We consider the task of weighted first-order model counting (WFOMC) used for probabilistic inference in the area of statistical relational learning. Given a formula $\phi$, domain size $n$ and a pair of weight functions, what is the weighted sum of all models of $\phi$ over a domain of size $n$? It was shown that computing WFOMC of any logical sentence with at most two logical variables can be done in time polynomial in $n$. However, it was also shown that the task is $\texttt{#}P_1$-complete once we add the third variable, which inspired the search for extensions of the two-variable fragment that would still permit a running time polynomial in $n$. One of such extension is the two-variable fragment with counting quantifiers. In this paper, we prove that adding a linear order axiom (which forces one of the predicates in $\phi$ to introduce a linear ordering of the domain elements in each model of $\phi$) on top of the counting quantifiers still permits a computation time polynomial in the domain size. We present a new dynamic programming-based algorithm which can compute WFOMC with linear order in time polynomial in $n$, thus proving our primary claim.

Sequence generation applications require satisfying semantic constraints, such as ensuring that programs are correct, using certain keywords, or avoiding undesirable content. Language models, whether fine-tuned or prompted with few-shot demonstrations, frequently violate these constraints, and lack a mechanism to iteratively revise their outputs. Moreover, some powerful language models are of extreme scale or inaccessible, making it inefficient, if not infeasible, to update their parameters for task-specific adaptation. We present Self-Correction, an approach that decouples an imperfect base generator (an off-the-shelf language model or supervised sequence-to-sequence model) from a separate corrector that learns to iteratively correct imperfect generations. To train the corrector, we propose an online training procedure that can use either scalar or natural language feedback on intermediate imperfect generations. We show that Self-Correction improves upon the base generator in three diverse generation tasks - mathematical program synthesis, lexically-constrained generation, and toxicity control - even when the corrector is much smaller than the base generator.

We conceptualize explainability in terms of logic and formula size, giving a number of related definitions of explainability in a very general setting. Our main interest is the so-called special explanation problem which aims to explain the truth value of an input formula in an input model. The explanation is a formula of minimal size that (1) agrees with the input formula on the input model and (2) transmits the involved truth value to the input formula globally, i.e., on every model. As an important example case, we study propositional logic in this setting and show that the special explainability problem is complete for the second level of the polynomial hierarchy. We also provide an implementation of this problem in answer set programming and investigate its capacity in relation to explaining answers to the n-queens and dominating set problems.

The satisfiability problem is one of the most famous problems in computer science. Its NP-completeness has been used to argue that SAT is intractable. However, there have been tremendous advances that allow SAT solvers to solve instances with millions of variables. A particularly successful paradigm is stochastic local search. In most cases, there are different ways of formulating the underlying problem. While it is known that this has an impact on the runtime of solvers, finding a helpful formulation is generally non-trivial. The recently introduced GapSAT solver [Lorenz and W\"orz 2020] demonstrated a successful way to improve the performance of an SLS solver on average by learning additional information which logically entails from the original problem. Still, there were cases in which the performance slightly deteriorated. This justifies in-depth investigations into how learning logical implications affects runtimes for SLS. In this work, we propose a method for generating logically equivalent problem formulations, generalizing the ideas of GapSAT. This allows a rigorous mathematical study of the effect on the runtime of SLS solvers. If the modification process is treated as random, Johnson SB distributions provide a perfect characterization of the hardness. Since the observed Johnson SB distributions approach lognormal distributions, our analysis also suggests that the hardness is long-tailed. As a second contribution, we theoretically prove that restarts are useful for long-tailed distributions. This implies that additional restarts can further refine all algorithms employing above mentioned modification technique. Since the empirical studies compellingly suggest that the runtime distributions follow Johnson SB distributions, we investigate this property theoretically. We succeed in proving that the runtimes for Sch\"oning's random walk algorithm are approximately Johnson SB.

