27,375 research outputs found
Unambiguous Languages Exhaust the Index Hierarchy
This work is a study of the expressive power of unambiguity in the case of automata over infinite trees. An automaton is called unambiguous if it has at most one accepting run on every input, the language of such an automaton is called an unambiguous language. It is known that not every regular language of infinite trees is unambiguous. Except that, very little is known about which regular tree languages are unambiguous.
This paper answers the question whether unambiguous languages are of bounded complexity among all regular tree languages. The notion of complexity is the canonical one, called the (parity or Rabin/Mostowski) index hierarchy. The answer is negative, as exhibited by a family of examples of unambiguous languages the cannot be recognised by any alternating parity tree automata of bounded range of priorities.
Hardness of the examples is based on the theory of signatures, previously studied by Walukiewicz. The technical core of the article is a definition of the canonical signatures together with a parity game that compares signatures of a given pair of parity games (of the same index)
Automatic Unbounded Verification of Alloy Specifications with Prover9
Alloy is an increasingly popular lightweight specification language based on
relational logic. Alloy models can be automatically verified within a bounded
scope using off-the-shelf SAT solvers. Since false assertions can usually be
disproved using small counter-examples, this approach suffices for most
applications. Unfortunately, it can sometimes lead to a false sense of
security, and in critical applications a more traditional unbounded proof may
be required. The automatic theorem prover Prover9 has been shown to be
particularly effective for proving theorems of relation algebras [7], a
quantifier-free (or point-free) axiomatization of a fragment of relational
logic. In this paper we propose a translation from Alloy specifications to fork
algebras (an extension of relation algebras with the same expressive power as
relational logic) which enables their unbounded verification in Prover9. This
translation covers not only logic assertions, but also the structural aspects
(namely type declarations), and was successfully implemented and applied to
several examples
Disjunctive bases: normal forms and model theory for modal logics
We present the concept of a disjunctive basis as a generic framework for
normal forms in modal logic based on coalgebra. Disjunctive bases were defined
in previous work on completeness for modal fixpoint logics, where they played a
central role in the proof of a generic completeness theorem for coalgebraic
mu-calculi. Believing the concept has a much wider significance, here we
investigate it more thoroughly in its own right. We show that the presence of a
disjunctive basis at the "one-step" level entails a number of good properties
for a coalgebraic mu-calculus, in particular, a simulation theorem showing that
every alternating automaton can be transformed into an equivalent
nondeterministic one. Based on this, we prove a Lyndon theorem for the full
fixpoint logic, its fixpoint-free fragment and its one-step fragment, a Uniform
Interpolation result, for both the full mu-calculus and its fixpoint-free
fragment, and a Janin-Walukiewicz-style characterization theorem for the
mu-calculus under slightly stronger assumptions.
We also raise the questions, when a disjunctive basis exists, and how
disjunctive bases are related to Moss' coalgebraic "nabla" modalities. Nabla
formulas provide disjunctive bases for many coalgebraic modal logics, but there
are cases where disjunctive bases give useful normal forms even when nabla
formulas fail to do so, our prime example being graded modal logic. We also
show that disjunctive bases are preserved by forming sums, products and
compositions of coalgebraic modal logics, providing tools for modular
construction of modal logics admitting disjunctive bases. Finally, we consider
the problem of giving a category-theoretic formulation of disjunctive bases,
and provide a partial solution
No value restriction is needed for algebraic effects and handlers
We present a straightforward, sound Hindley-Milner polymorphic type system
for algebraic effects and handlers in a call-by-value calculus, which allows
type variable generalisation of arbitrary computations, not just values. This
result is surprising. On the one hand, the soundness of unrestricted
call-by-value Hindley-Milner polymorphism is known to fail in the presence of
computational effects such as reference cells and continuations. On the other
hand, many programming examples can be recast to use effect handlers instead of
these effects. Analysing the expressive power of effect handlers with respect
to state effects, we claim handlers cannot express reference cells, and show
they can simulate dynamically scoped state
On Algorithms and Complexity for Sets with Cardinality Constraints
Typestate systems ensure many desirable properties of imperative programs,
including initialization of object fields and correct use of stateful library
interfaces. Abstract sets with cardinality constraints naturally generalize
typestate properties: relationships between the typestates of objects can be
expressed as subset and disjointness relations on sets, and elements of sets
can be represented as sets of cardinality one. Motivated by these applications,
this paper presents new algorithms and new complexity results for constraints
on sets and their cardinalities. We study several classes of constraints and
demonstrate a trade-off between their expressive power and their complexity.
Our first result concerns a quantifier-free fragment of Boolean Algebra with
Presburger Arithmetic. We give a nondeterministic polynomial-time algorithm for
reducing the satisfiability of sets with symbolic cardinalities to constraints
on constant cardinalities, and give a polynomial-space algorithm for the
resulting problem.
In a quest for more efficient fragments, we identify several subclasses of
sets with cardinality constraints whose satisfiability is NP-hard. Finally, we
identify a class of constraints that has polynomial-time satisfiability and
entailment problems and can serve as a foundation for efficient program
analysis.Comment: 20 pages. 12 figure
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