We introduce a fragment of second-order unification, referred to as \emph{Second-Order Ground Unification (SOGU)}, with the following properties: (i) only one second-order variable is allowed, and (ii) first-order variables do not occur. We study an equational variant of SOGU where the signature contains \textit{associative} binary function symbols (ASOGU) and show that Hilbert's 10$^{th}$ problem is reducible to ASOGU unifiability, thus proving undecidability. Our reduction provides a new lower bound for the undecidability of second-order unification, as previous results required first-order variable occurrences, multiple second-order variables, and/or equational theories involving \textit{length-reducing} rewrite systems. Furthermore, our reduction holds even in the case when associativity of the binary function symbol is restricted to \emph{power associative}, i.e. f(f(x,x),x)= f(x,f(x,x)), as our construction requires a single constant.

Program synthesis is the task of automatically generating code based on a specification. In Syntax-Guided Synthesis (SyGuS) this specification is a combination of a syntactic template and a logical formula, and the result is guaranteed to satisfy both. We present a reinforcement-learning guided algorithm for SyGuS which uses Monte-Carlo Tree Search (MCTS) to search the space of candidate solutions. Our algorithm learns policy and value functions which, combined with the upper confidence bound for trees, allow it to balance exploration and exploitation. A common challenge in applying machine learning approaches to syntax-guided synthesis is the scarcity of training data. To address this, we present a method for automatically generating training data for SyGuS based on anti-unification of existing first-order satisfiability problems, which we use to train our MCTS policy. We implement and evaluate this setup and demonstrate that learned policy and value improve the synthesis performance over a baseline by over 26 percentage points in the training and testing sets. Our tool outperforms state-of-the-art tool cvc5 on the training set and performs comparably in terms of the total number of problems solved on the testing set (solving 23% of the benchmarks on which cvc5 fails). We make our data set publicly available, to enable further application of machine learning methods to the SyGuS problem.

Abstract Pre-trained Large Language Models (LLMs) are beginning to dominate the discourse around automatic code generation with natural language specifications. In contrast, the best-performing synthesizers in the domain of formal synthesis with precise logical specifications are still based on enumerative algorithms. In this paper, we evaluate the abilities of LLMs to solve formal synthesis benchmarks by carefully crafting a library of prompts for the domain. When one-shot synthesis fails, we propose a novel enumerative synthesis algorithm, which integrates calls to an LLM into a weighted probabilistic search. This allows the synthesizer to provide the LLM with information about the progress of the enumerator, and the LLM to provide the enumerator with syntactic guidance in an iterative loop. We evaluate our techniques on benchmarks from the Syntax-Guided Synthesis (SyGuS) competition. We find that GPT-3.5 as a stand-alone tool for formal synthesis is easily outperformed by state-of-the-art formal synthesis algorithms, but our approach integrating the LLM into an enumerative synthesis algorithm shows significant performance gains over both the LLM and the enumerative synthesizer alone and the winning SyGuS competition tool.

We propose infinite model finding as a new task for SMT-Solving. Model finding has a long-standing tradition in SMT and automated reasoning in general. Yet, most of the current tools are limited to finite models despite the fact that many theories only admit infinite models. This paper shows a variety of such problems and evaluates synthesis approaches on them. Interestingly, state-of-the-art SMT solvers fail even on very small and simple problems. We target such problems by SyGuS tools as well as heuristic approaches.

We introduce a novel approach to the automated termination analysis of computer programs: we use neural networks to represent ranking functions. Ranking functions map program states to values that are bounded from below and decrease as a program runs; the existence of a ranking function proves that the program terminates. We train a neural network from sampled execution traces of a program so that the network's output decreases along the traces; then, we use symbolic reasoning to formally verify that it generalises to all possible executions. Upon the affirmative answer we obtain a formal certificate of termination for the program, which we call a neural ranking function. We demonstrate that, thanks to the ability of neural networks to represent nonlinear functions, our method succeeds over programs that are beyond the reach of state-of-the-art tools. This includes programs that use disjunctions in their loop conditions and programs that include nonlinear expressions.

Abstract Applying machine learning to mathematical terms and formulas requires a suitable representation of formulas that is adequate for AI methods. In this paper, we develop an encoding that allows for logical properties to be preserved and is additionally reversible. This means that the tree shape of a formula including all symbols can be reconstructed from the dense vector representation. We do that by training two decoders: one that extracts the top symbol of the tree and one that extracts embedding vectors of subtrees. The syntactic and semantic logical properties that we aim to preserve include both structural formula properties, applicability of natural deduction steps and even more complex operations like unifiability. We propose datasets that can be used to train these syntactic and semantic properties. We evaluate the viability of the developed encoding across the proposed datasets as well as for the practical theorem proving problem of premise selection in the Mizar corpus.

Economic activity has always been a fundamental part of society. With recent social and political changes economics has gained even more influence on our lives. In this paper we formalize two economic models in Isabelle/HOL: the pure exchange economy, where the only economic actors are consumers, as well as a version of the Arrow-Debreu Model, a private ownership economy, which includes production facilities. Interestingly, the definitions of various components of the economic models differ in the economic literature. We therefore show the equivalences and implications between various presentations, which allows us to create an extensible foundation for formalizing microeconomics and game theory compatible with multiple economic theories. We prove the First Theorem of Welfare Economics in both economic models. The theorem is the mathematical formulation of Adam Smith’s famous invisible hand and states that a group of self-interested and rational actors will eventually achieve an efficient allocation of goods. The formal proofs allow us to find more precise assumptions than those found in the economic literature.

In this work we are interested in minimal complete sets of solutions for homogeneous linear diophantine equations. Such equations naturally arise during AC-unification—that is, unification in the presence of associative and commutative symbols. Minimal complete sets of solutions are for example required to compute AC-critical pairs. We present a verified solver for homogeneous linear diophantine equations that we formalized in Isabelle/HOL. Our work provides the basis for formalizing AC-unification and will eventually enable the certification of automated AC-confluence and AC-completion tools.

Utility functions form an essential part of game theory and economics. In order to guarantee the existence of these utility functions sufficient properties are assumed in an axiomatic manner. In this paper we discuss these axioms and the von-Neumann-Morgenstern Utility Theorem, which names precise assumptions under which expected utility functions exist. We formalize these results in Isabelle/HOL. The formalization includes formal definitions of the underlying concepts including continuity and independence of preferences. We make the dependencies more precise and highlight some consequences for a formalization of game theory.