First-order definable string transformations

The connection between languages defined by computational models and logic for languages is well-studied. Monadic second-order logic and finite automata are shown to closely correspond to each-other for the languages of strings, trees, and partial-orders. Similar connections are shown for first-order logic and finite automata with certain aperiodicity restriction. Courcelle in 1994 proposed a way to use logic to define functions over structures where the output structure is defined using logical formulas interpreted over the input structure. Engelfriet and Hoogeboom discovered the corresponding "automata connection" by showing that two-way generalised sequential machines capture the class of monadic-second order definable transformations. Alur and Cerny further refined the result by proposing a one-way deterministic transducer model with string variables---called the streaming string transducers---to capture the same class of transformations. In this paper we establish a transducer-logic correspondence for Courcelle's first-order definable string transformations. We propose a new notion of transition monoid for streaming string transducers that involves structural properties of both underlying input automata and variable dependencies. By putting an aperiodicity restriction on the transition monoids, we define a class of streaming string transducers that captures exactly the class of first-order definable transformations.Comment: 31 page

Piecewise testable tree languages

This paper presents a decidable characterization of tree languages that can be defined by a boolean combination of Sigma_1 sentences. This is a tree extension of the Simon theorem, which says that a string language can be defined by a boolean combination of Sigma_1 sentences if and only if its syntactic monoid is J-trivial

Frex: dependently-typed algebraic simplification

We present an extensible, mathematically-structured algebraic simplification library design. We structure the library using universal algebraic concepts: a free algebra -- fral -- and a free extension -- frex -- of an algebra by a set of variables. The library's dependently-typed API guarantees simplification modules, even user-defined ones, are terminating, sound, and complete with respect to a well-specified class of equations. Completeness offers intangible benefits in practice -- our main contribution is the novel design. Cleanly separating between the interface and implementation of simplification modules provides two new modularity axes. First, simplification modules share thousands of lines of infrastructure code dealing with term-representation, pretty-printing, certification, and macros/reflection. Second, new simplification modules can reuse existing ones. We demonstrate this design by developing simplification modules for monoid varieties: ordinary, commutative, and involutive. We implemented this design in the new Idris2 dependently-typed programming language, and in Agda

Deciding regular grammar logics with converse through first-order logic

We provide a simple translation of the satisfiability problem for regular grammar logics with converse into GF2, which is the intersection of the guarded fragment and the 2-variable fragment of first-order logic. This translation is theoretically interesting because it translates modal logics with certain frame conditions into first-order logic, without explicitly expressing the frame conditions. A consequence of the translation is that the general satisfiability problem for regular grammar logics with converse is in EXPTIME. This extends a previous result of the first author for grammar logics without converse. Using the same method, we show how some other modal logics can be naturally translated into GF2, including nominal tense logics and intuitionistic logic. In our view, the results in this paper show that the natural first-order fragment corresponding to regular grammar logics is simply GF2 without extra machinery such as fixed point-operators.Comment: 34 page

Tree Languages Defined in First-Order Logic with One Quantifier Alternation

We study tree languages that can be defined in \Delta_2 . These are tree languages definable by a first-order formula whose quantifier prefix is forall exists, and simultaneously by a first-order formula whose quantifier prefix is . For the quantifier free part we consider two signatures, either the descendant relation alone or together with the lexicographical order relation on nodes. We provide an effective characterization of tree and forest languages definable in \Delta_2 . This characterization is in terms of algebraic equations. Over words, the class of word languages definable in \Delta_2 forms a robust class, which was given an effective algebraic characterization by Pin and Weil