On Structuring Proof Search for First Order Linear Logic

Full first order linear logic can be presented as an abstract logic programming language in Miller's system Forum, which yields a sensible operational interpretation in the 'proof search as computation' paradigm. However, Forum still has to deal with syntactic details that would normally be ignored by a reasonable operational semantics. In this respect, Forum improves on Gentzen systems for linear logic by restricting the language and the form of inference rules. We further improve on Forum by restricting the class of formulae allowed, in a system we call G-Forum, which is still equivalent to full first order linear logic. The only formulae allowed in G-Forum have the same shape as Forum sequents: the restriction does not diminish expressiveness and makes G-Forum amenable to proof theoretic analysis. G-Forum consists of two (big) inference rules, for which we show a cut elimination procedure. This does not need to appeal to finer detail in formulae and sequents than is provided by G-Forum, thus successfully testing the internal symmetries of our system.Comment: Author website at http://alessio.guglielmi.name/res

Compiling Intensional Sets in CLP

Constructive negation has been proved to be a valid alternative to negation as failure, especially when negation is required to have, in a sense, an ‘active’ role. In this paper we analyze an extension of the original constructive negation in order to gracefully integrate with the management of set-constraints in the context of a Constraint Logic Programming Language dealing with finite sets. We show that the marriage between CLP with sets and constructive negation gives us the possibility of representing a general class of intensionally defined sets without any further extension to the operational semantics of the language. The presence of intensional sets allows a definite increase in the expressive power and abstraction level offered by the host logic language.

On structuring proof search for first order linear logic

AbstractFull first-order linear logic can be presented as an abstract logic programming language in Miller's system Forum, which yields a sensible operational interpretation in the ‘proof search as computation’ paradigm. However, Forum still has to deal with syntactic details that would normally be ignored by a reasonable operational semantics. In this respect, Forum improves on Gentzen systems for linear logic by restricting the language and the form of inference rules. We further improve on Forum by restricting the class of formulae allowed, in a system we call G-Forum, which is still equivalent to full first-order linear logic. The only formulae allowed in G-Forum have the same shape as Forum sequents: the restriction does not diminish expressiveness and makes G-Forum amenable to proof theoretic analysis. G-Forum consists of two (big) inference rules, for which we show a cut elimination procedure. This does not need to appeal to finer detail in formulae and sequents than is provided by G-Forum, thus successfully testing the internal symmetries of our system

On the Proof Complexity of Deep Inference

International audienceWe obtain two results about the proof complexity of deep inference: (1) Deep-inference proof systems are as powerful as Frege ones, even when both are extended with the Tseitin extension rule or with the substitution rule; (2) there are analytic deep-inference proof systems that exhibit an exponential speedup over analytic Gentzen proof systems that they polynomially simulate

On analytic inference rules in the calculus of structures

In this note, we discuss the notion of analytic inference rule for propositional logics in the calculus of structures (CoS) [3]. CoS generalises the sequent calculus and preserves all its proof-theoretic properties. There is no notion of analytic rule outside of the sequent calculus, and so we investigate what such a notion could be for CoS. In [2], Section 6.4, we write that it is easy to prove that an ‘analytic ’ restriction of CoS p-simulates CoS, and so Frege systems and sequent-calculus systems (including non-analytic ones), by resorting to the following, natural notion of analyticity. An inference rule would be analytic if all the atoms appearing in its premiss also appear in its conclusion. This way, a finitary version of the atomic cut rule would be analytic, and this would yield the p-simulation result. However, it seems like that notion of analyticity, while natural, does not lead to expected results, and it does not seem refined enough to bring to light interesting problems. So, we briefly discuss here a definition of analyticit