Algorithmic specifications in linear logic with subexponentials

  title={Algorithmic specifications in linear logic with subexponentials},
  author={Vivek Nigam and Dale Miller},
  booktitle={PPDP '09},
The linear logic exponentials !,? are not canonical: one can add to linear logic other such operators, say !l,?1, which may or may not allow contraction and weakening, and where l is from some pre-ordered set of labels. We shall call these additional operators subexponentials and use them to assign locations to multisets of formulas within a linear logic programming setting. Treating locations as subexponentials greatly increases the algorithmic expressiveness of logic. To illustrate this new… 

Figures from this paper

Subexponential concurrent constraint programming
An Adequate Compositional Encoding of Bigraph Structure in Linear Logic with Subexponentials
The encoding of the bigraph structure is adequate and therefore the operations of composition and juxtaposition can be performed on the logical level and all the proof-theoretical tools of [Figure not available: see fulltext.] become available for querying and proving properties of bigraph structures.
Computational logic based on linear logic and fixed points
This work presents various levels of restrictions on the roles of fixed points in proofs of μMALL, the logic that results from adding least and greatest fixed points to first-order multiplicative-additive linear logic, as a framework for presenting several topics in computational logic.
Subexponentials in non-commutative linear logic
Conditions for when cut elimination is admissible in the presence of non-commutative subexponentials are investigated, investigating the interaction of the exchange rule with the local and non-local contraction rules.
Soft Subexponentials and Multiplexing
This work introduces a non-commutative substructural system with subexponential modalities controlled by a minimalistic set of rules and employs Lambek’s non-emptiness restriction, which is incompatible with the standard (sub)exponential setting.
Hybrid and Subexponential Linear Logics Technical Report
This paper shows an encoding of the HyLL's logical rules into LL with the highest level of adequacy, hence showing that HyLL is as expressive as LL and proposes an encode of HyLL into SELL ⋓ (SELL plus quantification over locations) that gives better insights about the meaning of worlds in HyLL.
Hybrid and Subexponential Linear Logics
Expressing additives using multiplicatives and subexponentials
It is shown that a classical propositional multiplicative subexponential logic (MSEL) with one unrestricted and two linear subexponentials can encode the halting problem for two register Minsky machines, and is hence undecidable.
Super Exponentials in Linear Logic
A generic proof of cut elimination is given, which implies that each variant of Linear Logic which appears as a valid instance of superLL also satisfies cut elimination.


Logic Programming with Focusing Proofs in Linear Logic
It is shown that the syntactic restriction induced by LinLog is not performed at the cost of any expressive power: a mapping from full linear logic to LinLog, preserving focusing proofs, and analogous to the normalization to clausal form for classical logic, is presented.
Light Linear Logic
The authors are seeking a ``logic of polytime'', not yet one more axiomatization, but an intrinsically polytime system, which admits full induction on data types, which shows that, within LLL, induction is compatible with low complexity.
Cut-elimination for a logic with definitions and induction
Monadic concurrent linear logic programming
This paper extends Lolli with the remaining connectives of intuitionistic linear logic restricted to occur inside a monad, an idea taken from the concurrent logical framework (CLF) to create LolliMon, a language that has a natural forward chaining, committed choice operational semantics inside the monad.
Linear Logical Algorithms
A bottom-up logic programming language based on linear logic that is amenable to efficient execution is identified and a novel cost semantics is described that can be used for complexity analysis of algorithms expressed in linear logic.
Forum: A Multiple-Conclusion Specification Logic
Testing Concurrent Systems: An Interpretation of Intuitionistic Logic
A condition on proofs, operational derivability (OD), is presented, and it is shown that a sequent in this logic has a proof iff it has an operationally derivable proof.
From Proofs to Focused Proofs: A Modular Proof of Focalization in Linear Logic
A new proof of the completeness of focused proofs in terms of proof transformation is given, first proved for MALL and then extended to full linear logic, and it is shown how the proof can be extended to larger systems, such as logics with induction.
Incorporating Tables into Proofs
The problem of automating and checking the use of previously proved lemmas in the proof of some main theorem is considered and the collection of such previously proved results is called a table and a partial order on the table's entries is used to denote the (provability) dependency relationship between tabled items.
Focusing and Polarization in Intuitionistic Logic
This work presents a new, focused proof system for intuitionistic logic, called LJF, and shows how other proof systems can be mapped into the new system by inserting logical connectives that prematurely stop focusing.