Weak MSO+U with Path Quantifiers over Infinite Trees

This paper shows that over infinite trees, satisfiability is decidable for weak monadic second-order logic extended by the unbounding quantifier U and quantification over infinite paths. The proof is by reduction to emptiness for a certain automaton model, while emptiness for the automaton model is decided using profinite trees.Comment: version of an ICALP 2014 paper with appendice

Parameterized Linear Temporal Logics Meet Costs: Still not Costlier than LTL

We continue the investigation of parameterized extensions of Linear Temporal Logic (LTL) that retain the attractive algorithmic properties of LTL: a polynomial space model checking algorithm and a doubly-exponential time algorithm for solving games. Alur et al. and Kupferman et al. showed that this is the case for Parametric LTL (PLTL) and PROMPT-LTL respectively, which have temporal operators equipped with variables that bound their scope in time. Later, this was also shown to be true for Parametric LDL (PLDL), which extends PLTL to be able to express all omega-regular properties. Here, we generalize PLTL to systems with costs, i.e., we do not bound the scope of operators in time, but bound the scope in terms of the cost accumulated during time. Again, we show that model checking and solving games for specifications in PLTL with costs is not harder than the corresponding problems for LTL. Finally, we discuss PLDL with costs and extensions to multiple cost functions.Comment: In Proceedings GandALF 2015, arXiv:1509.0685

Delay Games with WMSO+U Winning Conditions

Delay games are two-player games of infinite duration in which one player may delay her moves to obtain a lookahead on her opponent's moves. We consider delay games with winning conditions expressed in weak monadic second order logic with the unbounding quantifier, which is able to express (un)boundedness properties. We show that it is decidable whether the delaying player has a winning strategy using bounded lookahead and give a doubly-exponential upper bound on the necessary lookahead. In contrast, we show that bounded lookahead is not always sufficient to win such a game.Comment: A short version appears in the proceedings of CSR 2015. The definition of the equivalence relation introduced in Section 3 is updated: the previous one was inadequate, which invalidates the proof of Lemma 2. The correction presented here suffices to prove Lemma 2 and does not affect our main theorem. arXiv admin note: text overlap with arXiv:1412.370

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

Church Synthesis on Register Automata over Linearly Ordered Data Domains

Register automata are finite automata equipped with a finite set of registers in which they can store data, i.e. elements from an unbounded or infinite alphabet. They provide a simple formalism to specify the behaviour of reactive systems operating over data ?-words. We study the synthesis problem for specifications given as register automata over a linearly ordered data domain (e.g. (?, ?) or (?, ?)), which allow for comparison of data with regards to the linear order. To that end, we extend the classical Church synthesis game to infinite alphabets: two players, Adam and Eve, alternately play some data, and Eve wins whenever their interaction complies with the specification, which is a language of ?-words over ordered data. Such games are however undecidable, even when the specification is recognised by a deterministic register automaton. This is in contrast with the equality case, where the problem is only undecidable for nondeterministic and universal specifications. Thus, we study one-sided Church games, where Eve instead operates over a finite alphabet, while Adam still manipulates data. We show they are determined, and deciding the existence of a winning strategy is in ExpTime, both for ? and ?. This follows from a study of constraint sequences, which abstract the behaviour of register automata, and allow us to reduce Church games to ?-regular games. Lastly, we apply these results to the transducer synthesis problem for input-driven register automata, where each output data is restricted to be the content of some register, and show that if there exists an implementation, then there exists one which is a register transducer

Profinite trees, through monads and the lambda-calculus

In its simplest form, the theory of regular languages is the study of sets of finite words recognized by finite monoids. The finiteness condition on monoids gives rise to a topological space whose points, called profinite words, encode the limiting behavior of words with respect to finite monoids. Yet, some aspects of the theory of regular languages are not particular to monoids and can be described in a general setting. On the one hand, Boja\'{n}czyk has shown how to use monads to generalize the theory of regular languages and has given an abstract definition of the free profinite structure, defined by codensity, given a fixed monad and a notion of finite structure. On the other hand, Salvati has introduced the notion of language of $\lambda$-terms, using denotational semantics, which generalizes the case of words and trees through the Church encoding. In recent work, the author and collaborators defined the notion of profinite $\lambda$-term using semantics in finite sets and functions, which extend the Church encoding to profinite words. In this article, we prove that these two generalizations, based on monads and denotational semantics, coincide in the case of trees. To do so, we consider the monad of abstract clones which, when applied to a ranked alphabet, gives the associated clone of ranked trees. This induces a notion of free profinite clone, and hence of profinite trees. The main contribution is a categorical proof that the free profinite clone on a ranked alphabet is isomorphic, as a Stone-enriched clone, to the clone of profinite $\lambda$-terms of Church type. Moreover, we also prove a parametricity theorem on families of semantic elements which provides another equivalent formulation of profinite trees in terms of Reynolds parametricity

Extending the WMSO+U Logic With Quantification Over Tuples

We study a new extension of the weak MSO logic, talking about boundedness. Instead of a previously considered quantifier U, expressing the fact that there exist arbitrarily large finite sets satisfying a given property, we consider a generalized quantifier U, expressing the fact that there exist tuples of arbitrarily large finite sets satisfying a given property. First, we prove that the new logic WMSO+U_tup is strictly more expressive than WMSO+U. In particular, WMSO+U_tup is able to express the so-called simultaneous unboundedness property, for which we prove that it is not expressible in WMSO+U. Second, we prove that it is decidable whether the tree generated by a given higher-order recursion scheme satisfies a given sentence of WMSO+K_tup.Comment: This is an extended version of a paper published at the CSL 2024 conferenc

Two-Player Boundedness Counter Games

We consider two-player zero-sum games with winning objectives beyond regular languages, expressed as a parity condition in conjunction with a Boolean combination of boundedness conditions on a finite set of counters which can be incremented, reset to 0, but not tested. A boundedness condition requires that a given counter is bounded along the play. Such games are decidable, though with non-optimal complexity, by an encoding into the logic WMSO with the unbounded and path quantifiers, which is known to be decidable over infinite trees. Our objective is to give tight or tighter complexity results for particular classes of counter games with boundedness conditions, and study their strategy complexity. In particular, counter games with conjunction of boundedness conditions are easily seen to be equivalent to Streett games, so, they are CoNP-c. Moreover, finite-memory strategies suffice for Eve and memoryless strategies suffice for Adam. For counter games with a disjunction of boundedness conditions, we prove that they are in solvable in NP?CoNP, and in PTime if the parity condition is fixed. In that case memoryless strategies suffice for Eve while infinite memory strategies might be necessary for Adam. Finally, we consider an extension of those games with a max operation. In that case, the complexity increases: for conjunctions of boundedness conditions, counter games are EXPTIME-c