We survey the properties of sets of integers recognizable by automata when they are written in p-ary expansions. We focus on Cobham’s theorem which characterizes the sets recognizable in different bases p and on its generalization to N m due to Semenov. We detail the remarkable proof recently given by Muchnik for the theorem of Cobham-Semenov, the original proof being published in Russian. 1

Present day systems, intelligent or otherwise, are limited by the conceptualizations of the world given to them by their designers. In this paper, we propose a novel, first-principles approach to performing incremental reformulations for computational efficiency. First, we define a reformulation to be a shift in conceptualization: a change in the basic objects, functions, and relations assumed in a formulation. We then analyze the requirements for automating reformulation and show the need for justifying shifts in conceptualization. Inefficient formulations make irrelevant distinctions. A new class of meta-theoretical justifications for a reformulation called irrelevance explanations, is presented. A logical irrelevance explanation demonstrates that certain distinctions made in the formulation are not necessary for the computation of a given class of problems. A computational irrelevance explanation shows that some distinctions are not useful with respect to a given problem solver fo...

Automata provide an effective mechanization of decision procedures for Presburger arithmetic. However, only crude lower and upper bounds are known on the sizes of the automata produced by this approach. In this paper, we prove that the number of states of the minimal deterministic automaton for a Presburger arithmetic formula is triple exponentially bounded in the length of the formula. This upper bound is established by comparing the automata for Presburger arithmetic formulas with the formulas produced by a quantifier elimination method. We also show that this triple exponential bound is tight (even for nondeterministic automata). Moreover, we provide optimal automata constructions for linear equations and inequations.

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We show that the set of realizations of dimension n of a max-plus linear sequence is a nite union of polyhedral sets, which can be computed from any realization of the sequence. This yields an (expensive) algorithm to solve the max-plus minimal realization problem. These results are derived from general facts on rational expressions over idempotent commutative semirings: we show more generally that the set of values of the coecients of a commutative rational expression in one letter that yield a given max-plus rational series, is a nite union of polyhedral sets.

3.97> v m (ph) = 1, p.51. Newbasax def = (Basaxnf Ax6; Ax3;AxE g)[f Ax6 00 ; Ax6 01 ; Ax3 0 ; AxE 0 g = f Ax1;Ax2;Ax3 0 ; Ax4;Ax5;Ax6 00 ; Ax6 01 ; AxE 0 g (cf. p.191), where: Ax6 00 (8m; k 2 Obs) wm [tr m (k)] Rng(w k ), p.190. Intuitively, observer k sees all those events which are seen by another observer m on k's life-line. Ax6 01 (8m; k

Abstract We have constructed a tool for using SMT (SAT Modulo Theories) solvers to discharge verification conditions (VCs) from programs written in the SPARK language. The tool has API interfaces for some solvers and can drive any solver supporting the SMT-LIB standard input language. SPARK is a subset of Ada used primarily in high-integrity systems in the aerospace, defence, rail and security industries. Formal verification of SPARK programs is supported by tools produced by the UK company Altran Praxis. We report in this paper on our experience in proving SPARK VCs using the popular SMT solvers CVC3, Yices, Z3 and Simplify, and compare these solvers with Praxis’s automatic prover. We find that the SMT solvers can prove virtually all the VCs that are discharged by Praxis’s prover, and sometimes more. Average run-times of the fastest SMT solvers are observed to be roughly 1 − 2 × that of the Praxis prover. Significant work is sometimes needed in translating VCs into a form suitable for input to the SMT solvers. A major contribution of the paper is a detailed presentation of the translations we implement. This is expected to be of interest to other users of SMT solvers.