A fundamental advantage of Petri net models is the possibility to automatically compute useful system invariants from the syntax of the net. Classical techniques used for this are place invariants, P-components, siphons or traps. Recently, Bozga et al. have presented a novel technique for the \emph{parameterized} verification of safety properties of systems with a ring or array architecture. They show that the statement \enquote{for every instance of the parameterized Petri net, all markings satisfying the linear invariants associated to all the P-components, siphons and traps of the instance are safe} can be encoded in \acs{WS1S} and checked using tools like MONA. However, while the technique certifies that this infinite set of linear invariants extracted from P-components, siphons or traps are strong enough to prove safety, it does not return an explanation of this fact understandable by humans. We present a CEGAR loop that constructs a \emph{finite} set of \emph{parameterized} P-components, siphons or traps, whose infinitely many instances are strong enough to prove safety. For this we design parameterization procedures for different architectures.

Developing neural architectures that are capable of logical reasoning has become increasingly important for a wide range of applications (e.g., natural language processing). Towards this grand objective, we propose a symbolic reasoning architecture that chains many join operators together to model output logical expressions. In particular, we demonstrate that such an ensemble of join-chains can express a broad subset of ''tree-structured'' first-order logical expressions, named FOET, which is particularly useful for modeling natural languages. To endow it with differentiable learning capability, we closely examine various neural operators for approximating the symbolic join-chains. Interestingly, we find that the widely used multi-head self-attention module in transformer can be understood as a special neural operator that implements the union bound of the join operator in probabilistic predicate space. Our analysis not only provides a new perspective on the mechanism of the pretrained models such as BERT for natural language understanding but also suggests several important future improvement directions.

The deep connections between logic and automata theory have led to extensive applications in formal specification and verification of systems. In recent years, research has focused on extensibility of such techniques to infinite-state systems. Logic has also been the ground for foundational research, revisiting classical model theory from computational perspectives. This has led to results in Skolem Löwenheim theorems in the finite, synthesis of Boolean functions and Skolem functions, definability in first-order theories of graphs, algebraic characterizations, block products of algebraic structures, and decidable fragments of first-order modal logics. The last few years has been a productive period for research in the field of computational complexity in India. The topics in which significant research has been done include algebraic complexity theory, communication complexity, research on codes and expanders arising out of connections to probabilistically checkable proofs, and the dynamic complexity of reachability. Notably, in July 2021, a break-through was achieved in the field of algebraic complexity, showing the first superpolynomial lower bounds against constant-depth arithmetic circuits over fields of characteristic zero or large characteristic.

In 2020, poor-quality software systems led to financial losses of approximately USD 2.08 trillion in the U.S. alone.19 Formal methods, such as bounded model checking (BMC), help to improve software quality, but they often fail to scale to the size and complexity of software.

We study automated test generation for verifying discrete decision-making modules in autonomous systems. We utilize linear temporal logic to encode the requirements on the system under test in the system specification and the behavior that we want to observe during the test is given as the test specification which is unknown to the system. First, we use the specifications and their corresponding non-deterministic B\"uchi automata to generate the specification product automaton. Second, a virtual product graph representing the high-level interaction between the system and the test environment is constructed modeling the product automaton encoding the system, the test environment, and specifications. The main result of this paper is an optimization problem, framed as a multi-commodity network flow problem, that solves for constraints on the virtual product graph which can then be projected to the test environment. Therefore, the result of the optimization problem is reactive test synthesis that ensures that the system meets the test specifications along with satisfying the system specifications. This framework is illustrated in simulation on grid world examples, and demonstrated on hardware with the Unitree A1 quadruped, wherein dynamic locomotion behaviors are verified in the context of reactive test environments.

We study the generalization abilities of language models when translating natural language into formal specifications with complex semantics. In particular, we fine-tune language models on three datasets consisting of English sentences and their corresponding formal representation: 1) regular expressions (regex), frequently used in programming and search; 2) First-order logic (FOL), commonly used in software verification and theorem proving; and 3) linear-time temporal logic (LTL), which forms the basis for industrial hardware specification languages. Our experiments show that, in these diverse domains, the language models maintain their generalization capabilities from pre-trained knowledge of natural language to generalize, e.g., to new variable names or operator descriptions. Additionally, they achieve competitive performance, and even outperform the state-of-the-art for translating into regular expressions, with the benefits of being easy to access, efficient to fine-tune, and without a particular need for domain-specific reasoning.