Decidability and Undecidability Results for Propositional Schemata

International audienceWe define a logic of propositional formula schemata adding to the syntax of propositional logic indexed propositions and iterated connectives ranging over intervals parameterized by arithmetic variables. The satisfiability problem is shown to be undecidable for this new logic, but we introduce a very general class of schemata, called bound-linear, for which this problem becomes decidable. This result is obtained by reduction to a particular class of schemata called regular, for which we provide a sound and complete terminating proof procedure. This schemata calculus allows one to capture proof patterns corresponding to a large class of problems specified in propositional logic. We also show that the satisfiability problem becomes again undecidable for slight extensions of this class, thus demonstrating that bound-linear schemata represent a good compromise between expressivity and decidability

The range of non-linear natural polynomials cannot be context-free

summary:Suppose that some polynomial $f$ with rational coefficients takes only natural values at natural numbers, i. e., $L=\{f(n)\mid n\in {\mathbb N}\}\subseteq {\mathbb N}$. We show that the base-$q$ representation of $L$ is a context-free language if and only if $f$ is linear, answering a question of Shallit. The proof is based on a new criterion for context-freeness, which is a combination of the Interchange lemma and a generalization of the Pumping lemma

The range of non-linear natural polynomials cannot be context-free

Suppose that some polynomial $f$ with rational coefficients takes only natural values at natural numbers, i.e., $L=\{f(n)\mid n\in \mathbb N\}\subset\mathbb N$. We show that the base-$q$ representation of $L$ is a context-free language if and only if $f$ is linear, answering a question of Shallit. The proof is based on a new criterion for context-freeness, which is a combination of the Interchange lemma and a generalization of the Pumping lemma.Comment: This paper should be assigned to cs.FL, but I'm not endorsed over ther

Dependency structures and lexicalized grammars

In this dissertation, we show that that both the generative capacity and the parsing complexity of lexicalized grammar formalisms are systematically related to structural properties of the dependency structures that these formalisms can induce. Dependency structures model the syntactic dependencies among the words of a sentence. We identify three empirically relevant classes of dependency structures, and show how they can be characterized both in terms of restrictions on the relation between dependency and word-order and within an algebraic framework. In the second part of the dissertation, we develop natural notions of automata and grammars for dependency structures, show how these yield infinite hierarchies of ever more powerful dependency languages, and classify several grammar formalisms with respect to the languages in these hierarchies that they are able to characterize. Our results provide fundamental insights into the relation between dependency structures and lexicalized grammars.In dieser Arbeit zeigen wir, dass sowohl die Ausdrucksmächtigkeit als auch die Verarbeitungskomplexität von lexikalisierten Grammatikformalismen auf systematische Art und Weise von strukturellen Eigenschaften der Dependenzstrukturen abhängen, die diese Formalismen induzieren. Dependenzstrukturen modellieren die syntaktischen Abhängigkeiten zwischen den Wörtern eines Satzes. Wir identifizieren drei empirisch relevante Klassen von Dependenzstrukturen und zeigen, wie sich diese sowohl durch Einschränkungen der Interaktion zwischen Dependenz und Wortstellung, als auch in einem algebraischen Rahmen charakterisieren lassen. Im zweiten Teil der Arbeit entwickeln wir natürliche Begriffe von Automaten und Grammatiken für Dependenzstrukturen, zeigen, wie diese zu unendlichen Hierarchien immer ausdrucksmächtigerer Dependenzsprachen führen, und klassifizieren mehrere Grammatikformalismen in Bezug auf die Sprachen in diesen Hierarchien, die von ihnen charakterisiert werden können. Unsere Resultate liefern grundlegende Einsichten in das Verhältnis zwischen Dependenzstrukturen und lexikalisierten Grammatiken

Grammars over the Lambek Calculus with Permutation: Recognizing Power and Connection to Branching Vector Addition Systems with States

In [Van Benthem, 1991] it is proved that all permutation closures of context-free languages can be generated by grammars over the Lambek calculus with the permutation rule (LP-grammars); however, to our best knowledge, it is not established whether converse holds or not. In this paper, we show that LP-grammars are equivalent to linearly-restricted branching vector addition systems with states and with additional memory (shortly, lBVASSAM), which are modified branching vector addition systems with states. Then an example of such an lBVASSAM is presented, which generates a non-semilinear set of vectors; this yields that LP-grammars generate more than permutation closures of context-free languages. Moreover, equivalence of LP-grammars and lBVASSAM allows us to present a normal form for LP-grammars and, as a consequence, prove that LP-grammars are equivalent to LP-grammars without product

