Probabilistic asynchronous automata

Mixed product and asynchronous automata

Asynchronous automata are a natural distributed machine model<br />for recognizing trace languages - languages defined over an alphabet<br />equipped with an independence relation.<br />To handle infinite traces, Gastin and Petit introduced Buchi asynchronous<br />automata, which accept precisely the class of omega-regular trace<br />languages. Like their sequential counterparts, these automata need to<br />be non-deterministic in order to capture all omega-regular languages. Thus<br />complementation of these automata is non-trivial. Complementation<br />is an important operation because it is fundamental for treating the<br />logical connective "not" in decision procedures for monadic second-order<br />logics. Subsequently, Diekert and Muscholl solved the complementation<br />problem by showing that with a Muller acceptance condition, deterministic<br />automata suffice for recognizing omega-regular trace languages.<br />However, a direct determinization procedure, extending the classical<br />subset construction, has proved elusive.<br />In this paper, we present a direct determinization procedure for<br />Buchi asynchronous automata, which generalizes Safra's construction<br />for sequential Buchi automata. As in the sequential case, the blow-up<br />in the state space is essentially that of the underlying subset construction.

Zielonka’s theorem shows that each regular set of Mazurkiewicz traces can be implemented as a system of synchronized processes provided with some distributed control structure called an asynchronous automaton. This paper gives a new algorithm for the synthesis of a non-deterministic asynchronous automaton from a regular Mazurkiewicz trace language. Our approach is based on an unfolding procedure that improves the complexity of Zielonka’s and Pighizzini’s techniques: Our construction is polynomial in terms of the number of states but still double-exponential in the size of the alphabet. As opposed to Métivier’s work, our algorithm does not restrict to acyclic dependence alphabets.

We investigate the possibility of extending Chrobak normal form to the probabilistic case. While in the nondeterministic case a unary automaton can be simulated by an automaton in Chrobak normal form without increasing the number of the states in the cycles, we show that in the probabilistic case the simulation is not possible by keeping the same number of ergodic states. This negative result is proved by considering the natural extension to the probabilistic case of Chrobak normal form, obtained by replacing nondeterministic choices with probabilistic choices. We then propose a different kind of normal form, namely, cyclic normal form, which does not suffer from the same problem: we prove that each unary probabilistic automaton can be simulated by a probabilistic automaton in cyclic normal form, with at most the same number of ergodic states. In the nondeterministic case there are trivial simulations between Chrobak normal form and cyclic normal form, preserving the total number of states in the automata and in their cycles.

Petri nets with generalized algebra: a comparison

Any program can be broken into its loop structure, plus acyclic dependence graphs representing the body of each loop or subroutine. The parallelism inherent in these acyclic graphs augments the loop- level parallelism available in the program. This paper presents two algorithms for dynamic scheduling of acy- clic task graphs containing both data and control dependences, and describes a microarchitecture which implements these algorithms efficiently.

We provide a generalization of Mundici’s equivalence between unital Abelian lattice-ordered groups and MV-algebras: the category of unital commutative lattice-ordered monoids is equivalent to the category of MV-monoidal algebras. Roughly speaking, unital commutative lattice-ordered monoids are unital Abelian lattice-ordered groups without the unary operation x mapsto -x. The primitive operations are +, vee , wedge , 0, 1, -1. A prime example of these structures is mathbb {R}, with the obvious interpretation of the operations. Analogously, MV-monoidal algebras are MV-algebras without the negation x mapsto lnot x. The primitive operations are oplus , odot , vee , wedge , 0, 1. A motivating example of MV-monoidal algebra is the negation-free reduct of the standard MV-algebra [0, 1]subseteq mathbb {R}. We obtain the original Mundici’s equivalence as a corollary of our main result.

We exhibit the cartesian differential categories of Blute, Cockett and Seely as a particular kind of enriched category. The base for the enrichment is the category of commutative monoids—or in a straightforward generalisation, the category of modules over a commutative rig k. However, the tensor product on this category is not the usual one, but rather a warping of it by a certain monoidal comonad Q. Thus the enrichment base is not a monoidal category in the usual sense, but rather a skew monoidal category in the sense of Szlachányi. Our first main result is that cartesian differential categories are the same as categories with finite products enriched over this skew monoidal base. The comonad Q involved is, in fact, an example of a differential modality. Differential modalities are a kind of comonad on a symmetric monoidal k-linear category with the characteristic feature that their co-Kleisli categories are cartesian differential categories. Using our first main result, we are able to prove our second one: that every small cartesian differential category admits a full, structure-preserving embedding into the cartesian differential category induced by a differential modality (in fact, a monoidal differential modality on a monoidal closed category—thus, a model of intuitionistic differential linear logic). This resolves an important open question in this area.

