Given a theory T and two formulas A and B jointly unsatisfiable in T , a theory interpolant of A ... more Given a theory T and two formulas A and B jointly unsatisfiable in T , a theory interpolant of A and B is a formula I such that (i) its non-theory symbols are shared by both A and B, (ii) it is entailed by A in T , and (iii) it is unsatisfiable with B in T. Theory interpolation has found several successful applications in model checking. We present a novel method for computing interpolants for ground formulas in the theory of equality. The method produces interpolants from colored congruence graphs representing derivations in that theory. These graphs can be produced by conventional congruence closure algorithms in a straightforward manner. By working with graphs, rather than at the level of individual proof steps, we are able to derive interpolants that are pleasingly simple (conjunctions of Horn clauses) and smaller than those generated by other tools. Our interpolation method can be seen as a theory-specific implementation of a cooperative interpolation game between two provers. We present a generic version of the interpolation game, parametrized by the theory T , and define a general method to extract runs of the game from proofs in T and then generate interpolants from these runs.
Crossing Matrices And Thurston's Canonical Form For Braids
this paper is to show the relation between this simple invariant and the canonical form of positi... more this paper is to show the relation between this simple invariant and the canonical form of positive braids (as explained in [1] and [5]), with special emphasis on positive braids whose crossing numbers are 1. For example, it is possible to test whether a factorization b = a 1 a p of such a braid b is canonical because the crossing matrices of the factors must have a maximality property (Proposition 5.2). It is known [1; Thm. 2.6] that a positive braid of canonical length one is determined by its crossing matrix. We extend that result to positive braids of canonical length 2 and show that for canonical length 3 this is no longer true. We give a characterization of all those matrices which are crossing matrices of a braid of length 2. This answers the question posed in [1; p. 496]. As an example, if every strand of a positive braid crosses over every other strand exactly once, this braid is the fundamental braid squared or the full twist braid (Example 5.5)
Proceedings of the American Mathematical Society, 1997
It is shown that if R R is an integral domain which is not a field, and U 2 ( R [ x ] ) U_2(R[x])... more It is shown that if R R is an integral domain which is not a field, and U 2 ( R [ x ] ) U_2(R[x]) is the subgroup of S L 2 ( R [ x ] ) SL_2(R[x]) generated by all unipotent elements, then the quotient group S L 2 ( R [ x ] ) / U 2 ( R [ x ] ) SL_2(R[x])/U_2(R[x]) has a free quotient of infinite rank.
Introduction In the usual mathematical practice, functions whose arguments are functions are comm... more Introduction In the usual mathematical practice, functions whose arguments are functions are commonplace. Functions whose arguments are functions whose arguments are functions are rarer, and everything beyond the third level is exotic. Innitely nested function spaces like H(A;B) = (((( ! A) ! B) ! A) ! B) cannot exist for cardinality reasons. In domain theory, however, such innite expressions can be given a precise meaning. For suitable categories D, including all reasonable cartesian closed categories of pointed domains, the theory provides canonical solutions in D to recursive domain equations, and our H(A;B) can be dened as the canonical solution of the equation X = (X ! A) ! B. Textbook expos
2014 IEEE International Symposium on Hardware-Oriented Security and Trust (HOST), 2014
ABSTRACT The security architecture of modern systems-on-a-chip (SoC) is complex and critical to b... more ABSTRACT The security architecture of modern systems-on-a-chip (SoC) is complex and critical to be done right and quickly. SoC security architects feel an acute need for new tool-supported specification and validation technologies. Aiming to stimulate research into creation of these technologies, in this paper we provide some industrial insights and initial solutions. Focusing on a concrete non-trivial example of security sensitive firmware load protocols, we show how to: (1) concisely specify the communication between IP blocks; (2) model the adversary; (3) debug and verify the protocol.
Simulations between processes can be understood in terms of coalgebra homomorphisms, with homomor... more Simulations between processes can be understood in terms of coalgebra homomorphisms, with homomorphisms to the final coalgebra exactly identifying bisimilar processes. The elements of the final coalgebra are thus natural representatives of bisimilarity classes, and a denotational semantics of processes can be developed in a final-coalgebraenriched category where arrows are processes, canonically represented. In the present paper, we describe a general framework for building finalcoalgebra-enriched categories. Every such category is constructed from a multivariant functor representing a notion of process, much like Moggi's categories of computations arising from monads as notions of computation. The "notion of process" functors are intended to capture different flavors of processes as dynamically extended computations. These functors may involve a computational (co)monad, so that a process category in many cases contains an associated computational category as a retract. We further discuss categories of resumptions and of hyperfunctions, which are the main examples of process categories. Very informally, the resumptions can be understood as computations extended in time, whereas hypercomputations are extended in space.
Proceedings of the Edinburgh Mathematical Society, 2001
Given finite sets of cyclic words $\{u_1,\dots,u_k\}$ and $\{v_1,\dots,v_k\}$ in a finitely gener... more Given finite sets of cyclic words $\{u_1,\dots,u_k\}$ and $\{v_1,\dots,v_k\}$ in a finitely generated free group $F$ and two finite groups $A$ and $B$ of outer automorphisms of $F$, we produce an algorithm to decide whether there is an automorphism which conjugates $A$ to $B$ and takes $u_i$ to $v_i$ for each $i$. If $A$ and $B$ are trivial, this is the classic algorithm due to Whitehead. We use this algorithm together with Cohen and Lustig’s solution to the conjugacy problem for Dehn twist automorphisms of $F$ to solve the conjugacy problem for outer automorphisms which have a power which is a Dehn twist. This settles the conjugacy problem for all automorphisms of $F$ which have linear growth.AMS 2000 Mathematics subject classification: Primary 20F32. Secondary 57M07
We establish ÿnite or inÿnite presentability of the general linear group GLn(R), the Steinberg gr... more We establish ÿnite or inÿnite presentability of the general linear group GLn(R), the Steinberg group Stn(R), and the elementary group En(R) for large classes of rings R. In particular, we obtain a complete answer in the case when R is a free associative algebra with ÿeld coe cients.
Let G + Aut F be an action of a finite group G on a free group F. The main result of the paper is... more Let G + Aut F be an action of a finite group G on a free group F. The main result of the paper is that the maximal free product decomposition F=Ft*F2*...*F,, with factors Ft,Fz.. . ..F. invariant under the action of G, is practically unique. As an application, a classification is obtained of all periodic automorphisms of free groups of rank ~5.
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Papers by Sava Krstic