Well structured program equivalence is highly undecidable

We show that strict deterministic propositional dynamic logic with intersection is highly undecidable, solving a problem in the Stanford Encyclopedia of Philosophy. In fact we show something quite a bit stronger. We introduce the construction of program equivalence, which returns the value $\mathsf{T}$ precisely when two given programs are equivalent on halting computations. We show that virtually any variant of propositional dynamic logic has $\Pi_1^1$-hard validity problem if it can express even just the equivalence of well-structured programs with the empty program \texttt{skip}. We also show, in these cases, that the set of propositional statements valid over finite models is not recursively enumerable, so there is not even an axiomatisation for finitely valid propositions.Comment: 8 page

Fragments of ML Decidable by Nested Data Class Memory Automata

The call-by-value language RML may be viewed as a canonical restriction of Standard ML to ground-type references, augmented by a "bad variable" construct in the sense of Reynolds. We consider the fragment of (finitary) RML terms of order at most 1 with free variables of order at most 2, and identify two subfragments of this for which we show observational equivalence to be decidable. The first subfragment consists of those terms in which the P-pointers in the game semantic representation are determined by the underlying sequence of moves. The second subfragment consists of terms in which the O-pointers of moves corresponding to free variables in the game semantic representation are determined by the underlying moves. These results are shown using a reduction to a form of automata over data words in which the data values have a tree-structure, reflecting the tree-structure of the threads in the game semantic plays. In addition we show that observational equivalence is undecidable at every third- or higher-order type, every second-order type which takes at least two first-order arguments, and every second-order type (of arity greater than one) that has a first-order argument which is not the final argument

The foundations of computable general equilibrium theory

general equilibrium theory,CGE models,mathematical economics,computability,constructivity

Equivalence-Checking on Infinite-State Systems: Techniques and Results

The paper presents a selection of recently developed and/or used techniques for equivalence-checking on infinite-state systems, and an up-to-date overview of existing results (as of September 2004)

Constraints for Semistructured Data and XML

Integrity constraints play a fundamental role in database design. We review initial work on the expression of integrity constraints for semistructured data and XML

On the Origin of Abstraction : Real and Imaginary Parts of Decidability-Making

International audienceThe behavioral tradition has largely anchored on Simon's early conception of bounded rationality, it is important to engage more explicitly cognitive approaches particularly ones that might link to the issue of identifying novel competitive positions. The purpose of the study is to describe the cognitive processes by which decision-makers manage to work, individually or collectively, through undecidable situations and design innovatively. Most widespread models of rationality developed for preference-making and based on a real dimension should be extended for abstraction-making by adding a visible imaginary one. A development of a core analytical/conceptual apparatus is proposed to purposely account this dual form of reasoning, deductive to prove (then make) equivalence and abstractive to represent (then unmake) it. Complex numbers, comfortable to describe repetitive, expansional and superimposing phenomena (like waves, envelope of waves, interferences or holograms, etc.) appear as generalizable to cognitive processes at work when redesigning a decidable space by abstraction (like relief vision to design a missing depth dimension, Loyd's problem to design a missing degree of freedom, etc.). This theoretical breakthrough may open up vistas capacity in the fields of information systems, knowledge and decision

Coarse-graining of cellular automata, emergence, and the predictability of complex systems

We study the predictability of emergent phenomena in complex systems. Using nearest neighbor, one-dimensional Cellular Automata (CA) as an example, we show how to construct local coarse-grained descriptions of CA in all classes of Wolfram's classification. The resulting coarse-grained CA that we construct are capable of emulating the large-scale behavior of the original systems without accounting for small-scale details. Several CA that can be coarse-grained by this construction are known to be universal Turing machines; they can emulate any CA or other computing devices and are therefore undecidable. We thus show that because in practice one only seeks coarse-grained information, complex physical systems can be predictable and even decidable at some level of description. The renormalization group flows that we construct induce a hierarchy of CA rules. This hierarchy agrees well with apparent rule complexity and is therefore a good candidate for a complexity measure and a classification method. Finally we argue that the large scale dynamics of CA can be very simple, at least when measured by the Kolmogorov complexity of the large scale update rule, and moreover exhibits a novel scaling law. We show that because of this large-scale simplicity, the probability of finding a coarse-grained description of CA approaches unity as one goes to increasingly coarser scales. We interpret this large scale simplicity as a pattern formation mechanism in which large scale patterns are forced upon the system by the simplicity of the rules that govern the large scale dynamics.Comment: 18 pages, 9 figure

Alternating register automata on finite words and trees

We study alternating register automata on data words and data trees in relation to logics. A data word (resp. data tree) is a word (resp. tree) whose every position carries a label from a finite alphabet and a data value from an infinite domain. We investigate one-way automata with alternating control over data words or trees, with one register for storing data and comparing them for equality. This is a continuation of the study started by Demri, Lazic and Jurdzinski. From the standpoint of register automata models, this work aims at two objectives: (1) simplifying the existent decidability proofs for the emptiness problem for alternating register automata; and (2) exhibiting decidable extensions for these models. From the logical perspective, we show that (a) in the case of data words, satisfiability of LTL with one register and quantification over data values is decidable; and (b) the satisfiability problem for the so-called forward fragment of XPath on XML documents is decidable, even in the presence of DTDs and even of key constraints. The decidability is obtained through a reduction to the automata model introduced. This fragment contains the child, descendant, next-sibling and following-sibling axes, as well as data equality and inequality tests