145,344 research outputs found
Minimal Proof Search for Modal Logic K Model Checking
Most modal logics such as S5, LTL, or ATL are extensions of Modal Logic K.
While the model checking problems for LTL and to a lesser extent ATL have been
very active research areas for the past decades, the model checking problem for
the more basic Multi-agent Modal Logic K (MMLK) has important applications as a
formal framework for perfect information multi-player games on its own.
We present Minimal Proof Search (MPS), an effort number based algorithm
solving the model checking problem for MMLK. We prove two important properties
for MPS beyond its correctness. The (dis)proof exhibited by MPS is of minimal
cost for a general definition of cost, and MPS is an optimal algorithm for
finding (dis)proofs of minimal cost. Optimality means that any comparable
algorithm either needs to explore a bigger or equal state space than MPS, or is
not guaranteed to find a (dis)proof of minimal cost on every input.
As such, our work relates to A* and AO* in heuristic search, to Proof Number
Search and DFPN+ in two-player games, and to counterexample minimization in
software model checking.Comment: Extended version of the JELIA 2012 paper with the same titl
Non-normal modalities in variants of Linear Logic
This article presents modal versions of resource-conscious logics. We
concentrate on extensions of variants of Linear Logic with one minimal
non-normal modality. In earlier work, where we investigated agency in
multi-agent systems, we have shown that the results scale up to logics with
multiple non-minimal modalities. Here, we start with the language of
propositional intuitionistic Linear Logic without the additive disjunction, to
which we add a modality. We provide an interpretation of this language on a
class of Kripke resource models extended with a neighbourhood function: modal
Kripke resource models. We propose a Hilbert-style axiomatization and a
Gentzen-style sequent calculus. We show that the proof theories are sound and
complete with respect to the class of modal Kripke resource models. We show
that the sequent calculus admits cut elimination and that proof-search is in
PSPACE. We then show how to extend the results when non-commutative connectives
are added to the language. Finally, we put the logical framework to use by
instantiating it as logics of agency. In particular, we propose a logic to
reason about the resource-sensitive use of artefacts and illustrate it with a
variety of examples
A Decision Procedure for Herbrand Formulas without Skolemization
This paper describes a decision procedure for disjunctions of conjunctions of anti-prenex normal forms of pure first-order logic (FOLDNFs) that do not contain V within the scope of quantifiers. The disjuncts of these FOLDNFs are equivalent to prenex normal forms whose quantifier-free parts are conjunctions of atomic and negated atomic formulae (= Herbrand formulae). In contrast to the usual algorithms for Herbrand formulae, neither skolemization nor unification algorithms with function symbols are applied. Instead, a procedure is described that rests on nothing but equivalence transformations within pure first-order logic (FOL). This procedure involves the application of a calculus for negative normal forms (the NNF-calculus) with
A -||- A & A (= &I) as the sole rule that increases the complexity of given FOLDNFs.
The described algorithm illustrates how, in the case of Herbrand formulae,
decision problems can be solved through a systematic search for proofs that
reduce the number of applications of the rule &I to a minimum in the NNF-calculus. In the case of Herbrand formulae, it is even possible to entirely abstain from applying &I.
Finally, it is shown how the described procedure can be used within an optimized general search for proofs of contradiction and what kind of questions arise for a &I-minimal proof strategy in the case of a general search for proofs of contradiction
Approximating Optimal Bounds in Prompt-LTL Realizability in Doubly-exponential Time
We consider the optimization variant of the realizability problem for Prompt
Linear Temporal Logic, an extension of Linear Temporal Logic (LTL) by the
prompt eventually operator whose scope is bounded by some parameter. In the
realizability optimization problem, one is interested in computing the minimal
such bound that allows to realize a given specification. It is known that this
problem is solvable in triply-exponential time, but not whether it can be done
in doubly-exponential time, i.e., whether it is just as hard as solving LTL
realizability.
We take a step towards resolving this problem by showing that the optimum can
be approximated within a factor of two in doubly-exponential time. Also, we
report on a proof-of-concept implementation of the algorithm based on bounded
LTL synthesis, which computes the smallest implementation of a given
specification. In our experiments, we observe a tradeoff between the size of
the implementation and the bound it realizes. We investigate this tradeoff in
the general case and prove upper bounds, which reduce the search space for the
algorithm, and matching lower bounds.Comment: In Proceedings GandALF 2016, arXiv:1609.0364
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
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