The Expressive Power, Satisfiability and Path Checking Problems of MTL and TPTL over Non-Monotonic Data Words

Recently, verification and analysis of data words have gained a lot of interest. Metric temporal logic (MTL) and timed propositional temporal logic (TPTL) are two extensions of Linear time temporal logic (LTL). In MTL, the temporal operator are indexed by a constraint interval. TPTL is a more powerful logic that is equipped with a freeze formalism. It uses register variables, which can be set to the current data value and later these register variables can be compared with the current data value. For monotonic data words, Alur and Henzinger proved that MTL and TPTL are equally expressive and the satisfiability problem is decidable. We study the expressive power, satisfiability problems and path checking problems for MLT and TPTL over all data words. We introduce Ehrenfeucht-Fraisse games for MTL and TPTL. Using the EF-game for MTL, we show that TPTL is strictly more expressive than MTL. Furthermore, we show that the MTL definability problem that whether a TPTL-formula is definable in MTL is not decidable. When restricting the number of register variables, we are able to show that TPTL with two register variables is strictly more expressive than TPTL with one register variable. For the satisfiability problem, we show that for MTL, the unary fragment of MTL and the pure fragment of MTL, SAT is not decidable. We prove the undecidability by reductions from the recurrent state problem and halting problem of two-counter machines. For the positive fragments of MTL and TPTL, we show that a positive formula is satisfiable if and only it is satisfied by a finite data word. Finitary SAT and infinitary SAT coincide for positive MTL and positive TPTL. Both of them are r.e.-complete. For existential TPTL and existential MTL, we show that SAT is NP-complete. We also investigate the complexity of path checking problems for TPTL and MTL over data words. These data words can be either finite or infinite periodic. For periodic words without data values, the complexity of LTL model checking belongs to the class AC^1(LogDCFL). For finite monotonic data words, the same complexity bound has been shown for MTL by Bundala and Ouaknine. We show that path checking for TPTL is PSPACE-complete, and for MTL is P-complete. If the number of register variables allowed is restricted, we obtain path checking for TPTL with only one register variable is P-complete over both infinite and finite data words; for TPTL with two register variables is PSPACE-complete over infinite data words. If the encoding of constraint numbers of the input TPTL-formula is in unary notation, we show that path checking for TPTL with a constant number of variables is P-complete over infinite unary encoded data words. Since the infinite data word produced by a deterministic one-counter machine is periodic, we can transfer all complexity results for the infinite periodic case to model checking over deterministic one-counter machines

LIPIcs, Volume 261, ICALP 2023, Complete Volume

LIPIcs, Volume 261, ICALP 2023, Complete Volum

Kant's cognitive architecture

Imagine a machine, equipped with sensors, receiving a stream of sensory information. It must, somehow, make sense of this stream of sensory data. But what, exactly, does this involve? We have an intuitive understanding of what is involved in “making sense” of sensory data – but can we specify precisely what is involved? Can this intuitive notion be formalized? In this thesis, we make three contributions. First, we provide a precise formalization of what it means to “make sense” of a sensory sequence. According to our definition, making sense means constructing a symbolic causal theory that explains the sensory sequence and satisfies a set of unity conditions that were inspired by Kant’s discussion in the first half of the Critique of Pure Reason. According to our interpretation, making sense of sensory input is a type of program synthesis, but it is unsupervised program synthesis. Our second contribution is a computer implementation, the Apperception Engine, that was designed to satisfy our requirements for making sense of a sensory sequence. Our system is able to produce interpretable human-readable causal theories from very small amounts of data, because of the strong inductive bias provided by the Kantian unity constraints. A causal theory produced by our system is able to predict future sensor readings, as well as retrodict earlier readings, and impute missing sensory readings. In fact, it is able to do all three tasks simultaneously. The engine is implemented in Answer Set Programming (ASP) and induces theories expressed in an extension of Datalog that includes causal rules and constraints. We test the engine in a diverse variety of domains, including cellular automata, rhythms and simple nursery tunes, multi-modal binding problems, occlusion tasks, and sequence induction IQ tests. In each domain, we test our engine’s ability to predict future sensor values, retrodict earlier sensor values, and impute missing sensory data. The Apperception Engine performs well in all these domains, significantly out-performing neural net baselines. These results are significant because neural nets typically struggle to solve the binding problem (where information from different modalities must somehow be combined together into different aspects of one unified object) and fail to solve occlusion tasks (in which objects are sometimes visible and sometimes obscured from view). We note in particular that in the sequence induction IQ tasks, our system achieves human-level performance. This is notable because the Apperception Engine was not designed to solve these IQ tasks; it is not a bespoke hand-engineered solution to this particular domain. – Rather, it is a general purpose system that attempts to make sense of any sensory sequence, that just happens to be able to solve these IQ tasks “out of the box”. Our third contribution is a major extension of the engine to handle noisy and ambiguous data. While the initial implementation assumes the sensory input has already been preprocessed into ground atoms of first-order logic, our extension makes sense of raw unprocessed input – a sequence of pixel images from a video camera, for example. The resulting system is a neuro-symbolic framework for distilling interpretable theories out of streams of raw, unprocessed sensory experience.Open Acces

Programming Languages and Systems

This open access book constitutes the proceedings of the 29th European Symposium on Programming, ESOP 2020, which was planned to take place in Dublin, Ireland, in April 2020, as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2020. The actual ETAPS 2020 meeting was postponed due to the Corona pandemic. The papers deal with fundamental issues in the specification, design, analysis, and implementation of programming languages and systems

Decidability of Downward XPath

International audienceWe investigate the satisfiability problem for downward-XPath, the fragment of XPath that includes the child and descendant axes, and tests for (in)equality of attributes' values. We prove that this problem is decidable, ExpTime-complete. These bounds also hold when path expressions allow closure under the Kleene star operator. To obtain these results, we introduce a Downward Data automata model (DD automata) over trees with data, which has a decidable emptiness problem. Satisfiability of downward-XPath can be reduced to the emptiness problem of DD automata and hence its decidability follows. Although downward-XPath does not include any horizontal axis, DD automata are more expressive and can perform some horizontal tests. Thus, we show that the satisfiability remains in ExpTime even in the presence of the regular constraints expressible by DD automata. However, the same problem in the presence of any regular constraint is known to have a non-primitive recursive complexity. Finally, we give the exact complexity of the satisfiability problem for several fragments of downward-XPath

LIPIcs, Volume 244, ESA 2022, Complete Volume

LIPIcs, Volume 244, ESA 2022, Complete Volum

28th Annual Symposium on Combinatorial Pattern Matching : CPM 2017, July 4-6, 2017, Warsaw, Poland

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