstract syntax. Eigenvariables and generic reasoningIn specifying and reasoning about computations involving abstractions, one needs to encode both the static structure of such abstractions and their dynamic structure during computation. One successful approach to such an encoding, generally called higher-order abstract syntax [22], uses λ-terms to encode the static structure of abstractions and universally quantified judgments to encode their dynamic structure.There are, of course, several ways to prove a universally quantified expression, ∀ γ x.B. An approach that can be called the extensional, attempts to prove B [t/x] for all (closed) terms t of type γ. This rule might involve an infinite number of premises if the domain of the type γ is infinite. If the type γ is defined inductively, a proof by induction can replace the need for infinite premises with finite premises (the base cases and inductive cases) but with the need to discover invariants. Another more intensional approach, however, involves introducing a new, generic variable, say, c : γ, that has not been introduced before in the proof, and to prove the formula B[c/x] instead. In natural deduction and sequent calculus proofs, such new variables are called eigenvariables.In Gentzen's original presentation of the sequent calculus [5], eigenvariables were immutable: reading proofs bottom-up, once an eigenvariable is introduced it is not used as a site for substitution. In other words, Gentzen's eigenvariables did not vary in proof construction: rather they acted more as fresh, scoped constants.The generic interpretation of quantifiers generally entails the extensional interpretation: this is a simple consequence of the cut-elimination theorem as follows. Assume that the sequent Γ −→ ∀x.B is proved using the introduction of ∀ on the right from the premise Γ −→ B [c/x], where c is an eigenvariable and Π(c) is a proof of this premise. Similarly, assume that the sequent Γ , ∀xB −→ C is proved using the introduction of ∀ on the left from the premise Γ , B[t/x] −→ C, where t is some term. To reduce the rank of the cut formula ∀x.B between the s...

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Encoding transition systems in sequent calculus

Cited by 32 publications

References 18 publications

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

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

Induction and Co-induction in Sequent Calculus

A proof theory for generic judgments: an extended abstract

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