Separation and Renaming in Nominal Sets

Nominal sets provide a foundation for reasoning about names. They are used primarily in syntax with binders, but also, e.g., to model automata over infinite alphabets. In this paper, nominal sets are related to nominal renaming sets, which involve arbitrary substitutions rather than permutations, through a categorical adjunction. In particular, the left adjoint relates the separated product of nominal sets to the Cartesian product of nominal renaming sets. Based on these results, we define the new notion of separated nominal automata. We show that these automata can be exponentially smaller than classical nominal automata, if the semantics is closed under substitutions

Area/latency optimized early output asynchronous full adders and relative-timed ripple carry adders

This article presents two area/latency optimized gate level asynchronous full adder designs which correspond to early output logic. The proposed full adders are constructed using the delay-insensitive dual-rail code and adhere to the four-phase return-to-zero handshaking. For an asynchronous ripple carry adder (RCA) constructed using the proposed early output full adders, the relative-timing assumption becomes necessary and the inherent advantages of the relative-timed RCA are: (1) computation with valid inputs, i.e., forward latency is data-dependent, and (2) computation with spacer inputs involves a bare minimum constant reverse latency of just one full adder delay, thus resulting in the optimal cycle time. With respect to different 32-bit RCA implementations, and in comparison with the optimized strong-indication, weak-indication, and early output full adder designs, one of the proposed early output full adders achieves respective reductions in latency by 67.8, 12.3 and 6.1 %, while the other proposed early output full adder achieves corresponding reductions in area by 32.6, 24.6 and 6.9 %, with practically no power penalty. Further, the proposed early output full adders based asynchronous RCAs enable minimum reductions in cycle time by 83.4, 15, and 8.8 % when considering carry-propagation over the entire RCA width of 32-bits, and maximum reductions in cycle time by 97.5, 27.4, and 22.4 % for the consideration of a typical carry chain length of 4 full adder stages, when compared to the least of the cycle time estimates of various strong-indication, weak-indication, and early output asynchronous RCAs of similar size. All the asynchronous full adders and RCAs were realized using standard cells in a semi-custom design fashion based on a 32/28 nm CMOS process technology

Typeful Normalization by Evaluation

We present the first typeful implementation of Normalization by Evaluation for the simply typed lambda-calculus with sums and control operators: we guarantee type preservation and eta-long (modulo commuting conversions), beta-normal forms using only Generalized Algebraic Data Types in a general-purpose programming language, here OCaml; and we account for sums and control operators with Continuation-Passing Style. First, we implement the standard NbE algorithm for the implicational fragment in a typeful way that is correct by construction. We then derive its call-by-value continuation-passing counterpart, that maps a lambda-term with sums and call/cc into a CPS term in normal form, which we express in a typed dedicated syntax. Beyond showcasing the expressive power of GADTs, we emphasize that type inference gives a smooth way to re-derive the encodings of the syntax and typing of normal forms in Continuation-Passing Style

Strategic Issues, Problems and Challenges in Inductive Theorem Proving

Abstract(Automated) Inductive Theorem Proving (ITP) is a challenging field in automated reasoning and theorem proving. Typically, (Automated) Theorem Proving (TP) refers to methods, techniques and tools for automatically proving general (most often first-order) theorems. Nowadays, the field of TP has reached a certain degree of maturity and powerful TP systems are widely available and used. The situation with ITP is strikingly different, in the sense that proving inductive theorems in an essentially automatic way still is a very challenging task, even for the most advanced existing ITP systems. Both in general TP and in ITP, strategies for guiding the proof search process are of fundamental importance, in automated as well as in interactive or mixed settings. In the paper we will analyze and discuss the most important strategic and proof search issues in ITP, compare ITP with TP, and argue why ITP is in a sense much more challenging. More generally, we will systematically isolate, investigate and classify the main problems and challenges in ITP w.r.t. automation, on different levels and from different points of views. Finally, based on this analysis we will present some theses about the state of the art in the field, possible criteria for what could be considered as substantial progress, and promising lines of research for the future, towards (more) automated ITP