23 research outputs found
A Framework for the Flexible Integration of a Class of Decision Procedures into Theorem Provers
The role of decision procedures is often essential in theorem proving. Decision procedures can reduce the search space of heuristic components of a prover and increase its abilities. However, in some applications only a small number of conjectures fall within the scope of the available decision procedures. Some of these conjectures could in an informal sense fall ājust outsideā that scope. In these situations a problem arises because lemmas have to be invoked or the decision procedure has to communicate with the heuristic component of a theorem prover. This problem is also related to the general problem of how to exibly integrate decision procedures into heuristic theorem provers. In this paper we address such problems and describe a framework for the exible integration of decision procedures into other proof methods. The proposed framework can be used in different theorem provers, for different theories and for different decision procedures. New decision procedures can be simply āplugged-inā to the system. As an illustration, we describe an instantiation of this framework within the Clam proof-planning system, to which it is well suited. We report on some results using this implementation
Complexity of short Presburger arithmetic
We study complexity of short sentences in Presburger arithmetic (Short-PA).
Here by "short" we mean sentences with a bounded number of variables,
quantifiers, inequalities and Boolean operations; the input consists only of
the integers involved in the inequalities. We prove that assuming Kannan's
partition can be found in polynomial time, the satisfiability of Short-PA
sentences can be decided in polynomial time. Furthermore, under the same
assumption, we show that the numbers of satisfying assignments of short
Presburger sentences can also be computed in polynomial time
Subclasses of Presburger Arithmetic and the Weak EXP Hierarchy
It is shown that for any fixed , the -fragment of
Presburger arithmetic, i.e., its restriction to quantifier alternations
beginning with an existential quantifier, is complete for
, the -th level of the weak EXP
hierarchy, an analogue to the polynomial-time hierarchy residing between
and . This result completes the
computational complexity landscape for Presburger arithmetic, a line of
research which dates back to the seminal work by Fischer & Rabin in 1974.
Moreover, we apply some of the techniques developed in the proof of the lower
bound in order to establish bounds on sets of naturals definable in the
-fragment of Presburger arithmetic: given a -formula
, it is shown that the set of non-negative solutions is an ultimately
periodic set whose period is at most doubly-exponential and that this bound is
tight.Comment: 10 pages, 2 figure
Inexpressibility Results for Regular Languages in Nonregular Settings
My ostensible purpose in this talk is to describe some new results (found in collaboration with Amitabha Roy) on expressibility of regular languages in certain generalizations of first-order logic. [10]. This provides me with a good excuse for describing some the work on the algebraic theory of regular languages in what one might call ānonregular settingsā. The syntactic monoid and syntactic morphism of a regular language provide a highly effective tool for proving that a given regular language is not expressible or recognizable in certain compuational models, as long as the model is guaranteed to produce only regular languages. This includes finite automata, of course. but also formulas of propositional temporal logic, and first-order logic, provided one is careful to restrict the expressive power of such logics. (For example, by only allowing the order relation in first-order formulas.) Things become much harder, and quite a bit more interesting, when we drop this kind of restriction on the model. The questions that arise are important (particularly in computational complexity), and most of them are unsolved. They all point to a rich theory that extends the reach of algebraic methods beyond the domain of finite automata 1 Uniformizing Nonuniform Automata with Ramseyās Theorem Letās start with an especially trivial application of the syntactic monoid: Let Ī£ = {0, 1}, and consider the two language
Symbolic diagnosis of partially observable concurrent systems
Abstract. Monitoring large distributed concurrent systems is a challenging task. In this paper we formulate (model-based) diagnosis by means of hidden state history reconstruction, from event (e.g. alarm) observations. We follow a so-called true concurrency approach: the model defines explicitly the causal and concurrency relations between the observable events, produced by the system under supervision on different points of observation. The problem is to compute on-the-fly the different partial order histories, which are the possible explanations of the observable events. In this paper we extend our first method based on Petri nets unfolding to high-level parameterized Petri nets. This allows the designer to model data aspects (even on infinite domains) and non deterministic actions. The observation of such an action gives only partial information and the supervisor has to introduce parameters to represent the hidden aspects of the reached state. This supposes that the possible values for the parameters are symbolically computed and refined during supervision. In practice, non deterministic actions can also be used as an approximation to deal with incomplete information about the system. In this case the refinement of the parameters during supervision improves the knowledge of the model.