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Register a new userProof-Based Synthesis of Sorting Algorithms Using Multisets in Theorema
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Publication date
2019
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Informatics Engineering
and research's language is
English
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Using multisets, we develop novel techniques for mechanizing the proofs of the synthesis conjectures for list-sorting algorithms, and we demonstrate them in the Theorema system. We use the classical principle of extracting the algorithm as a set of rewrite rules based on the witnesses found in the proof of the synthesis conjecture produced from the specification of the desired function (input and output conditions). The proofs are in natural style, using standard rules, but most importantly domain specific inference rules and strategies. In particular the use of multisets allows us to develop powerful strategies for the synthesis of arbitrarily structured recursive algorithms by general Noetherian induction, as well as for the automatic generation of the specifications of all necessary auxiliary functions (insert, merge, split), whose synthesis is performed using the same method.
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Since the proof of the four color theorem in 1976, computer-generated proofs have become a reality in mathematics and computer science. During the last decade, we have seen formal proofs using verified proof assistants being used to verify the validity of such proofs. In this paper, we describe a formalized theory of size-optimal sorting networks. From this formalization we extract a certified checker that successfully verifies computer-generated proofs of optimality on up to 8 inputs. The checker relies on an untrusted oracle to shortcut the search for witnesses on more than 1.6 million NP-complete subproblems.
In this work we provide algorithmic solutions to five fundamental problems concerning the verification, synthesis and correction of concurrent systems that can be modeled by bounded p/t-nets. We express concurrency via partial orders and assume that behavioral specifications are given via monadic second order logic. A c-partial-order is a partial order whose Hasse diagram can be covered by c paths. For a finite set T of transitions, we let P(c,T,phi) denote the set of all T-labelled c-partial-orders satisfying phi. If N=(P,T) is a p/t-net we let P(N,c) denote the set of all c-partially-ordered runs of N. A (b, r)-bounded p/t-net is a b-bounded p/t-net in which each place appears repeated at most r times. We solve the following problems: 1. Verification: given an MSO formula phi and a bounded p/t-net N determine whether P(N,c)subseteq P(c,T,phi), whether P(c,T,phi)subseteq P(N,c), or whether P(N,c)cap P(c,T,phi)=emptyset. 2. Synthesis from MSO Specifications: given an MSO formula phi, synthesize a semantically minimal (b,r)-bounded p/t-net N satisfying P(c,T,phi)subseteq P(N, c). 3. Semantically Safest Subsystem: given an MSO formula phi defining a set of safe partial orders, and a b-bounded p/t-net N, possibly containing unsafe behaviors, synthesize the safest (b,r)-bounded p/t-net N whose behavior lies in between P(N,c)cap P(c,T,phi) and P(N,c). 4. Behavioral Repair: given two MSO formulas phi and psi, and a b-bounded p/t-net N, synthesize a semantically minimal (b,r)-bounded p/t net N whose behavior lies in between P(N,c) cap P(c,T,phi) and P(c,T,psi). 5. Synthesis from Contracts: given an MSO formula phi^yes specifying a set of good behaviors and an MSO formula phi^no specifying a set of bad behaviors, synthesize a semantically minimal (b,r)-bounded p/t-net N such that P(c,T,phi^yes) subseteq P(N,c) but P(c,T,phi^no ) cap P(N,c)=emptyset.
We interpret Linear Logic Proof Nets in a term language based on Solos calculus. The system includes a synchronisation mechanism, obtained by a conservative extension of the logic, that enables to define non-deterministic behaviours and multiparty sessions.
Most proof systems for concurrent programs assume the underlying memory model to be sequentially consistent (SC), an assumption which does not hold for modern multicore processors. These processors, for performance reasons, implement relaxed memory models. As a result of this relaxation a program, proved correct on the SC memory model, might execute incorrectly. To ensure its correctness under relaxation, fence instructions are inserted in the code. In this paper we show that the SC proof of correctness of an algorithm, carried out in the proof system of [Sou84], identifies per-thread instruction orderings sufficient for this SC proof. Further, to correctly execute this algorithm on an underlying relaxed memory model it is sufficient to respect only these orderings by inserting fence instructions.
Many systems are naturally modeled as Markov Decision Processes (MDPs), combining probabilities and strategic actions. Given a model of a system as an MDP and some logical specification of system behavior, the goal of synthesis is to find a policy that maximizes the probability of achieving this behavior. A popular choice for defining behaviors is Linear Temporal Logic (LTL). Policy synthesis on MDPs for properties specified in LTL has been well studied. LTL, however, is defined over infinite traces, while many properties of interest are inherently finite. Linear Temporal Logic over finite traces (LTLf) has been used to express such properties, but no tools exist to solve policy synthesis for MDP behaviors given finite-trace properties. We present two algorithms for solving this synthesis problem: the first via reduction of LTLf to LTL and the second using native tools for LTLf. We compare the scalability of these two approaches for synthesis and show that the native approach offers better scalability compared to existing automaton generation tools for LTL.
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