Characterizing Formality

Complexity classes are defined by quantitative restrictions of resources available to a computational model, like for instance the Turing machine. Contrarily, there is no obvious commonality in the definition of families of formal languages - instead they are described by example. This thesis is about the characterization of what makes a set of languages a family of formal languages. Families of formal languages, like for example the regular, context-free languages and their sub-families exhibit properties that are contrasted by the ones of complexity classes. Two of the properties families of formal languages seem to have is closure of intersection with regular languages, another is the existence of pumping or iteration arguments which yield the decidability of the emptiness. Complexity classes do not generally have a decidable emptiness, which lead us to a first candidate for the notion of formality - the decidability of the emptiness of regular intersection (intreg). We refute the decidability of intreg as a criterion by hiding the difficulty of deciding the emptiness of regular intersection: We show that for every decidable language L there is a language L' of essentially the same complexity such that intreg(L') is decidable. This implies that every complexity class contains complete languages for which the emptiness of regular intersection is decidable. An intermediate result we show is that the set of true quantified Boolean formulae has a decidable emptiness of regular intersection. As the known families of formal languages are all contained in NP, this yields a language (probably) outside of NP for which intreg is decidable, which additionally is a natural language in contrast to the artificial ones obtained by the hiding process. We introduce the notion of protocol languages which capture in some sense the behavior of a data-structure underlying the model of a formal language. They are defined in a fragment of second order logic, where the second order variables are uniquely determined by each word in the language and each letter implies a determined sub-structure of a word. Viewing the letters of a word as vertices and the successor as edges between them, each word can be seen as a path. The binary second order variables can be viewed as additional edges between word positions. Therefore, each word in a protocol language defines some unique graph. These graphs can be recognized by covering them with a predefined set of tiles which are node and edge-labeld graphs. Additional numerical constraints on the amount of each tile-type yields shrinking-arguments for protocol languages. If a word w in a protocol language exceeds a certain length such that the numerical constraints are (over-)satisfied, one can constuctively generate a shorter word w' from w that is also contained in the protocol language. We define logical extensions of protocol languages by allowing the conjunction of additional first order or monadic second order definable formulae and analyze the extensions in regard to trio operations. Protocol languages for the regular, context-free and indexed languages are exhibited -- for the first two we give protocol languages which act as generators for the respective family of formal languages. Finally, we show that the emptiness of protocol languages is decidable

FICS 2010

International audienceInformal proceedings of the 7th workshop on Fixed Points in Computer Science (FICS 2010), held in Brno, 21-22 August 201

Partial (In)Completeness in Abstract Interpretation

In the abstract interpretation framework, completeness represents an optimal simulation by the abstract operators over the behavior of the concrete operators. This corresponds to an ideal (often rare) feature where there is no loss of information accumulated in abstract computations with respect to the properties encoded by the underlying abstract domains. In this thesis, we deal with the opposite notion of completeness in abstract interpretation, that is, incompleteness, applied to two different contexts: static program analysis and formal languages over the Chomsky's hierarchy. In static program analysis, completeness is a very rare condition to be satisfied in practice and only the straightforward abstractions are complete for all programs, thus, we usually deal with incompleteness. For this reason, we introduce the notion of partial completeness. Partial completeness is a weaker notion of completeness which requires the imprecision of the analysis to be limited. A partially complete abstract interpretation allows some false alarms to be reported, but their number is bounded by a constant. We collect in partial completeness classes all the programs whose abstract interpretations share the same upper bound of imprecision. We then focus on the investigation of the computational limits of the class of partially complete programs with respect to a given abstract domain. Moreover, we show that the class of all partially complete programs is non-recursively enumerable, and its complement is productive whenever we allow an unlimited imprecision in the abstract domain. Finally, we formalize the local partial completeness class within which we require partial completeness only on some specific inputs. We prove that this last class of programs is a recursively enumerable set under a structural hypothesis on the underlying abstract domain, by showing an algorithm capable of proving the local partial completeness of a program with respect to a given abstract domain and an upper bound of imprecision. In formal language theory, we want to study a possible reformulation, by abstract interpretation, of classes of languages in the Chomsky's hierarchy, and, by exploiting the incompleteness of languages abstractions, we want to define separation results between classes of languages. To this end, we do a first step into this direction by studying the relation between indexed languages (recognized by indexed grammars) and context-free languages. Indexed grammars are a generalization of context-free grammars which recognize a proper subset of context-sensitive languages, the so called indexed languages. %The class of languages recognized by indexed grammars is called indexed languages and they correspond to the languages recognized by nested stack automata. For example, indexed grammars can recognize the language ${a^nb^nc^n mid ngeq 1 }$ which is not context-free, but they cannot recognize ${ (ab^n)^n mid ngeq 1}$ which is context-sensitive. Indexed grammars identify a set of languages that are more expressive than context-free languages, while having decidability results that lie in between the ones of context-free and context-sensitive languages. We provide a fixpoint characterization of the languages recognized by an indexed grammar and we study possible ways to abstract, in the abstract interpretation sense, these languages and their grammars into context-free and regular languages. We formalize the separation class between indexed and context-free languages, i.e., all the languages that cannot be generated by a context-free grammar, as an instance of incompleteness of stack elimination abstraction over indexed grammars