Indexed linear logic and higher-order model checking

In recent work, Kobayashi observed that the acceptance by an alternating tree automaton A of an infinite tree T generated by a higher-order recursion scheme G may be formulated as the typability of the recursion scheme G in an appropriate intersection type system associated to the automaton A. The purpose of this article is to establish a clean connection between this line of work and Bucciarelli and Ehrhard's indexed linear logic. This is achieved in two steps. First, we recast Kobayashi's result in an equivalent infinitary intersection type system where intersection is not idempotent anymore. Then, we show that the resulting type system is a fragment of an infinitary version of Bucciarelli and Ehrhard's indexed linear logic. While this work is very preliminary and does not integrate key ingredients of higher-order model-checking like priorities, it reveals an interesting and promising connection between higher-order model-checking and linear logic.Comment: In Proceedings ITRS 2014, arXiv:1503.0437

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

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

A Type-Directed Negation Elimination

In the modal mu-calculus, a formula is well-formed if each recursive variable occurs underneath an even number of negations. By means of De Morgan's laws, it is easy to transform any well-formed formula into an equivalent formula without negations -- its negation normal form. Moreover, if the formula is of size n, its negation normal form of is of the same size O(n). The full modal mu-calculus and the negation normal form fragment are thus equally expressive and concise. In this paper we extend this result to the higher-order modal fixed point logic (HFL), an extension of the modal mu-calculus with higher-order recursive predicate transformers. We present a procedure that converts a formula into an equivalent formula without negations of quadratic size in the worst case and of linear size when the number of variables of the formula is fixed.Comment: In Proceedings FICS 2015, arXiv:1509.0282

LambdaY-Calculus With Priorities

International audienceThe lambdaY-calculus with priorities is a variant of the simply-typed lambda calculus designed for higher-order model-checking. The higher-order model-checking problem asks if a given parity tree automaton accepts the Böhm tree of a given term of the simply-typed lambda calculus with recursion. We show that this problem can be reduced to the same question but for terms of lambdaY-calculus with priorities and visibly parity automata; a subclass of parity automata. The latter question can be answered by evaluating terms in a simple powerset model with least and greatest fixpoints. We prove that the recognizing power of powerset models and visibly parity automata are the same. So, up to conversion to the lambdaY-calculus with priorities, powerset models with least and greatest fixpoints are indeed the right semantic framework for the model-checking problem. The reduction to lambdaY-calculus with priorities is also efficient algorithmically: it gives an algorithm of the same complexity as direct approaches to the higher-order model-checking problem. This indicates that the task of calculating the value of a term in a powerset model is a central algo-rithmic problem for higher-order model-checking

Cost Automata, Safe Schemes, and Downward Closures

Higher-order recursion schemes are an expressive formalism used to define languages of possibly infinite ranked trees. They extend regular and context-free grammars, and are equivalent to simply typed ?Y-calculus and collapsible pushdown automata. In this work we prove, under a syntactical constraint called safety, decidability of the model-checking problem for recursion schemes against properties defined by alternating B-automata, an extension of alternating parity automata for infinite trees with a boundedness acceptance condition. We then exploit this result to show how to compute downward closures of languages of finite trees recognized by safe recursion schemes

Collapsible Pushdown Automata and Recursion Schemes

International audienceWe consider recursion schemes (not assumed to be homogeneously typed, and hence not necessarily safe) and use them as generators of (possibly infinite) ranked trees. A recursion scheme is essentially a finite typed {deterministic term} rewriting system that generates, when one applies the rewriting rules ad infinitum, an infinite tree, called its value tree. A fundamental question is to provide an equivalent description of the trees generated by recursion schemes by a class of machines. In this paper we answer this open question by introducing collapsible pushdown automata (CPDA), which are an extension of deterministic (higher-order) pushdown automata. A CPDA generates a tree as follows. One considers its transition graph, unfolds it and contracts its silent transitions, which leads to an infinite tree which is finally node labelled thanks to a map from the set of control states of the CPDA to a ranked alphabet. Our contribution is to prove that these two models, higher-order recursion schemes and collapsible pushdown automata, are equi-expressive for generating infinite ranked trees. This is achieved by giving an effective transformations in both directions

Higher-Order Nonemptiness Step by Step

We show a new simple algorithm that checks whether a given higher-order grammar generates a nonempty language of trees. The algorithm amounts to a procedure that transforms a grammar of order n to a grammar of order n-1, preserving nonemptiness, and increasing the size only exponentially. After repeating the procedure n times, we obtain a grammar of order 0, whose nonemptiness can be easily checked. Since the size grows exponentially at each step, the overall complexity is n-EXPTIME, which is known to be optimal. More precisely, the transformation (and hence the whole algorithm) is linear in the size of the grammar, assuming that the arity of employed nonterminals is bounded by a constant. The same algorithm allows to check whether an infinite tree generated by a higher-order recursion scheme is accepted by an alternating safety (or reachability) automaton, because this question can be reduced to the nonemptiness problem by taking a product of the recursion scheme with the automaton. A proof of correctness of the algorithm is formalised in the proof assistant Coq. Our transformation is motivated by a similar transformation of Asada and Kobayashi (2020) changing a word grammar of order n to a tree grammar of order n-1. The step-by-step approach can be opposed to previous algorithms solving the nonemptiness problem "in one step", being compulsorily more complicated

A model for divergence insensitive properties of lambdaY-terms

A term of a simply typed λ-calculus with fixpoints can be considered as an abstraction of a higher-order functional program. The result of the computation of a term is its Böhm tree. Given a tree automaton describing a property of Böhm trees, we are interested in constructing a model recognizing the property, in a sense that the value of a term determines if its Böhm tree satisfies the property. We show how to construct models recognizing properties expressed by parity automata that cannot detect divergence. We call them Ω-blind parity automata, as the symbol Ω is used in Böhm trees to represent divergence; an automaton is Ω-blind when it has to accept Ω from every state. The models we construct resemble standard Scott models of latices of monotone functions, but application needs to be modified and the the fixpoint operator should be interpreted as a particular non-extremal fixpoint in a lattice