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From linear proofs to direct logic with exponentials [chapter]

Enno Sandner
1997 Lecture Notes in Computer Science  
Following the idea of Linear Proofs presented in 4] we introduce the Direct Logic 14] with exponentials (DLE). The logic combines Direct Logic with the exponentials of Linear Logic.  ...  For a well-chosen subclass of formulas of this logic we provide a matrix-characterization which can be used as a foundation for proof-search methods based on the connection calculus.  ...  Completeness and Soundness. Let 2 (F ) s and m 2 M . For every provable sequent ) the matrix i( ) is linear complementary and for every linear complementary matrix m the sequent ) j(m) is provable.  ... 
doi:10.1007/3540634932_10 fatcat:fi62ng247fgghmcygtnfa6tgly

Extending the Family of Intuitionistic Many-Valued Logics Introduced by Rousseau

1986 Annals of the Japan Association for Philosophy of Science  
A matrix is provable if and only if it is valid. so the proof of (1) and hence of Proposition 6 is completed. • 3.3.  ...  A matrix K is provable ("¥K) iff it is the end-matrix of some proof-figure. A matrix K is cut free provable (_??  ... 
doi:10.4288/jafpos1956.7.47 fatcat:a7bobiyfpvh6rera4aytvyetzi

From QBFs to MALL and Back via Focussing

Anupam Das
2020 Journal of automated reasoning  
Thus we have that non-Π f k+1 -provability is computable in Σ p k+1 and so, by Dfn. 5, Π f k+1 -provability is computable in Π p k+1 .  ...  Das For completeness, we proceed by induction on the size of a FMALL proof, by essentially a rule permutation argument.  ...  For instance here are the cases when A is a or formula, where the derivations marked IH are obtained from the inductive hypothesis.  ... 
doi:10.1007/s10817-020-09564-x fatcat:qth4tkmaujfjpa2vqikxhuqkpi

Matrix-Based Constructive Theorem Proving [chapter]

Christoph Kreitz, Jens Otten, Stephan Schmitt, Brigitte Pientka
2000 Applied Logic Series  
We present a coherent account of matrix methods for constructive theorem proving and show how to extend them to inductive theorem proving by integrating rippling techniques into the unification process  ...  Matrix methods for intuitionistic logic also are the central inference engine for program synthesis and verification.  ...  of linear logic [14, 17] .  ... 
doi:10.1007/978-94-015-9383-0_12 fatcat:psdwy6yocnhtve5oitvdf7lsxy


Zhijiang He, Mustafa Celik, Lawrence Pileggi
1997 Proceedings of the 34th annual conference on Design automation conference - DAC '97  
The minimally required mutual inductances are extracted for a provably stable model.  ...  Furthermore, it is well known that simply discarding smallest terms to sparsify the inductance matrix can render the partial inductance matrix indefinite and result in an unstable circuit model.  ...  The complete partial inductance matrix is positive definite. It has been shown that simply discarding mutual terms can render this matrix indefinite.  ... 
doi:10.1145/266021.266050 dblp:conf/dac/HeCP97 fatcat:2xjvtg4ccvhnpdisl7fwpcjlli

From QBFs to MALL and back via focussing: fragments of multiplicative additive linear logic for each level of the polynomial hierarchy [article]

Anupam Das
2020 arXiv   pre-print
This refines the well-known results that both MALLw and MALL are PSPACE-complete.  ...  Our main result is the obtention of fragments of MALLw (MALL with weakening) complete for each level of the polynomial hierarchy.  ...  Thus we have that non-Π f k+1 -provability is computable in Σ p k+1 and so, by Dfn. 5, Π f k+1 -provability is computable in Π p k+1 .  ... 
arXiv:1906.03611v3 fatcat:swjxpfzzerg6pjx6ob7gj3b2ua

Decision problems for propositional linear logic

Patrick Lincoln, John Mitchell, Andre Scedrov, Natarajan Shankar
1992 Annals of Pure and Applied Logic  
propositional linear logic.  ...  Further, we prove that without the modal storage operator, which indicates unboundedness of resources, the decision problem becomes PSPACE-Complete.  ...  Our most significant results are that provability for full propositional linear logic is undecidable, but that provability becomes PSPACE complete when the modal storage operator is removed.  ... 
doi:10.1016/0168-0072(92)90075-b fatcat:rgznokvv2vcj7ax6hbk3ozg6ve

