Author name cluster

P. Madhusudan

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9 papers

2 author rows

IJCAI Conference 2022 Conference Paper

Composing Neural Learning and Symbolic Reasoning with an Application to Visual Discrimination

Adithya Murali
Atharva Sehgal
Paul Krogmeier
P. Madhusudan

We consider the problem of combining machine learning models to perform higher-level cognitive tasks with clear specifications. We propose the novel problem of Visual Discrimination Puzzles (VDP) that requires finding interpretable discriminators that classify images according to a logical specification. Humans can solve these puzzles with ease and they give robust, verifiable, and interpretable discriminators as answers. We propose a compositional neurosymbolic framework that combines a neural network to detect objects and relationships with a symbolic learner that finds interpretable discriminators. We create large classes of VDP datasets involving natural and artificial images and show that our neurosymbolic framework performs favorably compared to several purely neural approaches.

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CSL Conference 2018 Conference Paper

A Decidable Fragment of Second Order Logic With Applications to Synthesis

P. Madhusudan
Umang Mathur 0001
Shambwaditya Saha
Mahesh Viswanathan 0001

We propose a fragment of many-sorted second order logic called EQSMT and show that checking satisfiability of sentences in this fragment is decidable. EQSMT formulae have an exists^*forall^* quantifier prefix (over variables, functions and relations) making EQSMT conducive for modeling synthesis problems. Moreover, EQSMT allows reasoning using a combination of background theories provided that they have a decidable satisfiability problem for the exists^*forall^* FO-fragment (e. g. , linear arithmetic). Our decision procedure reduces the satisfiability of EQSMT formulae to satisfiability queries of exists^*forall^* formulae of each individual background theory, allowing us to use existing efficient SMT solvers supporting exists^*forall^* reasoning for these theories; hence our procedure can be seen as effectively quantified SMT (EQSMT) reasoning.

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MFCS Conference 2018 Conference Paper

Lagrange's Theorem for Binary Squares

P. Madhusudan
Dirk Nowotka
Aayush Rajasekaran
Jeffrey O. Shallit

We show how to prove theorems in additive number theory using a decision procedure based on finite automata. Among other things, we obtain the following analogue of Lagrange's theorem: every natural number > 686 is the sum of at most 4 natural numbers whose canonical base-2 representation is a binary square, that is, a string of the form xx for some block of bits x. Here the number 4 is optimal. While we cannot embed this theorem itself in a decidable theory, we show that stronger lemmas that imply the theorem can be embedded in decidable theories, and show how automated methods can be used to search for these stronger lemmas.

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CSL Conference 2011 Conference Paper

Synthesizing Reactive Programs

P. Madhusudan

Current theoretical solutions to the classical Church's synthesis problem are focused on synthesizing transition systems and not programs. Programs are compact and often the true aim in many synthesis problems, while the transition systems that correspond to them are often large and not very useful as synthesized artefacts. Consequently, current practical techniques first synthesize a transition system, and then extract a more compact representation from it. We reformulate the synthesis of reactive systems directly in terms of program synthesis, and develop a theory to show that the problem of synthesizing programs over a fixed set of Boolean variables in a simple imperative programming language is decidable for regular omega-specifications. We also present results for synthesizing programs with recursion against both regular specifications as well as visibly-pushdown language specifications. Finally, we show applications to program repair, and conclude with open problems in synthesizing distributed programs.

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MFCS Conference 2009 Conference Paper

Query Automata for Nested Words

P. Madhusudan
Mahesh Viswanathan 0001

Abstract We study visibly pushdown automata (VPA) models for expressing and evaluating queries on words with a nesting structure. We define a query VPA model, which is a 2-way deterministic VPA that can mark in one run all positions in a document that satisfy a query, and show that it is equi-expressive as unary monadic queries. This surprising result parallels a classic result by Hopcroft and Ullman for queries on regular word languages. We also compare our model to query models on unranked trees, and show that our result is fundamentally different from those known for automata on trees.

