Quantified CTL: Expressiveness and Complexity

While it was defined long ago, the extension of CTL with quantification over atomic propositions has never been studied extensively. Considering two different semantics (depending whether propositional quantification refers to the Kripke structure or to its unwinding tree), we study its expressiveness (showing in particular that QCTL coincides with Monadic Second-Order Logic for both semantics) and characterise the complexity of its model-checking and satisfiability problems, depending on the number of nested propositional quantifiers (showing that the structure semantics populates the polynomial hierarchy while the tree semantics populates the exponential hierarchy)

On the Expressiveness of QCTL

QCTL extends the temporal logic CTL with quantification over atomic propositions. While the algorithmic questions for QCTL and its fragments with limited quantification depth are well-understood (e.g. satisfiability of QkCTL, with at most k nested blocks of quantifiers, is (k+1)-EXPTIME-complete), very few results are known about the expressiveness of this logic. We address such expressiveness questions in this paper. We first consider the distinguishing power of these logics (i.e., their ability to separate models), their relationship with behavioural equivalences, and their ability to capture the behaviours of finite Kripke structures with so-called characteristic formulas. We then consider their expressive power (i.e., their ability to express a property), showing that in terms of expressiveness the hierarchy QkCTL collapses at level 2 (in other terms, any QCTL formula can be expressed using at most two nested blocks of quantifiers)

From Quantified CTL to QBF

QCTL extends the temporal logic CTL with quantifications over atomic propositions. This extension is known to be very expressive: QCTL allows us to express complex properties over Kripke structures (it is as expressive as MSO). Several semantics exist for the quantifications: here, we work with the structure semantics, where the extra propositions label the Kripke structure (and not its execution tree), and the model-checking problem is known to be PSPACE-complete in this framework. We propose a model-checking algorithm for QCTL based on a reduction to QBF. We consider several reduction strategies, and we compare them with a prototype (based on the SMT-solver Z3) on several examples

Automata for the mu-calculus and Related Results

The propositional mu-calculus as introduced by Kozen in [4] isconsidered. The notion of disjunctive formula is defined and it is shownthat every formula is semantically equivalent to a disjunctive formula.For these formulas many difficulties encountered in the general case maybe avoided. For instance, satisfiability checking is linear for disjunctiveformulas. This kind of formula gives rise to a new notion of finite automatonwhich characterizes the expressive power of the mu-calculus overall transition systems

On using compressibility to detect when slime mould completed computation

© 2016 Wiley Periodicals, Inc. Slime mould Physarum polycephalum is a single cell visible by an unaided eye. The slime mould optimizes its network of protoplasmic tubes in gradients of attractants and repellents. This behavior is interpreted as computation. Several prototypes of the slime mould computers were designed to solve problems of computation geometry, graphs, transport networks, and to implement universal computing circuits. Being a living substrate, the slime mould does not halt its behavior when a task is solved but often continues foraging the space thus masking the solution found. We propose to use temporal changes in compressibility of the slime mould patterns as indicators of the halting of the computation. Compressibility of a pattern characterizes the pattern's morphological diversity, that is, a number of different local configurations. At the beginning of computation the slime explores the space, thus generating less compressible patterns. After gradients of attractants and repellents are detected the slime spans data sites with its protoplasmic network and retracts scouting branches, thus generating more compressible patterns. We analyze the feasibility of the approach on results of laboratory experiments and computer modelling

Syntactic complexity in the modal μ calculus

This thesis studies how to eliminate syntactic complexity in Lμ, the modal μ calculus. Lμ is a verification logic in which a least fixpoint operator μ, and its dual v, add recursion to a simple modal logic. The number of alternations between μ and v is a measure of complexity called the formula’s index: the lower the index, the easier a formula is to model-check. The central question of this thesis is a long standing one, the Lμ index problem: given a formula, what is the least index of any equivalent formula, that is to say, its semantic index? I take a syntactic approach, focused on simplifying formulas. The core decidability results are (i) alternative, syntax-focused decidability proofs for ML and Pμ 1 , the low complexity classes of μ; and (ii) a proof that Ʃμ 2 , the fragment of Lμ with one alternation, is decidable for formulas in the dual class Pμ 2 . Beyond its algorithmic contributions, this thesis aims to deepen our understanding of the index problem and the tools at our disposal. I study disjunctive form and related syntactic restrictions, and how they affect the index problem. The main technical results are that the transformation into disjunctive form preserves Pμ 2 -indices but not μ 2 -indices, and that some properties of binary trees are expressible with a lower index using disjunctive formulas than non-deterministic automata. The latter is part of a thorough account of how the Lμ index problem and the Rabin–Mostowski index problem for parity automata are related. In the final part of the thesis, I revisit the relationship between the index problem and parity games. The syntactic index of a formula is an upper bound on the descriptive complexity of its model-checking parity games. I show that the semantic index of a formula Ψ is bounded above by the descriptive complexity of the model-checking games for Ψ. I then study whether this bound is strict: if a formula Ψ is equivalent to a formula in an alternation class C, does a formula of C suffice to describe the winning regions of the model-checking games of Ψ? I prove that this is the case for ML, Pμ 1 , Ʃμ 2 , and the disjunctive fragment of any alternation class. I discuss the practical implications of these results and propose a uniform approach to the index problem, which subsumes the previously described decision procedures for low alternation classes. In brief, this thesis can be read as a guide on how to approach a seemingly complex Lμ formula. Along the way it studies what makes this such a difficult problem and proposes novel approaches to both simplifying individual formulas and deciding further fragments of the alternation hierarchy