Relational semantics of linear logic and higher-order model-checking

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 his 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.Comment: 24 pages. Submitte

On Model-Checking Higher-Order Effectful Programs (Long Version)

Model-checking is one of the most powerful techniques for verifying systems and programs, which since the pioneering results by Knapik et al., Ong, and Kobayashi, is known to be applicable to functional programs with higher-order types against properties expressed by formulas of monadic second-order logic. What happens when the program in question, in addition to higher-order functions, also exhibits algebraic effects such as probabilistic choice or global store? The results in the literature range from those, mostly positive, about nondeterministic effects, to those about probabilistic effects, in the presence of which even mere reachability becomes undecidable. This work takes a fresh and general look at the problem, first of all showing that there is an elegant and natural way of viewing higher-order programs producing algebraic effects as ordinary higher-order recursion schemes. We then move on to consider effect handlers, showing that in their presence the model checking problem is bound to be undecidable in the general case, while it stays decidable when handlers have a simple syntactic form, still sufficient to capture so-called generic effects. Along the way we hint at how a general specification language could look like, this way justifying some of the results in the literature, and deriving new ones

Recursion Schemes and the WMSO+U Logic

We study the weak MSO logic extended by the unbounding quantifier (WMSO+U), expressing the fact that there exist arbitrarily large finite sets satisfying a given property. We prove that it is decidable whether the tree generated by a given higher-order recursion scheme satisfies a given sentence of WMSO+U

Streett Automata Model Checking of Higher-Order Recursion Schemes

We propose a practical algorithm for Streett automata model checking of higher-order recursion schemes (HORS), which checks whether the tree generated by a given HORS is accepted by a given Streett automaton. The Streett automata model checking of HORS is useful in the context of liveness verification of higher-order functional programs. The previous approach to Streett automata model checking converted Streett automata to parity automata and then invoked a parity tree automata model checker. We show through experiments that our direct approach outperforms the previous approach. Besides being able to directly deal with Streett automata, our algorithm is the first practical Streett or parity automata model checking algorithm that runs in time polynomial in the size of HORS, assuming that the other parameters are fixed. Previous practical fixed-parameter polynomial time algorithms for HORS could only deal with the class of trivial tree automata. We have confirmed through experiments that (a parity automata version of) our model checker outperforms previous parity automata model checkers for HORS

Lambda-calculus and formal language theory

Formal and symbolic approaches have offered computer science many application fields. The rich and fruitful connection between logic, automata and algebra is one such approach. It has been used to model natural languages as well as in program verification. In the mathematics of language it is able to model phenomena ranging from syntax to phonology while in verification it gives model checking algorithms to a wide family of programs. This thesis extends this approach to simply typed lambda-calculus by providing a natural extension of recognizability to programs that are representable by simply typed terms. This notion is then applied to both the mathematics of language and program verification. In the case of the mathematics of language, it is used to generalize parsing algorithms and to propose high-level methods to describe languages. Concerning program verification, it is used to describe methods for verifying the behavioral properties of higher-order programs. In both cases, the link that is drawn between finite state methods and denotational semantics provide the means to mix powerful tools coming from the two worlds

Relational Semantics of Linear Logic and Higher-order Model Checking

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

Bindings as bounded natural functors

We present a general framework for specifying and reasoning about syntax with bindings. Abstract binder types are modeled using a universe of functors on sets, subject to a number of operations that can be used to construct complex binding patterns and binding-aware datatypes, including non-well-founded and infinitely branching types, in a modular fashion. Despite not committing to any syntactic format, the framework is “concrete” enough to provide definitions of the fundamental operators on terms (free variables, alpha-equivalence, and capture-avoiding substitution) and reasoning and definition principles. This work is compatible with classical higher-order logic and has been formalized in the proof assistant Isabelle/HOL

SVS-JOIN : efficient spatial visual similarity join for geo-multimedia

In the big data era, massive amount of multimedia data with geo-tags has been generated and collected by smart devices equipped with mobile communications module and position sensor module. This trend has put forward higher request on large-scale geo-multimedia retrieval. Spatial similarity join is one of the significant problems in the area of spatial database. Previous works focused on spatial textual document search problem, rather than geo-multimedia retrieval. In this paper, we investigate a novel geo-multimedia retrieval paradigm named spatial visual similarity join (SVS-JOIN for short), which aims to search similar geo-image pairs in both aspects of geo-location and visual content. Firstly, the definition of SVS-JOIN is proposed and then we present the geographical similarity and visual similarity measurement. Inspired by the approach for textual similarity join, we develop an algorithm named SVS-JOIN B by combining the PPJOIN algorithm and visual similarity. Besides, an extension of it named SVS-JOIN G is developed, which utilizes spatial grid strategy to improve the search efficiency. To further speed up the search, a novel approach called SVS-JOIN Q is carefully designed, in which a quadtree and a global inverted index are employed. Comprehensive experiments are conducted on two geo-image datasets and the results demonstrate that our solution can address the SVS-JOIN problem effectively and efficiently