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CSL Conference 2008 Conference Paper

An Infinite Automaton Characterization of Double Exponential Time

Salvatore La Torre
P. Madhusudan
Gennaro Parlato

Abstract Infinite-state automata are a new invention: they are automata that have an infinite number of states represented by words, transitions defined using rewriting, and with sets of initial and final states. Infinite-state automata have gained recent interest due to a remarkable result by Morvan and Stirling, which shows that automata with transitions defined using rational rewriting precisely capture context-sensitive ( NLinSpace) languages. In this paper, we show that infinite automata defined using a form of multi-stack rewriting precisely defines double exponential time (more precisely, 2ETime, the class of problems solvable in \(2^{2^{O(n)}}\) time). The salient aspect of this characterization is that the automata have no ostensible limits on time nor space, and neither direction of containment with respect to 2ETime is obvious. In this sense, the result captures the complexity class qualitatively, by restricting the power of rewriting.

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TCS Journal 2006 Journal Article

Modular strategies for recursive game graphs

Rajeev Alur
Salvatore La Torre
P. Madhusudan

Many problems in formal verification and program analysis can be formalized as computing winning strategies for two-player games on graphs. In this paper, we focus on solving games in recursive game graphs which can model the control flow in sequential programs with recursive procedure calls. While such games can be viewed as the pushdown games studied in the literature, the natural notion of winning in our framework requires the strategies to be modular with only local memory; that is, resolution of choices within a module does not depend on the context in which the module is invoked, but only on the history within the current invocation of the module. While reachability in (global) pushdown games is known to be EXPTIME-complete, we show reachability in modular games to be NP-complete. We present a fixed-point computation algorithm for solving modular games such that in the worst case the number of iterations is exponential in the total number of returned values from the modules. If the strategy within a module does not depend on the global history, but can remember the history of the past invocations of this module, that is, if memory is local but persistent, we show that reachability becomes undecidable.

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STOC Conference 2004 Conference Paper

Visibly pushdown languages

Rajeev Alur
P. Madhusudan

We propose the class of visibly pushdown languages as embeddings of context-free languages that is rich enough to model program analysis questions and yet is tractable and robust like the class of regular languages. In our definition, the input symbol determines when the pushdown automaton can push or pop, and thus the stack depth at every position. We show that the resulting class Vpl of languages is closed under union, intersection, complementation, renaming, concatenation, and Kleene-*, and problems such as inclusion that are undecidable for context-free languages are Exptime-complete for visibly pushdown automata. Our framework explains, unifies, and generalizes many of the decision procedures in the program analysis literature, and allows algorithmic verification of recursive programs with respect to many context-free properties including access control properties via stack inspection and correctness of procedures with respect to pre and post conditions. We demonstrate that the class Vpl is robust by giving two alternative characterizations: a logical characterization using the monadic second order (MSO) theory over words augmented with a binary matching predicate, and a correspondence to regular tree languages. We also consider visibly pushdown languages of infinite words and show that the closure properties, MSO-characterization and the characterization in terms of regular trees carry over. The main difference with respect to the case of finite words turns out to be determinizability: nondeterministic Büchi visibly pushdown automata are strictly more expressive than deterministic Muller visibly pushdown automata.

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TCS Journal 2002 Journal Article

Branching time controllers for discrete event systems

P. Madhusudan
P.S. Thiagarajan

We study the problem of synthesizing controllers for discrete event systems in a branching time framework. We use a class of labelled transition systems to model both plants and specifications. We use first simulations and later bisimulations to capture the role of a controller; the controlled behaviour of the plant should be related via a simulation (bisimulation) to the specification. For both simulations and bisimulations we show that the problem of checking if a pair of finite transition systems – one modelling the plant and the other the specification – admits a controller is decidable in polynomial time. We also show that the size of the controller, if one exists, can be bounded by a polynomial in the sizes of the plant and the specification and can be effectively constructed in polynomial time. Finally, we prove that in the case of simulations, the problem of checking for the existence of a controller is undecidable in a natural concurrent setting.

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