Verified decision procedures for MSO on words based on derivatives of regular expressions

Monadic second-order logic on finite words is a decidable yet expressive logic into which many decision problems can be encoded. Since MSO formulas correspond to regular languages, equivalence of MSO formulas can be reduced to the equivalence of some regular structures (e.g., automata). This paper presents a verified functional decision procedure for MSO formulas that is not based on automata but on regular expressions. Functional languages are ideally suited for this task: regular expressions are data types and functions on them are defined by pattern matching and recursion and are verified by structural induction. Decision procedures for regular expression equivalence have been formalized before, usually based on Brzozowski derivatives. Yet, for a straightforward embedding of MSO formulas into regular expressions, an extension of regular expressions with a projection operation is required. We prove total correctness and completeness of an equivalence checker for regular expressions extended in that way. We also define a language-preserving translation of formulas into regular expressions with respect to two different semantics of MSO. Our results have been formalized and verified in the theorem prover Isabelle. Using Isabelle's code generation facility, this yields purely functional, formally verified programs that decide equivalence of MSO formula

On the Use of Quasiorders in Formal Language Theory

In this thesis we use quasiorders on words to offer a new perspective on two well-studied problems from Formal Language Theory: deciding language inclusion and manipulating the finite automata representations of regular languages. First, we present a generic quasiorder-based framework that, when instantiated with different quasiorders, yields different algorithms (some of them new) for deciding language inclusion. We then instantiate this framework to devise an efficient algorithm for searching with regular expressions on grammar-compressed text. Finally, we define a framework of quasiorder-based automata constructions to offer a new perspective on residual automata.Comment: PhD thesi

Verification of real-time systems: improving tool support

We address a number of limitations of Timed Automata and real-time model-checkers, which undermine the reliability of formal verification. In particular, we focus on the model-checker Uppaal as a representative of this technology. Timelocks and Zeno runs represent anomalous behaviours in a timed automaton, and may invalidate the verification of safety and liveness properties. Currently, model-checkers do not offer adequate support to prevent or detect such behaviours. In response, we develop new methods to guarantee timelock-freedom and absence of Zeno runs, which improve and complement the existent support. We implement these methods in a tool to check Uppaal specifications. The requirements language of model-checkers is not well suited to express sequence and iteration of events, or past computations. As a result, validation problems may arise during verification (i.e., the property that we verify may not accurately reflect the intended requirement). We study the logic PITL, a rich propositional subset of Interval Temporal Logic, where these requirements can be more intuitively expressed than in model-checkers. However, PITL has a decision procedure with a worst-case non-elementary complexity, which has hampered the development of efficient tool support. To address this problem, we propose (and implement) a translation from PITL to the second-order logic WS1S, for which an efficient decision procedure is provided by the tool MONA. Thanks to the many optimisations included in MONA, we obtain an efficient decision procedure for PITL, despite its non-elementary complexity. Data variables in model-checkers are restricted to bounded domains, in order to obtain fully automatic verification. However, this may be too restrictive for certain kinds of specifications (e.g., when we need to reason about unbounded buffers). In response, we develop the theory of Discrete Timed Automata as an alternative formalism for real-time systems. In Discrete Timed Automata, WS1S is used as the assertion language, which enables MONA to assist invariance proofs. Furthermore, the semantics of urgency and synchronisation adopted in Discrete Timed Automata guarantee, by construction, that specifications are free from a large class of timelocks. Thus, we argue that well-timed specifications are easier to obtain in Discrete Timed Automata than in Timed Automata and most other notations for real-time systems

Verification of real-time systems : improving tool support

We address a number of limitations of Timed Automata and real-time model-checkers, which undermine the reliability of formal verification. In particular, we focus on the model-checker Uppaal as a representative of this technology. Timelocks and Zeno runs represent anomalous behaviours in a timed automaton, and may invalidate the verification of safety and liveness properties. Currently, model-checkers do not offer adequate support to prevent or detect such behaviours. In response, we develop new methods to guarantee timelock-freedom and absence of Zeno runs, which improve and complement the existent support. We implement these methods in a tool to check Uppaal specifications. The requirements language of model-checkers is not well suited to express sequence and iteration of events, or past computations. As a result, validation problems may arise during verification (i.e., the property that we verify may not accurately reflect the intended requirement). We study the logic PITL, a rich propositional subset of Interval Temporal Logic, where these requirements can be more intuitively expressed than in model-checkers. However, PITL has a decision procedure with a worst-case non-elementary complexity, which has hampered the development of efficient tool support. To address this problem, we propose (and implement) a translation from PITL to the second-order logic WS1S, for which an efficient decision procedure is provided by the tool MONA. Thanks to the many optimisations included in MONA, we obtain an efficient decision procedure for PITL, despite its non-elementary complexity. Data variables in model-checkers are restricted to bounded domains, in order to obtain fully automatic verification. However, this may be too restrictive for certain kinds of specifications (e.g., when we need to reason about unbounded buffers). In response, we develop the theory of Discrete Timed Automata as an alternative formalism for real-time systems. In Discrete Timed Automata, WS1S is used as the assertion language, which enables MONA to assist invariance proofs. Furthermore, the semantics of urgency and synchronisation adopted in Discrete Timed Automata guarantee, by construction, that specifications are free from a large class of timelocks. Thus, we argue that well-timed specifications are easier to obtain in Discrete Timed Automata than in Timed Automata and most other notations for real-time systems.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

Efficient Automata Techniques and Their Applications

Tato práce se zabývá vývojem efektivních technik pro konečné automaty a jejich aplikace. Zejména se věnujeme konečným automatům použitých pří detekci útoků v síťovém provozu a automatům v rozhodovacích procedurách a verifikaci. V první části práce navrhujeme techniky přibližné redukce nedeterministických automatů, které snižují spotřebu zdrojů v hardwarově akcelerovaném zkoumání obsahu paketů. Druhá část práce je je věnována automatům v rozhodovacích procedurách, zejména slabé monadické logice druhého řádů k následníků (WSkS) a teorie nad řetězci. Navrhujeme novou rozhodovací proceduru pro WS2S založenou na automatových termech, umožňující efektivně prořezávat stavový prostor. Dále studujeme techniky předzpracování WSkS formulí za účelem snížení velikosti konstruovaných automatů. Automaty jsme také aplikovali v rozhodovací proceduře teorie nad řetězci pro efektivní reprezentaci důkazového stromu. V poslední části práce potom navrhujeme optimalizace rank-based komplementace Buchiho automatů, které snižuje počet generovaných stavů během konstrukce komplementu.This thesis develops efficient techniques for finite automata and their applications. In particular, we focus on finite automata in network intrusion detection and automata in decision procedures and verification. In the first part of the thesis, we propose techniques of approximate reduction of nondeterministic automata decreasing consumption of resources of hardware-accelerated deep packet inspection. The second part is devoted to automata in decision procedures, in particular, to weak monadic second-order logic of k successors (WSkS) and the theory of strings. We propose a novel decision procedure for WS2S based on automata terms allowing one to effectively prune the state space. Further, we study techniques of WSkS formulae preprocessing intended to reduce the sizes of constructed intermediate automata. Moreover, we employ automata in a decision procedure of the theory of strings for efficient handling of the proof graph. The last part of the thesis then proposes optimizations in rank-based Buchi automata complementation reducing the number of generated states during the construction.