On Model-Checking Higher-Order Effectful Programs

Vertex-minors, monadic second-order logic, and a conjecture by Seese

Formulas of monadic second-order logic can be used to specify graph transductions, i.e., multivalued functions from graphs to graphs. We obtain in this way classes of graph transductions, called monadic second-order definable graph transductions (or more simply definable transductions) that are closed under composition and preserve the two known classes of context-free sets of graphs, namely the class of Hyperedge Replacement (HR) and the class of Vertex Replacement (VR) sets. These two classes can be characterized in terms of definable transductions and recognizable sets of finite trees. These characterizations are independent of the rewriting mechanisms used to define the HR and VR grammars. When restricted to words, the definable transductions are strictly more powerful than the rational transductions with finite image; they do not preserve context-free languages. We also describe the sets of discrete (edgeless) labeled graphs that are the images of HR and VR sets under definable transductions: this gives a version of Parikh's Theorem (i.e., the characterization of the commutative images of context-free languages) which extends the classical one and applies to HR and VR sets of graphs.

We establish that every equational graph can be characterized, up to isomorphism, by a formula of monadic second-order logic. It follows that the isomorphism of two equational graphs is decidable. Equational graphs can be used to describe the behaviour of recursive applicative program schemes. We obtain a sufficient and decidable condition for the equivalence of these program schemes.

It is known that a language is context-free iff it is the set of borders of the trees of recognizable set, where the border of a (labelled) tree is the word consisting of its leaf labels read from left to right.We give a generalization of this result in terms of planar graphs of bounded tree-width. Here the border of a planar graph is the word of edge labels of a path which borders a face for some planar embedding. We prove that a language is context-free iff it is the set of borders of the graphs of a set of (labelled) planar graphs of bounded tree-width which is definable by a formula of monadic second-order logic.

Programming with user-defined effects and effect handlers has many practical use cases involving imperative effects. Additionally, it is natural and powerful to use multi-shot effect handlers for non-deterministic or probabilistic programs that allow backtracking to compute a comprehensive outcome. Existing works for verifying effect handlers are restricted in one of three ways: i) permitting multi-shot continuations under pure setting; ii) allowing heap manipulation for only one-shot continuations; or iii) allowing multi-shot continuations with heap-manipulation but under a restricted frame rule. This work proposes a novel calculus called Effectful Specification Logic (ESL) to support unrestricted effect handlers, where zero-/one-/multi-shot continuations can co-exist with imperative effects and higher-order constructs. ESL captures behaviors in stages, and provides precise models to support invoked effects, handlers and continuations. To show its feasibility, we prototype an automated verification system for this novel specification logic, prove its soundness, report on useful case studies, and present experimental results. With this proposal, we have provided an extended specification logic that is capable of modeling arbitrary imperative higher-order programs with algebraic effects and continuation-enabled handlers.

We describe ongoing work on building an environment to support reasoning in proof assistants that represent formal systems using higher-order abstract syntax (HOAS). We use a simple and general specification language whose syntax supports HOAS. Using this language, we can encode the syntax and inference rules of a variety of formal systems, such as programming languages and logics. We describe our tool, implemented in OCaml, which parses this syntax, and translates it to a Coq library that includes definitions and hints for aiding automated proof in the Hybrid system. Hybrid itself is implemented in Coq, and designed specifically to reason about such formal systems. Given an input specification, the library that is automatically generated by our tool imports the general Hybrid library and adds definitions and hints for aiding automated proof in Hybrid about the specific programming language or logic defined in the specification. This work is part of a larger project to compare reasoning in systems supporting HOAS. Our current work focuses on Hybrid, Abella, Twelf, and Beluga, and the specification language is designed to be general enough to allow the automatic generation of libraries for all of these systems from a single specification.

In this article, we develop a new and somewhat unexpected connection between higher-order model-checking and linear logic. Our starting point is the observation that once embedded in the relational semantics of linear logic, the Church encoding of any higher-order recursion scheme (HORS) comes together with a dual Church encoding of an alternating tree automata (ATA) of the same signature. Moreover, the interaction between the relational interpretations of the HORS and of the ATA identifies the set of accepting states of the tree automaton against the infinite tree generated by the recursion scheme. We show how to extend this result to alternating parity automata (APT) by introducing a parametric version of the exponential modality of linear logic, capturing the formal properties of colors (or priorities) in higher-order model-checking. We show in particular how to reunderstand in this way the type-theoretic approach to higher-order model-checking developed by Kobayashi and Ong. We briefly explain in the end of the paper how this analysis driven by linear logic results in a new and purely semantic proof of decidability of the formulas of the monadic second-order logic for higher-order recursion schemes.