Optimization complexity of linear logic proof games

Patrick D. Lincoln, John C. Mitchell, Andre Scedrov
1999 Theoretical Computer Science  
linear logic.  ...  A class of linear logic proof games is developed, each with a numeric score that depends on the number of preferred axioms used in a complete or partial proof tree.  ...  Acknowledgements We would like to thank Mitsu Okada and the local organizers of the Linear Logic '96 Tokyo Meeting for the splendid arrangements at the conference.  ... 
doi:10.1016/s0304-3975(99)00057-2 fatcat:veu4bd5pznan3ncml3cwj2fod4

On intuitionistic many-valued logics

1986 Journal of the Mathematical Society of Japan  
A g-matrix is maximal unprovable if it is unprovable and any proper extension of it is provable. 3.61. Any matrix is a g-matrix. A matrix is provable i ff it is provable as a g-matrix. D 3.62.  ...  A generalized matrix (abbreviated by `g-matrix') is a finite or infinite set of signed formulas. A g-matrix is provable if it contains a provable matrix.  ... 
doi:10.2969/jmsj/03830409 fatcat:2icqjupktjevhez2l4jgosj6g4

Provable Repair of Deep Neural Networks [article]

Matthew Sotoudeh, Aditya V. Thakur
2021 arXiv   pre-print
using piecewise-linear activation functions.  ...  The key insight behind both of these algorithms is the introduction of a Decoupled DNN architecture, which allows us to reduce provable repair to a linear programming problem.  ...  and non-linear functions.  ... 
arXiv:2104.04413v2 fatcat:zef2xrr2zbedzdmvwrgthnto7u

Proof-search in type-theoretic languages: an introduction

Didier Galmiche, David J. Pym
2000 Theoretical Computer Science  
The usual completeness theorems are stated with respect to provability.  ...  judgements of provability of the form S proves L : Just as satisfaction relations can be deÿned by induction on the structure of propositions, so judgements of provability can be deÿned by induction on  ... 
doi:10.1016/s0304-3975(99)00169-3 fatcat:gy4nmmsupfeype23pw3jzbrvwu

Connection methods in linear logic and proof nets construction

D. Galmiche
2000 Theoretical Computer Science  
Aiming at using proof nets as a tool for automated deduction in linear logic, we deÿne a connection-based characterization of provability in Multiplicative Linear Logic (MLL).  ...  This central result is illustrated with a speciÿc algorithm that is able to construct, for a provable MLL sequent, a set of connections, a proof net and a sequent proof.  ...  Theorem 5.2 (Completeness). If S is a sequent provable in MLL then the algorithm returns a proof net of S. Proof.  ... 
doi:10.1016/s0304-3975(99)00176-0 fatcat:qvmalztc2vanzc5wjc4xvcjzuu

Substructural Verification and Computational Feasibility [chapter]

Daniel Leivant
2002 Foundations of Information Technology in the Era of Network and Mobile Computing  
When induction is restricted to positive formulas, a generalization of I:~ formulas, exactly the primitive recursive functions are provable.  ...  We show here that induction over arbitrary formulas does not add new provable functions if we disallow in derivations the closing of multiple data-complex assumptions.  ...  For ~nstance, linear logic leads to polytime [6, 5] , second-order existential database queries are poly-time if the matrix is Horn [12, 8] , monotone inductive definitions (where each object is inserted  ... 
doi:10.1007/978-0-387-35608-2_41 dblp:conf/ifipTCS/Leivant02 fatcat:pq322qal2rdvxm7s3xmocmuz3u

Page 1309 of Mathematical Reviews Vol. , Issue 96c [page]

1996 Mathematical Reviews  
One of the results which can be easily understood is that, if a [],-sentence whose matrix is an equation is provable in Peano arithmetic, then there is a < €o- DR function which realizes it.  ...  Bistructures form a categorical model of Girard’s classical linear . logic in which the involution of linear logic is modelled, roughly speaking, by a reversal of the roles of input and output.  ... 

Linear objects: Logical processes with built-in inheritance

Jean-Marc Andreoli, Remo Pareschi
1991 New generation computing  
Event spaces and their linear  ...  Is there a use for linear logic? To Appear ACM/IFIP PEPM, 1991. 92] P. Wadler. Linear types can change the world! IFIP TC 2 Conf. on Prog. Concepts and Methods, 1991. 93] P. Wadler.  ...  Theorem 2.3.4 (Cut-Elimination) If a sequent is provable in linear logic, then it is provable in linear logic without using the Cut rule. Proof. By induction on the degree of the assumed proof.  ... 
doi:10.1007/bf03037173 fatcat:qhsukwvkdresrlwwu6s2kvvosa
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