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Verifying quantum systems has attracted a lot of interest in the last decades.In this paper, we study the quantitative model-checking of quantum continuous-time Markov chains (quantum CTMCs). The branching-time properties of quantum CTMCs are specified by continuous stochastic logic (CSL), which is well-known for verifying real-time systems, including classical CTMCs. The core of checking the CSL formulas lies in tackling multiphase until formulas. We develop an algebraic method using proper projection, matrix exponentiation, and definite integration to symbolically calculate the probability measures of path formulas. Thus the decidability of CSL is established. To be efficient, numerical methods are incorporated to guarantee that the time complexity is polynomial in the encoding size of the input model and linear in the size of the input formula. A running example of Apollonian networks is further provided to demonstrate our method.

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In relational verification, judicious alignment of computational steps facilitates proof of relations between programs using simple relational assertions. Relational Hoare logics (RHL) provide compositional rules that embody various alignments of executions. Seemingly more flexible alignments can be expressed in terms of product automata based on program transition relations. A single degenerate alignment rule (sequential composition), atop a complete Hoare logic, comprises a RHL for $\forall\forall$ properties that is complete in the sense of Cook. The notion of alignment completeness was previously proposed as an additional measure, and some rules were shown to be alignment complete with respect to a few ad hoc forms of alignment automata. This paper proves alignment completeness with respect to a general class of $\forall\forall$ alignment automata, for a RHL comprised of standard rules together with a rule of semantics-preserving rewrites based on Kleene algebra with tests. A new logic for $\forall\exists$ properties is introduced and shown to be sound and alignment complete for a new general class of automata. The $\forall\forall$ and $\forall\exists$ automata are shown to be semantically complete. Thus both logics are complete in the sense of Cook. The paper includes discussion of why alignment is not the only important principle for relational reasoning and proposes entailment completeness as further means to evaluate RHLs.

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In this paper, we explore the descriptive complexity theory of finite groups by examining the power of the second Ehrenfeucht--Fraïssé bijective pebble game in Hella's (Ann. Pure Appl. Log., 1989) hierarchy. This is a Spoiler--Duplicator game in which Spoiler can place up to two pebbles each round. While it trivially solves graph isomorphism, it may be nontrivial for finite groups, and other ternary relational structures. We first provide a novel generalization of Weisfeiler--Leman (WL) coloring, which we call 2-ary WL. We then show that 2-ary WL is equivalent to the second Ehrenfeucht--Fraïssé bijective pebble game in Hella's hierarchy. Our main result is that, in the pebble game characterization, only $O(1)$ pebbles and $O(1)$ rounds are sufficient to identify all groups without Abelian normal subgroups (a class of groups for which isomorphism testing is known to be in $\mathsf{P}$; Babai, Codenotti, & Qiao, ICALP 2012). We actually show that $7$ pebbles and $7$ rounds suffice. In particular, we show that within the first few rounds, Spoiler can force Duplicator to select an isomorphism between two such groups at each subsequent round. By Hella's results (ibid.), this is equivalent to saying that these groups are identified by formulas in first-order logic with generalized 2-ary quantifiers, using only $7$ variables and $7$ quantifier depth.

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The class of type-two basic feasible functionals ($\mathtt{BFF}_2$) is the analogue of $\mathtt{FP}$ (polynomial time functions) for type-2 functionals, that is, functionals that can take (first-order) functions as arguments. $\mathtt{BFF}_2$ can be defined through Oracle Turing machines with running time bounded by second-order polynomials. On the other hand, higher-order term rewriting provides an elegant formalism for expressing higher-order computation. We address the problem of characterizing $\mathtt{BFF}_2$ by higher-order term rewriting. Various kinds of interpretations for first-order term rewriting have been introduced in the literature for proving termination and characterizing first-order complexity classes. In this paper, we consider a recently introduced notion of cost-size interpretations for higher-order term rewriting and see second order rewriting as ways of computing type-2 functionals. We then prove that the class of functionals represented by higher-order terms admitting polynomially bounded cost-size interpretations exactly corresponds to $\mathtt{BFF}_2$.

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We study the computational expressivity of proof systems with fixed point operators, within the 'proofs-as-programs' paradigm. We start with a calculus muLJ (due to Clairambault) that extends intuitionistic logic by least and greatest positive fixed points. Based in the sequent calculus, muLJ admits a standard extension to a 'circular' calculus CmuLJ. Our main result is that, perhaps surprisingly, both muLJ and CmuLJ represent the same first-order functions: those provably total in $Π^1_2$-$\mathsf{CA}_0$, a subsystem of second-order arithmetic beyond the 'big five' of reverse mathematics and one of the strongest theories for which we have an ordinal analysis (due to Rathjen). This solves various questions in the literature on the computational strength of (circular) proof systems with fixed points. For the lower bound we give a realisability interpretation from an extension of Peano Arithmetic by fixed points that has been shown to be arithmetically equivalent to $Π^1_2$-$\mathsf{CA}_0$ (due to Möllerfeld). For the upper bound we construct a novel computability model in order to give a totality argument for circular proofs with fixed points. In fact we formalise this argument itself within $Π^1_2$-$\mathsf{CA}_0$ in order to obtain the tight bounds we are after. Along the way we develop some novel reverse mathematics for the Knaster-Tarski fixed point theorem.

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