This article studies the logical properties of a very general class of infinite ranked trees, namely, those generated by higher-order recursion schemes. We consider, for both monadic second-order logic and modal -calculus, three main problems: model-checking, logical reflection (a.k.a. global model-checking, that asks for a finite description of the set of elements for which a formula holds), and selection (that asks, if exists, for some finite description of a set of elements for which an MSO formula with a second-order free variable holds). For each of these problems, we provide an effective solution. This is obtained, thanks to a known connection between higher-order recursion schemes and collapsible pushdown automata and on previous work regarding parity games played on transition graphs of collapsible pushdown automata.

Recently, speech assistants have brought convenience to our lives from many aspects. In the education field, speech assistants can also help teachers to reduce their burdens. However, there is no suitable solution to synthesize speeches for mathematical formulas although there have been lots of good techniques for text-to-speech (TTS) in the general domain. One possible solution is that we can convert mathematical formulas expressed in the LaTeX format to spelling texts and synthesize them into speech. In this paper, we investigated text generation methods that translate mathematical formulas in LaTex into Chinese spelling texts. For this purpose, we first constructed a parallel corpus of mathematical formulas and Chinese spelling texts, then compared the existing commonly used text generation methods, such as rule-based, Seq2Seq, Transformer and Graph2Seq, and finally proposed a novel model. As far as we know, this is the first study for Chinese spelling text generation of mathematical formulas. Experiment results on the annotated corpus show that our proposed model significantly outperforms the existing commonly used generation models.

In the last two decades, there has been much progress on model checking of both probabilistic systems and higher-order programs. In spite of the emergence of higher-order probabilistic programming languages, not much has been done to combine those two approaches. In this paper, we initiate a study on the probabilistic higher-order model checking problem, by giving some first theoretical and experimental results. As a first step towards our goal, we introduce PHORS, a probabilistic extension of higher-order recursion schemes (HORS), as a model of probabilistic higher-order programs. The model of PHORS may alternatively be viewed as a higher-order extension of recursive Markov chains. We then investigate the probabilistic termination problem -- or, equivalently, the probabilistic reachability problem. We prove that almost sure termination of order-2 PHORS is undecidable. We also provide a fixpoint characterization of the termination probability of PHORS, and develop a sound (but possibly incomplete) procedure for approximately computing the termination probability. We have implemented the procedure for order-2 PHORSs, and confirmed that the procedure works well through preliminary experiments that are reported at the end of the article.

M. Bezem defined an extensional semantics for positive higher-order logic programs. Recently, it was demonstrated by Rondogiannis and Symeonidou that Bezem's technique can be extended to higher-order logic programs with negation, retaining its extensional properties, provided that it is interpreted under a logic with an infinite number of truth values. Rondogiannis and Symeonidou also demonstrated that Bezem's technique, when extended under the stable model semantics, does not in general lead to extensional stable models. In this paper, we consider the problem of extending Bezem's technique under the well-founded semantics. We demonstrate that the well-founded extensionfailsto retain extensionality in the general case. On the positive side, we demonstrate that for stratified higher-order logic programs, extensionality is indeed achieved. We analyze the reasons of the failure of extensionality in the general case, arguing that a three-valued setting cannot distinguish between certain predicates that appear to have a different behaviour inside a program context, but which happen to be identical as three-valued relations.

Type-based model checking algorithms for higher-order recursion schemes have recently emerged as a promising approach to the verification of functional programs. We introduce pattern-matching recursion schemes (PMRS) as an accurate model of computation for functional programs that manipulate algebraic data-types. PMRS are a natural extension of higher-order recursion schemes that incorporate pattern-matching in the defining rules.This paper is concerned with the following (undecidable) verification problem: given a correctness property φ, a functional program ℘ (qua PMRS) and a regular input set ℑ, does every term that is reachable from ℑ under rewriting by ℘ satisfy φ? To solve the PMRS verification problem, we present a sound semi-algorithm which is based on model-checking and counterexample guided abstraction refinement. Given a no-instance of the verification problem, the method is guaranteed to terminate.From an order-n PMRS and an input set generated by a regular tree grammar, our method constructs an order-n weak PMRS which over-approximates only the first-order pattern-matching behaviour, whilst remaining completely faithful to the higher-order control flow. Using a variation of Kobayashi's type-based approach, we show that the (trivial automaton) model-checking problem for weak PMRS is decidable. When a violation of the property is detected in the abstraction which does not correspond to a violation in the model, the abstraction is automatically refined by `unfolding' the pattern-matching rules in the program to give successively more and more accurate weak PMRS models.

