Combining algebra and higher-order types

We present Zenon, an automated theorem prover for first order classical logic (with equality), based on the tableau method. Zenon is intended to be the dedicated prover of the Focal environment, an object-oriented algebraic specification and proof system, which is able to produce OCaml code for execution and Coq code for certification. Zenon can directly generate Coq proofs (proof scripts or proof terms), which can be reinserted in the Coq specifications produced by Focal. Zenon can also be extended, which makes specific (and possibly local) automation possible in Focal.

The static dependency pair method is a method for proving the termination of higher-order rewrite systems a la Nipkow. It combines the dependency pair method introduced for first-order rewrite systems with the notion of strong computability introduced for typed lambda-calculi. Argument filterings and usable rules are two important methods of the dependency pair framework used by current state-of-the-art first-order automated termination provers. In this paper, we extend the class of higher-order systems on which the static dependency pair method can be applied. Then, we extend argument filterings and usable rules to higher-order rewriting, hence providing the basis for a powerful automated termination prover for higher-order rewrite systems.

Polymorphic Rewriting Conserves Algebraic Confluence

Higher-order rewrite systems are an extension of term-rewriting to (simply typed) A-terms. This yields a formalism with a built-in notion of variable binding (λ-abstraction) and substitution (β-reduction) which is capable of describing the manipulation of terms with bound variables. Typical examples are various λ-calculi themselves, logical formulae, proof terms and programs.In this talk we survey the growing literature in this area with an emphasis on confluence results. In particular we examine some recent results by van Oostrom on very general confluence criteria which go beyond orthogonality and even generalize a result by Huet on parallel reduction between critical pairs. We also present a new critical pair condition for higher-order rewrite systems with arbitrary left-hand sides.

We relate different approaches for proving the unsatisfiability of a system of real polynomial equations over Boolean variables. On the one hand, there are the static proof systems Sherali-Adams and sum-of-squares (a.k.a. Lasserre), which are based on linear and semi-definite programming relaxations. On the other hand, we consider polynomial calculus, which is a dynamic algebraic proof system that models Gröbner basis computations. Our first result is that sum-of-squares simulates polynomial calculus: any polynomial calculus refutation of degree d can be transformed into a sum-of-squares refutation of degree 2d and only polynomial increase in size. In contrast, our second result shows that this is not the case for Sherali-Adams: there are systems of polynomial equations that have polynomial calculus refutations of degree 3 and polynomial size, but require Sherali-Adams refutations of large degree and exponential size. A corollary of our first result is that the proof systems Positivstellensatz and Positivstellensatz Calculus, which have been separated over non-Boolean polynomials, simulate each other in the presence of Boolean axioms.

It is known that the set of all redexes for a left-linear term rewriting system is recognizable by a tree automaton, which means that we can construct a tree automaton that accepts redexes. The present paper extends this result to Nipkow's higher-order rewrite systems, in which every left-hand side is a linear fully-extended pattern. A naive extension of the first-order method causes the automata to have infinitely many states in order to distinguish bound variables in λ-terms, even if they are closed. To avoid this problem, it is natural to adopt de Bruijn notation, in which bound variables are represented as natural numbers (possibly finite symbols, such as 0, s(0), and s(s(0))). We propose a variant of de Bruijn notation in which only bound variables are represented as natural numbers because it is not necessary to represent free variables as natural numbers.

We study whether lower bounds against constant-depth algebraic circuits computing the Permanent over finite fields (Limaye-Srinivasan-Tavenas, J. ACM 2025; Forbes, CCC 2024) are hard to prove in certain proof systems. We focus on a DNF formula that expresses that such lower bounds are hard for constant-depth algebraic proofs. Using an adaptation of the diagonalization framework of Santhanam and Tzameret (SIAM J. Comput. 2025), we show unconditionally that this family of DNF formulas does not admit polynomial-size propositional AC0[p]-Frege proofs infinitely often. This rules out the possibility that the DNF family is easy, and establishes that its status is either that of a hard tautology for AC0[p]-Frege or else unprovable (not a tautology). While it remains open whether the DNFs in question are tautologies, we provide evidence in this direction. In particular, under the plausible assumption that certain weak properties of multilinear algebra, specifically those involving tensor rank, do not admit short constant-depth algebraic proofs, the DNFs are tautologies. We also observe that several weaker variants of the DNF formula are provably tautologies, and we show that the question of whether the DNFs are tautologies connects to conjectures of Razborov (ICALP 1996) and Krajicek (J. Symb. Log. 2004). Our result has two additional features. (i) Existential depth amplification: the DNF formula is parameterised by a constant depth d bounding the depth of the algebraic proofs. We show that there exists some fixed depth d such that if there are no small depth-d algebraic proofs of certain circuit lower bounds for the Permanent, then there are no such small algebraic proofs in any constant depth. (ii) Necessity: we show that our result is a necessary step towards establishing lower bounds against constant-depth algebraic proofs, and more generally against any sufficiently strong proof system.

