1,200 research outputs found
Complexity Bounds for Ordinal-Based Termination
`What more than its truth do we know if we have a proof of a theorem in a
given formal system?' We examine Kreisel's question in the particular context
of program termination proofs, with an eye to deriving complexity bounds on
program running times.
Our main tool for this are length function theorems, which provide complexity
bounds on the use of well quasi orders. We illustrate how to prove such
theorems in the simple yet until now untreated case of ordinals. We show how to
apply this new theorem to derive complexity bounds on programs when they are
proven to terminate thanks to a ranking function into some ordinal.Comment: Invited talk at the 8th International Workshop on Reachability
Problems (RP 2014, 22-24 September 2014, Oxford
Abstract Canonical Inference
An abstract framework of canonical inference is used to explore how different
proof orderings induce different variants of saturation and completeness.
Notions like completion, paramodulation, saturation, redundancy elimination,
and rewrite-system reduction are connected to proof orderings. Fairness of
deductive mechanisms is defined in terms of proof orderings, distinguishing
between (ordinary) "fairness," which yields completeness, and "uniform
fairness," which yields saturation.Comment: 28 pages, no figures, to appear in ACM Trans. on Computational Logi
The complexity of linear-time temporal logic over the class of ordinals
We consider the temporal logic with since and until modalities. This temporal
logic is expressively equivalent over the class of ordinals to first-order
logic by Kamp's theorem. We show that it has a PSPACE-complete satisfiability
problem over the class of ordinals. Among the consequences of our proof, we
show that given the code of some countable ordinal alpha and a formula, we can
decide in PSPACE whether the formula has a model over alpha. In order to show
these results, we introduce a class of simple ordinal automata, as expressive
as B\"uchi ordinal automata. The PSPACE upper bound for the satisfiability
problem of the temporal logic is obtained through a reduction to the
nonemptiness problem for the simple ordinal automata.Comment: Accepted for publication in LMC
Middle-Out Reasoning for Logic Program Synthesis
We propose a novel approach to automating the synthesis of logic programs: Logic programs are synthesized as a by-product of the planning of a verification proof. The approach is a two-level one: At the object level, we prove program verification conjectures in a sorted, first-order theory. The conjectures are of the form 8args \Gamma\Gamma\Gamma\Gamma! : prog(args \Gamma\Gamma\Gamma\Gamma! ) $ spec(args \Gamma\Gamma\Gamma\Gamma! ). At the meta-level, we plan the object-level verification with an unspecified program definition. The definition is represented with a (second-order) meta-level variable, which becomes instantiated in the course of the planning
Reasoning about transfinite sequences
We introduce a family of temporal logics to specify the behavior of systems
with Zeno behaviors. We extend linear-time temporal logic LTL to authorize
models admitting Zeno sequences of actions and quantitative temporal operators
indexed by ordinals replace the standard next-time and until future-time
operators. Our aim is to control such systems by designing controllers that
safely work on -sequences but interact synchronously with the system in
order to restrict their behaviors. We show that the satisfiability problem for
the logics working on -sequences is EXPSPACE-complete when the
integers are represented in binary, and PSPACE-complete with a unary
representation. To do so, we substantially extend standard results about LTL by
introducing a new class of succinct ordinal automata that can encode the
interaction between the different quantitative temporal operators.Comment: 38 page
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