From Deep Inference to Proof Nets via Cut Elimination

Straßburger, Lutz

doi:10.1093/logcom/exp047

Cited by 11 publications

(18 citation statements)

References 28 publications

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“…Contextual closure just corresponds to equivalence being a congruence, it is a necessary ingredient of the calculus of structures. All other equations can be removed and replaced by rules (see, e.g., [Str05]), as in the sequent calculus. This might prove necessary for certain applications.…”

Section: The Systemmentioning

confidence: 99%

A system of interaction and structure V: the exponentials and splitting

Guglielmi¹,

Straβburger²

2011

Math. Struct. Comp. Sci.

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System NEL is the mixed commutative/non-commutative linear logic BV augmented with linear logic's exponentials, or, equivalently, it is MELL augmented with the non-commutative self-dual connective seq. System NEL is Turingcomplete, it is able to directly express process algebra sequential composition and it faithfully models causal quantum evolution. In this paper, we show cut elimination for NEL, based on a property that we call splitting. NEL is presented in the calculus of structures, which is a deep-inference formalism, because no Gentzen formalism can express it analytically. The splitting theorem shows how and to what extent we can recover a sequent-like structure in NEL proofs. Together with the decomposition theorem, proved in the previous paper of the series, this immediately leads to a cut-elimination theorem for NEL.

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Section: The Systemmentioning

confidence: 99%

A system of interaction and structure V: the exponentials and splitting

Guglielmi¹,

Straβburger²

2011

Math. Struct. Comp. Sci.

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“…2 Note the parallelism induced by independent branches could be illuminated further by the proposed formalisms that are more explicit about concurrent proof search in deep inference, such as formalism B [47]. It may also be interesting to revisit the above problem in the context of the interactive proof class IP [42,34].…”

Section: Proposition 4 Proof Search In Mav Is In Pspacementioning

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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“…For examplē Depending on which cut we reduce first, we get either a aā a orā aā a If we reduce the remaining cut, we get a a orā a respectively. The solution for circumventing this problem is to reduce atomic cuts only in unproblematic situations like (15) and (17), and leave all atomic cuts like (16) unreduced, as it is done for C-nets in [24]. C-nets are a variant of N-nets that are considered cut-free if they contain only atomic cuts that touch a contraction on both sides.…”

Section: Flow Graph Based Proof Netsmentioning

confidence: 99%