Gabriele Puppis

Bounded repairability of word languages

What do you do if a computational object (e.g. program trace) fails a specification? An obvious approach is to perform a repair: modify the object minimally to get something that satisfies the constraints. This approach has been investigated in the database community, for integrity constraints, and in the AI community for propositional logics. Here we study how difficult it is to repair a document in the form of a string. Specifically, we consider number of edits that must be applied to an input string in order to satisfy a given target language. This number may be unbounded; our main contribution is to isolate the complexity of the bounded repair problem based on a characterization of the regular languages that admit bounded repairr. We consider the settings where the repair strategy is unconstrained and when the editing must be produced in a streaming way, i.e. by a letter-to-letter transducer.

A decidable weakening of Compass Logic based on cone-shaped cardinal directions

We introduce a modal logic, called Cone Logic, whose formulas describe properties of points in the plane and spatial relationships between them. Points are labelled by proposition letters and spatial relations are induced by the four cone-shaped cardinal directions. Cone Logic can be seen as a weakening of Venemas Compass Logic. We prove that, unlike Compass Logic and other projection-based spatial logics, its satisfiability problem is decidable (precisely, PSPACE-complete). We also show that it is expressive enough to capture meaningful interval temporal logics - in particular, the interval temporal logic of Allens relations Begins, During, and Later, and their transposes.

The per-character cost of repairing word languages

We show how to calculate the maximum number of edits per character needed to convert any string in one regular language to a string in another language. Our algorithm makes use of a local determinization procedure applicable to a subclass of distance automata. We then show how to calculate the same property when the editing needs to be done in streaming fashion, by a finite state transducer, using a reduction to mean-payoff games. In this case, we show that the optimal streaming editor can be produced in P.

Walking on Data Words

We see data words as sequences of letters with additional edges that connect pairs of positions carrying the same data value. We consider a natural model of automaton walking on data words, called Data Walking Automaton, and study its closure properties, expressiveness, and the complexity of paradigmatic problems. We prove that deterministic DWA are strictly included in non-deterministic DWA, that the former subclass is closed under all boolean operations, and that the latter class enjoys a decidable containment problem.

Untwisting two-way transducers in elementary time

Functional transductions realized by two-way transducers (equivalently, by streaming transducers and by MSO transductions) are the natural and standard notion of “regular” mappings from words to words. It was shown recently (LICS13) that it is decidable if such a transduction can be implemented by some one-way transducer, but the given algorithm has non-elementary complexity. We provide an algorithm of different flavor solving the above question, that has double exponential space complexity. We further apply our technique to decide whether the transduction realized by a two-way transducer can be implemented by a sweeping transducer, with either known or unknown number of passes.

Decidability of the Interval Temporal Logic \(\mathsf{A\bar{A}B\bar{B}}\) over the Rationals

The classification of the fragments of Halpern and Shoham’s logic with respect to decidability/undecidability of the satisfiability problem is now very close to the end. We settle one of the few remaining questions concerning the fragment (mathsf{Abar{A}Bbar{B}}), which comprises Allen’s interval relations “meets” and “begins” and their symmetric versions. We already proved that (mathsf{Abar{A}Bbar{B}}) is decidable over the class of all finite linear orders and undecidable over ordered domains isomorphic to ℕ. In this paper, we first show that (mathsf{Abar{A}Bbar{B}}) is undecidable over ℝ and over the class of all Dedekind-complete linear orders. We then prove that the logic is decidable over ℚ and over the class of all linear orders.

Which DTDs are streaming bounded repairable

Integrity constraint management concerns both checking whether data is valid and taking action to restore correctness when invalid data is discovered. In XML the notion of valid data can be captured by schema languages such as Document Type Definitions (DTDs) and more generally XML schemas. DTDs have the property that constraint checking can be done in streaming fashion. In this paper we consider when the corresponding action to restore validity -- repair -- can be done in streaming fashion. We formalize this as the problem of determining, given a DTD, whether or not a streaming procedure exists that transforms an input document so as to satisfy the DTD, using a number of edits independent of the document. We show that this problem is decidable. In fact, we show the decidability of a more general problem, allowing a more general class of schemas than DTDs, and requiring a repair procedure that works only for documents that are already known to satisfy another class of constraints. The decision procedure relies on a new analysis of the structure of DTDs, reducing to a novel notion of game played on pushdown systems associated with the schemas.

Bounded Repairability for Regular Tree Languages

We study the problem of bounded repairability of a given restriction tree language R into a target tree language T. More precisely, we say that R is bounded repairable with respect to T if there exists a bound on the number of standard tree editing operations necessary to apply to any tree in R to obtain a tree in T. We consider a number of possible specifications for tree languages: bottom-up tree automata (on curry encoding of unranked trees) that capture the class of XML schemas and document type definitions (DTDs). We also consider a special case when the restriction language R is universal (i.e., contains all trees over a given alphabet).n We give an effective characterization of bounded repairability between pairs of tree languages represented with automata. This characterization introduces two tools—synopsis trees and a coverage relation between them—allowing one to reason about tree languages that undergo a bounded number of editing operations. We then employ this characterization to provide upper bounds to the complexity of deciding bounded repairability and show that these bounds are tight. In particular, when the input tree languages are specified with arbitrary bottom-up automata, the problem is coNExp-complete. The problem remains coNExp-complete even if we use deterministic nonrecursive DTDs to specify the input languages. The complexity of the problem can be reduced if we assume that the alphabet, the set of node labels, is fixed: the problem becomes PSpace-complete for nonrecursive DTDs and coNP-complete for deterministic nonrecursive DTDs. Finally, when the restriction tree language R is universal, we show that the bounded repairability problem becomes Exp-complete if the target language is specified by an arbitrary bottom-up tree automaton and becomes tractable (P-complete, in fact) when a deterministic bottom-up automaton is used.

Which XML Schemas are Streaming Bounded Repairable

In this paper we consider the problem of repairing, that is, restoring validity of, documents with respect to XML schemas. We formalize this as the problem of determining, given an XML schema, whether or not a streaming procedure exists that transforms an input document so as to satisfy the XML schema, using a number of edits independent of the document. We show that this problem is decidable. In fact, we show the decidability of a more general problem, which allows the repair procedure to work on documents that are already known to satisfy another XML schema. The decision procedure relies on the analysis of the structure of an automaton model specifying the restriction and target XML schemas and reduces te problem to a novel notion of game played on pushdown systems associated with the schemas.

Logics with Rigidly Guarded Data Tests

The notion of orbit finite data monoid was recently introduced by Bojanczyk as an algebraic object for defining recognizable languages of data words. Following Buchis approach, we introduce a variant of monadic second-order logic with data equality tests that captures precisely the data languages recognizable by orbit finite data monoids. We also establish, following this time the approach of Schutzenberger, McNaughton and Papert, that the first-order fragment of this logic defines exactly the data languages recognizable by aperiodic orbit finite data monoids. Finally, we consider another variant of the logic that can be interpreted over generic structures with data. The data languages defined in this variant are also recognized by unambiguous finite memory automata.