We define a transformation from a left-linear constructor-based conditional rewrite system into an overlapping inductively sequential rewrite system. This transformation is sound and complete for the computations in the source system. Since there exists a sound and complete narrowing strategy for the target system, the combination of these results offers the first procedure for provably sound and complete narrowing computations for the whole class of the leftlinear constructor-based conditional rewrite systems. We address the differences between demand driven and lazy strategies and between narrowing strategies and narrowing calculi. In this context, we analyze the efficiency and practicality of using our transformation for the implementation of functional logic programming languages. The results of this paper complement, extend, and occasionally rectify, previously published results in this area.

In recent years there has been an increasing interest in studying proof systems stronger than Resolution, with the aim of building more efficient SAT solvers based on them. In defining these proof systems, we try to find a balance between the power of the proof system (the size of the proofs required to refute a formula) and the difficulty of finding the proofs. In this paper we consider the proof systems circular Resolution, Sherali-Adams, Nullstellensatz and Weighted Resolution and we study their relative power from a theoretical perspective. We prove that circular Resolution, Sherali-Adams and Weighted Resolution are polynomially equivalent proof systems. We also prove that Nullstellensatz is polynomially equivalent to a restricted version of Weighted Resolution. The equivalences carry on also for versions of the systems where the coefficients/weights are expressed in unary. The practical interest in these systems comes from the fact that they admit efficient algorithms to find proofs in case these have small width/degree.

The Trends in Functional Programming Symposia are an annual event dedicated to promoting new research directions in functional programming and to investigating the relationship between functional programming and other branches of Computer Science. The TFP series started in 1999 and the 2008 meeting marked the first decade of the symposium. In this paper we identify and discuss the trends presented at TFP using a macro-vision lens that looks at topics in the first two volumes of TFP and the two most recent volumes of TFP. The goal is to calibrate how trends have changed over a decade instead of analyzing trends with a micro-vision lens that focuses on changes from year to year. The major trends identified are work on parallel functional programming, types, formal verification, implementation of functional languages, and application development.

We study confluence and the Church-Rosser property in term rewriting and λ-calculus with explicit bounds on term sizes and reduction lengths. Given a system R , we are interested in the lengths of the reductions in the smallest valleys t → * s ′ * ← t ′ expressed as a function: —for confluence a function vs R ( m , n ) where the valleys are for peaks t ← s → * t ′ with s of size at most m and the reductions of maximum length n , and —for the Church-Rosser property a function cvs R ( m , n ) where the valleys are for conversions t ↔ * t ′ with t and t ′ of size at most m and the conversion of maximum length n . For confluent Term Rewriting Systems (TRSs), we prove that vs R is a total computable function, and for linear such systems that cvs R is a total computable function. Conversely, we show that every total computable function is the lower bound on the functions vs R ( m , n ) and cvs R ( m , n ) for some TRS R : In particular, we show that for every total computable function φ: N → N there is a TRS R with a single term s such that vs R (| s |, n ) ≥ φ( n ) and cvs R ( n , n ) ≥ φ( n ) for all n . For orthogonal TRSs R we prove that there is a constant k such that: (a) vs R ( m , n ) is bounded from above by a function exponential in k and (b) cvs R ( m , n ) is bounded from above by a function in the fourth level of the Grzegorczyk hierarchy. Similarly, for λ-calculus, we show that vs R ( m , n ) is bounded from above by a function in the fourth level of the Grzegorczyk hierarchy.

Encoding TLA+ into unsorted and many-sorted first-order logic

A frequency domain proof for the equivalence of the maximally flat FIR fractional delay filter and the Lagrange interpolator

Semantics of order-sorted specifications

A framework for compiler construction of functional programming languages is described. The framework has been designed to allow experimenting with functional programming languages and their implementations. Its three main characteristics are: (1) given the existing implementation of a functional programming language (not necessary in a source code), the implementation of a new functional programming language can be easily and efficiently developed, using the framework; (2) the framework enables enhancement of already implemented languages with new features; and (3) implementations of the languages in the framework are transparent, which can be freely exchanged between the languages.