153 research outputs found
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
Tower-Complete Problems in Contraction-Free Substructural Logics
We investigate the non-elementary computational complexity of a family of
substructural logics without contraction. With the aid of the technique
pioneered by Lazi\'c and Schmitz (2015), we show that the deducibility problem
for full Lambek calculus with exchange and weakening
() is not in Elementary (i.e., the class of decision
problems that can be decided in time bounded by an elementary recursive
function), but is in PR (i.e., the class of decision problems that can be
decided in time bounded by a primitive recursive function). More precisely, we
show that this problem is complete for Tower, which is a non-elementary
complexity class forming a part of the fast-growing complexity hierarchy
introduced by Schmitz (2016). The same complexity result holds even for
deducibility in BCK-logic, i.e., the implicational fragment of
. We furthermore show the Tower-completeness of the
provability problem for elementary affine logic, which was proved to be
decidable by Dal Lago and Martini (2004).Comment: The full version of the paper accepted to CSL 202
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 , while
model checking for First Order Logic is PSPACE-complete
Phase Semantics for Linear Logic with Least and Greatest Fixed Points
The truth semantics of linear logic (i.e. phase semantics) is often overlooked despite having a wide range of applications and deep connections with several denotational semantics. In phase semantics, one is concerned about the provability of formulas rather than the contents of their proofs (or refutations). Linear logic equipped with the least and greatest fixpoint operators (?MALL) has been an active field of research for the past one and a half decades. Various proof systems are known viz. finitary and non-wellfounded, based on explicit and implicit (co)induction respectively.
In this paper, we extend the phase semantics of multiplicative additive linear logic (a.k.a. MALL) to ?MALL with explicit (co)induction (i.e. ?MALL^{ind}). We introduce a Tait-style system for ?MALL called ?MALL_? where proofs are wellfounded but potentially infinitely branching. We study its phase semantics and prove that it does not have the finite model property
Alternating Vector Addition Systems with States
International audienceAlternating vector addition systems are obtained by equipping vector addition systems with states (VASS) with 'fork' rules, and provide a natural setting for infinite-arena games played over a VASS. Initially introduced in the study of propositional linear logic, they have more recently gathered attention in the guise of multi-dimensional energy games for quantitative verification and synthesis. We show that establishing who is the winner in such a game with a state reachability objective is 2-ExpTime-complete. As a further application, we show that the same complexity result applies to the problem of whether a VASS is simulated by a finite-state system
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