Some Observations on the Proof Theory of Second Order Propositional Multiplicative Linear Logic

Straßburger, Lutz

doi:10.1007/978-3-642-02273-9_23

Cited by 14 publications

(13 citation statements)

References 17 publications

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“…MAV is a finite calculus in which only finite structures can be presented. In future work, we aim to investigate extensions for second-order quantifiers [48] and infinite structures [3] that are likely to be undecidable [46,32].…”

Section: Discussionmentioning

confidence: 99%

The Consistency and Complexity of Multiplicative Additive System Virtual

Horne¹

2015

SACS

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This paper investigates the proof theory of multiplicative additive system virtual (MAV). MAV combines two established proof calculi: multiplicative additive linear logic (MALL) and basic system virtual (BV). Due to the presence of the self-dual non-commutative operator from BV, the calculus MAV is defined in the calculus of structuresa generalisation of the sequent calculus where inference rules can be applied in any context. A generalised cut elimination result is proven for MAV, thereby establishing the consistency of linear implication defined in the calculus. The cut elimination proof involves a termination measure based on multisets of multisets of natural numbers to handle subtle interactions between operators of BV and MAV. Proof search in MAV is proven to be a PSPACE-complete decision problem. The study of this calculus is motivated by observations about applications in computer science to the verification of protocols and to querying.

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Section: Discussionmentioning

confidence: 99%

The Consistency and Complexity of Multiplicative Additive System Virtual

Horne¹

2015

SACS

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“…Surprisingly, to date, the only direct proof of cut elimination involving quantifiers in the calculus of structures is for BVQ [43]. Related cut elimination results involving first-order quantifiers in the calculus of structures rely on a correspondence with the sequent calculus [5,47]. However, due to the presence of the non-commutative operator seq there is no sequent calculus presentation [49] for MAV1; hence we pursue here a direct proof.…”

Section: Cut Elimination and Its Consequencesmentioning

confidence: 97%

De Morgan Dual Nominal Quantifiers Modelling Private Names in Non-Commutative Logic

Horne

Tiu

Aman

et al. 2019

ACM Trans. Comput. Logic

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This paper explores the proof theory necessary for recommending an expressive but decidable first-order system, named MAV1, featuring a de Morgan dual pair of nominal quantifiers. These nominal quantifiers called 'new' and 'wen' are distinct from the self-dual Gabbay-Pitts and Miller-Tiu nominal quantifiers. The novelty of these nominal quantifiers is they are polarised in the sense that 'new' distributes over positive operators while 'wen' distributes over negative operators. This greater control of bookkeeping enables private names to be modelled in processes embedded as predicates in MAV1. The technical challenge is to establish a cut elimination result, from which essential properties including the transitivity of implication follow. Since the system is defined using the calculus of structures, a generalisation of the sequent calculus, novel techniques are employed. The proof relies on an intricately designed multiset-based measure of the size of a proof, which is used to guide a normalisation technique called splitting. The presence of equivariance, which swaps successive quantifiers, induces complex inter-dependencies between nominal quantifiers, additive conjunction and multiplicative operators in the proof of splitting. Every rule is justified by an example demonstrating why the rule is necessary for soundly embedding processes and ensuring that cut elimination holds.2.1 An established extension of BV with a self-dual quantifier An abstract syntax for predicates, and term-rewriting system for BVQ are defined in Fig 1. The rules can be applied to rewrite a predicate of the form on the left of the long right arrow to the predicate on the right (note rewriting -the direction of proof search -is in the opposite direction to linear implication). The key feature of the calculus of structures is deep inference, with is the ability to apply all rewrite rules in any context, i.e. predicates with a hole of the following form:The term-rewriting system is defined modulo a congruence, where a congruence is an equivalence relation that holds in any context. The congruence, ≡ in Fig. 1, makes par and times commutative and seq non-commutative in general. For the nominal quantifier ∇, the congruence enables: α-conversion for renaming bound names; equivariance which allows names bound by successive Technical report. Publication date: November 2017.Definition 2.2. A proof is a derivation P −→ I from a predicate P to the unit I. When such a derivation exists, we say that P is provable, and write ⊢ P holds.As a basic property of linear implication ⊢ P ⊸ P must hold for any P. Now assume that ⊢ Q ⊸ Q is provable in BVQ (hence, by the above definitions, there exists a derivation Q Q −→ I), and consider a formula of the form ∇x .Q. Using the unify rule, we can construct the following proof.Thus unify is necessary in order to guarantee reflexivity -the most basic property of implication -for an extension of BV with a self-dual nominal quantifier. In the next section, we explain why the unify rule is problematic for modelling proc...

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“…• Second-order: The COS has been extended to second-order propositional multiplicative linear logic [18]. There is a natural embedding of inductive and coinductive fixpoints into secondorder MALL [2].…”

Section: Conclusion Related Work and Perspectivesmentioning

confidence: 99%

Equality and fixpoints in the calculus of structures

Chaudhuri

Guenot

2014

Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Nin

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International audienceThe standard proof theory for logics with equality and fixpoints suffers from limitations of the sequent calculus, where reasoning is separated from computational tasks such as unification or rewriting. We propose in this paper an extension of the calculus of structures, a deep inference formalism, that supports incremental and contextual reasoning with equality and fixpoints in the setting of linear logic. This system allows deductive and computational steps to mix freely in a continuum which integrates smoothly into the usual versatile rules of multiplicative-additive linear logic in deep inference

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Some Observations on the Proof Theory of Second Order Propositional Multiplicative Linear Logic

Abstract: Abstract

Cited by 14 publications

References 17 publications

The Consistency and Complexity of Multiplicative Additive System Virtual

The Consistency and Complexity of Multiplicative Additive System Virtual

De Morgan Dual Nominal Quantifiers Modelling Private Names in Non-Commutative Logic

Equality and fixpoints in the calculus of structures

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