The pebble game on directed acyclic graphs is commonly encountered as an abstract model for register allocation problems. The traditional move rule of the game asserts that one may put a pebble on node x once all its immediate predecessors have a pebble, leaving it open whether the pebble to be placed on x should be taken from some predecessor of x or from the free pool (the strict interpretation). We show that allowing pebbles to slide along an edge as a legal move enables one to save precisely one pebble over the strict interpretation. However, in the worst case the saving may be obtained only at the cost of squaring the time needed to pebble the dag. It shows that one has to be very careful in describing properties of pebblings; the interpretation of the rules can seriously affect the results. As a main result we prove a linear to exponential time trade-off for any fixed interpretation of the rules when a single pebble is saved. There exist families of dags with indegrees ≤2, with the property that they can be pebbled in linear time when one more pebble than the minimum needed is available but which require exponential time when the extra pebble is dropped.

The problem of hypothesis testing against independence for a Gauss-Markov random field (GMRF) is analyzed. Assuming an acyclic dependency graph, an expression for the log-likelihood ratio of detection is derived. Assuming random placement of nodes over a large region according to the Poisson or uniform distribution and nearest-neighbor dependency graph, the error exponent of the Neyman-Pearson detector is derived using large-deviations theory. The error exponent is expressed as a dependency-graph functional and the limit is evaluated through a special law of large numbers for stabilizing graph functionals. The exponent is analyzed for different values of the variance ratio and correlation. It is found that a more correlated GMRF has a higher exponent at low values of the variance ratio whereas the situation is reversed at high values of the variance ratio.

Free partially commutative inverse monoids are investigated. Analogously to free partially commutative monoids (trace monoids), free partially commutative inverse monoid are the quotients of free inverse monoids modulo a partially defined commutation relation on the generators. An O(n log(n)) algorithm on a RAM for the word problem is presented, and NP-completeness of the generalized word problem and the membership problem for rational sets is shown. Moreover, free partially commutative inverse monoids modulo a finite idempotent presentation are studied. For these monoids, the word problem is decidable if and only if the complement of the commutation relation is transitive.KeywordsWord ProblemInverse SemigroupCayley GraphDependence GraphDependence RelationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In this article, we give a new formulation of acyclic scheduling problem under registers and resources constraints in multiple instructions issuing processors (VLIW and superscalar) Given a direct acyclic data dependence graph G = (V, E), the complexity of our integer linear programming model is bounded by O(|V|2) variables and O(|E| + |V|2) constraints according to a target architecture description. This complexity is better than the complexity of the existing techniques which includes a worst total schedule time factor.

A modified data dependency graph (DDG) for modeling loop-optimizing restrictions is proposed. By using it, pipelining, loop-carried data dependencies and folding of individual multicycle operations over iteration boundaries can be modeled. Properties of semantically correct software pipelining transformations are first defined in the acyclic DDG of the unrolled loop. The case when an operation in a loop body is dependent on that same operation in some previous iteration is analyzed in the acyclic DDG of the unrolled loop. The algorithm for finding a real machine optimal schedule is split into two independent tasks: finding an optimal acyclic data dependency graph of one loop body and then finding an optimal schedule for a real machine by implementing basic block methods. A method for loop optimizing named 'pipelining of strongly connected components' is proposed. It is proved that the method is semantically correct. The loop body optimization is converted into sub-tasks of optimizing individual strongly connected components of the modified data dependency graph of a loop. >

We consider generalized metric spaces taking distances in an arbitrary ordered commutative monoid, and investigate when a class $\mathcal{K}$ of finite generalized metric spaces satisfies the Hrushovski extension property: for any $A\in\mathcal{K}$ there is some $B\in\mathcal{K}$ such that $A$ is a subspace of $B$ and any partial isometry of $A$ extends to a total isometry of $B$. Our main result is the Hrushovski property for the class of finite generalized metric spaces over a semi-archimedean monoid $\mathcal{R}$. When $\mathcal{R}$ is also countable, this can be used to show that the isometry group of the Urysohn space over $\mathcal{R}$ has ample generics. Finally, we prove the Hrushovski property for classes of integer distance metric spaces omitting triangles of uniformly bounded odd perimeter. As a corollary, given odd $n\geq 3$, we obtain ample generics for the automorphism group of the universal, existentially closed graph omitting cycles of odd length bounded by $n$.

Trace monoids are obtained from free monoids by defining a subset I of pairs of letters that are allowed to commute. Most of the work of this theory is an attempt to relate the properties of these monoids to the properties of I. Following the work initiated by Buchi we show that when I is an equivalence relation (the trace monoid is then a free product of free commutative monoids) it is possible to define a second order logic whose models are the traces viewed as dependence graphs and which characterizes exactly the sets of traces that are rational. This logic essentially utilizes a predicate based on the ordering defined by the dependence graph and a predicate related to a restricted use of the comparison of cardinality.