Syntactical properties of unbounded nets of processors (original) (raw)
Finite-valued Logics for Information Processing
Fundamenta Informaticae
We examine the issue of collecting and processing information from various sources, which involves handling incomplete and inconsistent information. Inspired by the framework first proposed by Belnap, we consider structures consisting of information sources which provide information about the values of formulas of classical propositional logic, and a processor which collects that information and extends it by deriving conclusions following from it according to the truth tables of classical logic, applied forward and backward. Our model extends Belnap's in allowing the sources to provide information also about complex formulas. As that framework cannot be captured by finite ordinary logical matrices, we use Nmatrices for that purpose. In opposition to the approach proposed in our earlier work, we assume that the information sources are reasonable, i.e. that they provide information consistent with certain coherence rules. We provide sound and complete sequent calculi admitting strong cut elimination for the logic of a single information source, and (several variants of) the logic generated by the source and processor structures described above. In doing this, we also provide new characterizations for some known logics. We prove that, in opposition to the variant with unconstrained information sources considered earlier, the latter logic cannot be generated by structures with any bounded number of sources.
Linear Logic for Nets with Bounded Resources
Annals of Pure and Applied Logic, 1996
In this paper we introduce a new type of nets with bounded types of distributed resources (BR-nets). Linear Logic to describe the behaviour of BR-nets is defined. It is based on Girard's Linear Logic but captures not only consumption of resources but their presence as well. Theorem of soundness and completeness of the proposed axiomatization is proved and the complexity of the provability problem is established for the general case and some particular ones.
A semantic measure of the execution time in linear logic
Theoretical Computer Science, 2011
We give a semantic account of the execution time (i.e. the number of cut elimination steps leading to the normal form) of an untyped M ELL net. We first prove that: 1) a net is head-normalizable (i.e. normalizable at depth 0) if and only if its interpretation in the multiset based relational semantics is not empty and 2) a net is normalizable if and only if its exhaustive interpretation (a suitable restriction of its interpretation) is not empty. We then give a semantic measure of execution time: we prove that we can compute the number of cut elimination steps leading to a cut free normal form of the net obtained by connecting two cut free nets by means of a cut link, from the interpretations of the two cut free nets. These results are inspired by similar ones obtained by the first author for the untyped lambda-calculus.
From truth to computability II
Theoretical Computer Science, 2007
Computability logic is a formal theory of computational tasks and resources. Formulas in it represent interactive computational problems, and "truth" is understood as algorithmic solvability. Interactive computational problems, in turn, are defined as a certain sort games between a machine and its environment, with logical operators standing for operations on such games. Within the ambitious program of finding axiomatizations for incrementally rich fragments of this semantically introduced logic, the earlier article "From truth to computability I" proved soundness and completeness for system CL3, whose language has the so called parallel connectives (including negation), choice connectives, choice quantifiers, and blind quantifiers. The present paper extends that result to the significantly more expressive system CL4 with the same collection of logical operators. What makes CL4 expressive is the presence of two sorts of atoms in its language: elementary atoms, representing elementary computational problems (i.e. predicates, i.e. problems of zero degree of interactivity), and general atoms, representing arbitrary computational problems. CL4 conservatively extends CL3, with the latter being nothing but the general-atom-free fragment of the former. Removing the blind (classical) group of quantifiers from the language of CL4 is shown to yield a decidable logic despite the fact that the latter is still first-order.
On the relation between coherence semantics and multiplicative proof nets
1994
It is known that (mix) proof nets admit a coherence semantics, computed as a set of experiments. We prove here the converse: a proof structure is shown to be a proof net whenever its set of experiments is a semantical object -a clique of the corresponding coherence space. Moreover the interpretation of atomic formulae can be restricted to a given coherent space with four tokens in its web. This is done by transforming cut-links into tensor-links.
A graph-theoretic account of logics
A graph-theoretic account of logics is explored based on the general notion of m-graph (that is, a graph where each edge can have a finite sequence of nodes as source). Signatures, interpretation structures and deduction systems are seen as m-graphs. After defining a category freely generated by a m-graph, formulas and expressions in general can be seen as morphisms. Moreover, derivations involving rule instantiation are also morphisms. Soundness and completeness theorems are proved. As a consequence of the generality of the approach our results apply to very different logics encompassing, among others, substructural logics as well as logics with nondeterministic semantics, and subsume all logics endowed with an algebraic semantics.
07441 Abstracts Collection--Algorithmic-Logical Theory of Infinite Structures}
Abstract From 28.10. to 02.11. 2007, the Dagstuhl Seminar 07441``Algorithmic-Logical Theory of Infinite Structures''was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper.
Proceedings of the 14th conference on Computational …, 1992
Proof-Nets ) are a good device for processing with eategorial grammars, mainly because they avoid spurious ambiguities. Nevertheless, they do not provide easily readable structures and they hide the true proximity between Categorial Grammars and Dependency Grammars. We give here an other kind of Proof-Nets which is much related to Dependency Structures similar to those we meet in, for instance . These new Proof-Nets are called Connection Nets. We show that Connection Nets provide not only easily interpretable structures, but also that processing with them is more efficient. 1