The four papers in this special issue apply tools and techniques from computer performance evaluation in the very different domain of modelling biological systems. This might seem to be a very odd thing to do but the practice of analysing biological systems in this way is becoming increasing common. As data about the internal components of biological systems is becoming more readily available, biologists are increasingly asking questions about how systems function. In addition to conducting laboratory experiments, they are supported in this exploration by in silico experimentation based on models. The view taken of the biological processes focusses on the stimuli and responses, a view akin to that taken of engineered systems in systems engineering. Thus this new endeavour in biology is known as Systems Biology . Performance analysts have a long tradition of modelling systems in order to understand and predict their function. Their focus is particularly on the dynamic aspects of the system, the use of, and contention for, resources, and the impact of uncertainty or randomness. These issues are important in the biological setting also, and so it is perhaps inevitable that we see some people and techniques from performance modelling being applied in systems biology. In particular some of the high-level modelling formalisms which have supported Markovian performance modelling in the last few decades (stochastic Petri nets, stochastic process algebras, etc.) are being applied in the biological domain. Furthermore analysis techniques, such as Markovian analysis, Monte Carlo simulation and probabilistic model checking have also been adoped. In this volume we have sought to give a snapshot of a variety of work which is going on at this interface between systems biology and more traditional quantitative analysis techniques. It is by no means an exhaustive account of this exciting area, but rather a taster which will hopefully whet your appetite to find out more. To open the volume, the editors provide a survey paper describing the motivations and goals of the systems biology endeavour, summarising the existing modelling techniques and outlining some instances of cross-over between performance modelling and systems biology. This includes an account of the use of ordinary differential equations (ODE) and stochastic simulation to analyse biological systems, and the adoption of high-level modelling formalisms such as Petri nets and process algebras to drive these ODE models and simulations. In their paper Kwiatkowska, Norman and Parker show the application of logic and probabilistic model checking to the analysis of biological signalling pathways. They use the PRISM probabilistic model-checker to check formulae of the CSL logic against CTMC-based models of the MAPK cascade, a sequence of biochemical reactions which sends a message within a cell. The paper provides an introduction to the CSL logic as well as the reactive modules language implemented by the PRISM model checker. Performance measures of interest are described using reward structures and the analysis achieved by PRISM is able to show how the percentage of activated MAPK, a key component of the pathway, and the number of MAPK-MAPKK reactions, vary as a function of time, for different values of the initial number of MAPKs. The paper by Jeschke, Ewald, Park, Fujimoto and Uhrmacher addresses the drive for increased physical accuracy in simulation models which represent the spatial aspects of cell biology. Standard approaches to stochastic simulation of cellular systems assume that the cell is a homogeneous soup of biochemical components. The truth is far removed from this, as the cell has a lot of internal structure which can have a profound effect on the dynamics of reactions. Setting aside the assumption that the reacting chemical species are well-stirred, spatial approaches divide the volume into sub-volumes and apply a structured method which identifies the next reaction to occur in each subvolume. The cost of such an increase in accuracy in the simulation model is a much increased running time so the authors use a parallel and distributed approach to improve performance. To close this special issue we have a paper by DemattB, Priami and Romanel which uses the BlenX language and the Beta Workbench software to analyse the MAPK pathway considered also by Kwiatkowska, Norman and Parker. The BlenX language, and the Beta-binders process calculus which was its inspiration, are examples of a new generation of languages which have been designed specifically for the biological domain, as an alternative to using existing languages designed for modelling computer systems. The paper shows how a well-designed platform for modelling and simulation can lift the user's experience and make their use of process calculi more valuable, delivering insights which would not have been seen otherwise.

The Theorem of Three Moments in a simple form, referring to bending moments at any three consecutive supports A, B, C, of a continuous beam uniform in each bay, AB = a, BG = c, takes the formw and being constant load intensities over AB and CB respectively.On p. 159, Vol. 42 (1938), of this journal, Capt. J. Morris discussed a modification in this case (and in the more general case when the beam is in thrust) when the load intensity varies according to a parabolic law.

A frailty model for interaction between multiple events