Formal language

About: Formal language is a research topic. Over the lifetime, 5763 publications have been published within this topic receiving 154114 citations.

A unified approach for showing language containment and equivalence between various types of Ω-automata

Abstract: We consider the language containment and equivalence problems for six different types of ω-automata: Buchi, Muller, Rabin, Streett, the L-automata of Kurshan, and the ∀-automata of Manna and Pnueli. We give a six by six matrix in which each row and column is associated with one of these types of automata. The entry in the i th row and j th column is the complexity of showing containment between the i th type of automaton and the j th . Thus, for example, we give the complexity of showing language containment and equivalence between a Buchi automaton and a Muller or Streett automaton. Our results are obtained by a uniform method that associates a formula of the logic CTL* with each type of automaton. Our algorithms use a model checking procedure for the logic with the formulas obtained from the automata. The results of our paper are important for verification of finite state concurrent systems with fairness constraints. A natural way of reasoning about such systems is to model the finite state program by one ω-automaton and its specification by another.

The Mizar Mathematical Library in OMDoc: Translation and Applications

Abstract: The Mizar Mathematical Library is one of the largest libraries of formalized and mechanically verified mathematics. Its language is highly optimized for authoring by humans. As in natural languages, the meaning of an expression is influenced by its (mathematical) context in a way that is natural to humans, but harder to specify for machine manipulation. Thus its custom file format can make the access to the library difficult. Indeed, the Mizar system itself is currently the only system that can fully operate on the Mizar library. This paper presents a translation of the Mizar library into the OMDoc format (Open Mathematical Documents), an XML-based representation format for mathematical knowledge. OMDoc is geared towards machine support and interoperability by making formula structure and context dependencies explicit. Thus, the Mizar library becomes accessible for a wide range of OMDoc-based tools for formal mathematics and knowledge management. We exemplify interoperability by indexing the translated library in the MathWebSearch engine, which provides an "applicable theorem search" service (almost) out of the box.

A Formal Hierarchy of RNN Architectures

Abstract: We develop a formal hierarchy of the expressive capacity of RNN architectures. The hierarchy is based on two formal properties: space complexity, which measures the RNN's memory, and rational recurrence, defined as whether the recurrent update can be described by a weighted finite-state machine. We place several RNN variants within this hierarchy. For example, we prove the LSTM is not rational, which formally separates it from the related QRNN (Bradbury et al., 2016). We also show how these models' expressive capacity is expanded by stacking multiple layers or composing them with different pooling functions. Our results build on the theory of ``saturated" RNNs (Merrill, 2019). While formally extending these findings to unsaturated RNNs is left to future work, we hypothesize that the practical learnable capacity of unsaturated RNNs obeys a similar hierarchy. We provide empirical results to support this conjecture. Experimental findings from training unsaturated networks on formal languages support this conjecture.

Metamorphism, Formal Grammars and Undecidable Code Mutation

Abstract: This paper presents a formalisation of the different existing code mutation techniques (polymorphism and metamor- phism) by means of formal grammars. While very few theoretical results are known about the detection complexity of viral mutation techniques, we exhaustively address this critical issue by considering the Chomsky classification of formal grammars. This enables us to determine which family of code mutation techniques are likely to be detected or on the contrary are bound to remain undetected. As an illustration we then present, on a formal basis, a proof-of-concept metamorphic mutation engine denoted PB MOT, whose detection has been proven to be undecidable. Keywords—Polymorphism, Metamorphism, Formal Grammars, Formal Languages, Language Decision, Code Mutation, Word Prob- lem. that the set Di of polymorphic viruses with an infinite number of forms is a Σ3-complete set. Unfortunately, no results is known for other classes of polymorphic viruses and for the general case of metamorphism. Many open problems still remain. Up to now, only very few examples of metamorphic codes are known to exist. The most sophisticated one is the MetaPHOR engine whose essential feature is a certain amount of non-determinism. Experiments in our laboratory showed that existing antivirus software can be very easily defeated by MetaPHOR-like technology. However, the analysis of this engine (9, Chap. 4) has proved that its metamorphic techniques still belong to trivial classes. Our research thus focused on the formalisation of metamor- phism by means of formal grammar and languages. We aimed at identifying the different possible classes of possible code mutation techniques. The first results, which are presented in this paper, enable to assert that detection complexity of code mutation techniques can be far higher that NP-complete and that for some well-chosen classes, detection is an undecidable problem.

Theory of Finite Automata With an Introduction to Formal Languages

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