Logics with rigidly guarded data tests

The notion of orbit finite data monoid was recently introduced by Bojanczyk as an algebraic object for defining recognizable languages of data words. Following Buchi's approach, we introduce a variant of monadic second-order logic with data equality tests that captures precisely the data languages recognizable by orbit finite data monoids. We also establish, following this time the approach of Schutzenberger, McNaughton and Papert, that the first-order fragment of this logic defines exactly the data languages recognizable by aperiodic orbit finite data monoids. Finally, we consider another variant of the logic that can be interpreted over generic structures with data. The data languages defined in this variant are also recognized by unambiguous finite memory automata

Using ACL2 to Verify Loop Pipelining in Behavioral Synthesis

Behavioral synthesis involves compiling an Electronic System-Level (ESL) design into its Register-Transfer Level (RTL) implementation. Loop pipelining is one of the most critical and complex transformations employed in behavioral synthesis. Certifying the loop pipelining algorithm is challenging because there is a huge semantic gap between the input sequential design and the output pipelined implementation making it infeasible to verify their equivalence with automated sequential equivalence checking techniques. We discuss our ongoing effort using ACL2 to certify loop pipelining transformation. The completion of the proof is work in progress. However, some of the insights developed so far may already be of value to the ACL2 community. In particular, we discuss the key invariant we formalized, which is very different from that used in most pipeline proofs. We discuss the needs for this invariant, its formalization in ACL2, and our envisioned proof using the invariant. We also discuss some trade-offs, challenges, and insights developed in course of the project.Comment: In Proceedings ACL2 2014, arXiv:1406.123

Decidability of graph neural networks via logical characterizations

We present results concerning the expressiveness and decidability of a popular graph learning formalism, graph neural networks (GNNs), exploiting connections with logic. We use a family of recently-discovered decidable logics involving ``Presburger quantifiers''. We show how to use these logics to measure the expressiveness of classes of GNNs, in some cases getting exact correspondences between the expressiveness of logics and GNNs. We also employ the logics, and the techniques used to analyze them, to obtain decision procedures for verification problems over GNNs. We complement this with undecidability results for static analysis problems involving the logics, as well as for GNN verification problems

Towards a complete transformational toolkit for compilers

PIM is an equational logic designed to function as a ``transformational toolkit'' for compilers and other programming tools that analyze and manipulate imperative languages.It has been applied to such problems as program slicing, symbolic evaluation, conditional constant propagation, and dependence analysis.PIM consists of the untyped lambda calculus extended with an algebraic data type that characterizes the behavior of lazy stores and generalized conditionals.A graph form of PIM terms is by design closely related to several intermediate representations commonly used in optimizing compilers. In this paper, we show that PIM's core algebraic component, PIM$_t$, possesses a complete equational axiomatization (under the assumption of certain reasonable restrictions on term formation). This has the practical consequence of guaranteeing that every semantics-preserving transformation on a program representable in PIM$_t$ can be derived by application of PIM$_t$ rules. We systematically derive the complete PIM$_t$ logic as the culmination of a sequence of increasingly powerful equational systems starting from a straightforward ``interpreter'' for closed PIM$_t$ terms. This work is an intermediate step in a larger program to develop a set of well-founded tools for manipulation of imperative programs by compilers and other systems that perform program analysis

A complete transformational toolkit for compilers

In an earlier paper, one of the present authors presented a preliminary account of an equational logic called PIM. PIM is intended to function as a 'transformational toolkit' to be used by compilers and analysis tools for imperative languages, and has been applied to such problems as program slicing, symbolic evaluation, conditional constant propagation, and dependence analysis. PIM consists of the untyped lambda calculus extended with an algebraic rewriting system that characterizes the behavior of lazy stores and generalized conditionals. A major question left open in the earlier paper was whether there existed a complete equational axiomatization of PIM's semantics. In this paper, we answer this question in the affirmative for PIM's core algebraic component, PIMt, under the assumption of certain reasonable restrictions on term formation. We systematically derive the complete PIM logic as the culmination of a sequence of increasingly powerful equational systems starting from a straightforward 'interpreter' for closed PIM terms

On computing fixpoints in well-structured regular model checking, with applications to lossy channel systems

We prove a general finite convergence theorem for "upward-guarded" fixpoint expressions over a well-quasi-ordered set. This has immediate applications in regular model checking of well-structured systems, where a main issue is the eventual convergence of fixpoint computations. In particular, we are able to directly obtain several new decidability results on lossy channel systems.Comment: 16 page