Phase semantics and decidability of elementary affine logic

AbstractLight, elementary and soft linear logics are formal systems derived from Linear Logic, enjoying remarkable normalization properties. In this paper, we prove decidability of Elementary Affine Logic, EAL. The result is obtained by semantical means, first defining a class of phase models for EAL and then proving soundness and (strong) completeness, following Okada's technique. Phase models for Light Affine Logic and Soft Linear Logic are also defined and shown complete

How hard is it to verify flat affine counter systems with the finite monoid property ?

We study several decision problems for counter systems with guards defined by convex polyhedra and updates defined by affine transformations. In general, the reachability problem is undecidable for such systems. Decidability can be achieved by imposing two restrictions: (i) the control structure of the counter system is flat, meaning that nested loops are forbidden, and (ii) the set of matrix powers is finite, for any affine update matrix in the system. We provide tight complexity bounds for several decision problems of such systems, by proving that reachability and model checking for Past Linear Temporal Logic are complete for the second level of the polynomial hierarchy $\Sigma^P_2$, while model checking for First Order Logic is PSPACE-complete

Recurrence with affine level mappings is P-time decidable for CLP(R)

In this paper we introduce a class of constraint logic programs such that their termination can be proved by using affine level mappings. We show that membership to this class is decidable in polynomial time.Comment: To appear in Theory and Practice of Logic Programming (TPLP

Gluing together proof environments: Canonical extensions of LF type theories featuring locks

© F. Honsell, L. Liquori, P. Maksimovic, I. Scagnetto This work is licensed under the Creative Commons Attribution License.We present two extensions of the LF Constructive Type Theory featuring monadic locks. A lock is a monadic type construct that captures the effect of an external call to an oracle. Such calls are the basic tool for gluing together diverse Type Theories and proof development environments. The oracle can be invoked either to check that a constraint holds or to provide a suitable witness. The systems are presented in the canonical style developed by the CMU School. The first system, CLLF/p,is the canonical version of the system LLF p, presented earlier by the authors. The second system, CLLF p?, features the possibility of invoking the oracle to obtain a witness satisfying a given constraint. We discuss encodings of Fitch-Prawitz Set theory, call-by-value λ-calculi, and systems of Light Linear Logic. Finally, we show how to use Fitch-Prawitz Set Theory to define a type system that types precisely the strongly normalizing terms

Deciding Conditional Termination

We address the problem of conditional termination, which is that of defining the set of initial configurations from which a given program always terminates. First we define the dual set, of initial configurations from which a non-terminating execution exists, as the greatest fixpoint of the function that maps a set of states into its pre-image with respect to the transition relation. This definition allows to compute the weakest non-termination precondition if at least one of the following holds: (i) the transition relation is deterministic, (ii) the descending Kleene sequence overapproximating the greatest fixpoint converges in finitely many steps, or (iii) the transition relation is well founded. We show that this is the case for two classes of relations, namely octagonal and finite monoid affine relations. Moreover, since the closed forms of these relations can be defined in Presburger arithmetic, we obtain the decidability of the termination problem for such loops.Comment: 61 pages, 6 figures, 2 table

Diophantine Undecidability of Holomorphy Rings of Function Fields of Characteristic 0

Let $K$ be a one-variable function field over a field of constants of characteristic 0. Let $R$ be a holomorphy subring of $K$, not equal to $K$. We prove the following undecidability results for $R$: If $K$ is recursive, then Hilbert's Tenth Problem is undecidable in $R$. In general, there exist $x_1,...,x_n \in R$ such that there is no algorithm to tell whether a polynomial equation with coefficients in \Q(x_1,...,x_n) has solutions in $R$.Comment: This version contains minor revisions and will appear in Annales de l Institut Fourie