Slot Machines: an approach to the Strategy Challenge in SMT solving (presentation only)

International audienceIn this short introduction we briefly describe the relevance of Psyche's system description [GL13] to the Strategy Challenge set by L. de Moura and G. O. Passmore for SMT solving

Polarities & Focussing: a journey from Realisability to Automated Reasoning

This dissertation explores the roles of polarities and focussing in various aspects of Computational Logic.These concepts play a key role in the the interpretation of proofs as programs, a.k.a. the Curry-Howard correspondence, in the context of classical logic. Arising from linear logic, they allow the construction of meaningful semantics for cut-elimination in classical logic, some of which relate to the Call-by-Name and Call-by-Value disciplines of functional programming. The first part of this dissertation provides an introduction to these interpretations, highlighting the roles of polarities and focussing. For instance: proofs of positive formulae provide structured data, while proofs of negative formulae consume such data; focussing allows the description of the interaction between the two kinds of proofs as pure pattern-matching. This idea is pushed further in the second part of this dissertation, and connected to realisability semantics, where the structured data is interpreted algebraically, and the consumption of such data is modelled with the use of an orthogonality relation. Most of this part has been proved in the Coq proof assistant.Polarities and focussing were also introduced with applications to logic programming in mind, where computation is proof-search. In the third part of this dissertation, we push this idea further by exploring the roles that these concepts can play in other applications of proof-search, such as theorem proving and more particularly automated reasoning. We use these concepts to describe the main algorithm of SAT-solvers and SMT-solvers: DPLL. We then describe the implementation of a proof-search engine called Psyche. Its architecture, based on the concept of focussing, offers a platform where smart techniques from automated reasoning (or a user interface) can safely and trustworthily be implemented via the use of an API

Sequent Calculi with procedure calls

In this paper, we introduce two focussed sequent calculi, LKp(T) and LK+(T), that are based on Miller-Liang's LKF system for polarised classical logic. The novelty is that those sequent calculi integrate the possibility to call a decision procedure for some background theory T, and the possibility to polarise literals "on the fly" during proof-search. These features are used in our other works to simulate the DPLL(T) procedure as proof-search in the extension of LKp(T) with a cut-rule. In this report we therefore prove cut-elimination in LKp(T). Contrary to what happens in the empty theory, the polarity of literals affects the provability of formulae in presence of a theory T. On the other hand, changing the polarities of connectives does not change the provability of formulae, only the shape of proofs. In order to prove this, we introduce a second sequent calculus, LK+(T) that extends LKp(T) with a relaxed focussing discipline, but we then show an encoding of LK+(T) back into the more restrictive system LK(T). We then prove completeness of LKp(T) (and therefore of LK+(T)) with respect to first-order reasoning modulo the ground propositional lemmas of the background theory T

L'intéropérabilité entre des preuves d'arithmétique en utilisan Dedukti

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Machine-checked ZKP for NP-relations: Formally Verified Security Proofs and Implementations of MPC-in-the-Head

MPC-in-the-Head (MitH) is a general framework that enables constructing efficient zero- knowledge (ZK) protocols for NP relations from secure multiparty computation (MPC) protocols. In this paper we present the first machine-checked implementations of this transformation. We begin with an EasyCrypt formalization that preserves modular structure of the original MitH construction and can be instantiated with arbitrary MPC protocols, secret sharing and commitment schemes satisfying standard notions of security. We then formalize various suitable components, which we use to obtain full-fledged ZK protocols for general relations. We compare two approaches for obtaining verified executable implementations. The first approach realizes a fully automated extraction from EasyCrypt to OCaml. The second one reduces the trusted computing base (TCB) and provides better performance for the extracted executable by combining code extraction with manual formal verification of low-level components implemented in the Jasmin language. We conclude the paper with a discussion of the trade-off between formal verification effort and performance, and also discuss how our approach opens the way for fully verified implementations of state-of the-art optimized protocols based on MitH

Realisability semantics of abstract focussing, formalised

International audienceWe present a sequent calculus for abstract focussing, equipped with proof-terms: in the tradition of Zeilberger's work, logical connectives and their introduction rules are left as a parameter of the system, which collapses the synchronous and asynchronous phases of focussing as macro rules. We go further by leaving as a parameter the operation that extends a context of hypotheses with new ones, which allows us to capture both classical and intuitionistic focussed sequent calculi. We then define the realisability semantics of (the proofs of) the system, on the en basis of Munch-Maccagnoni's orthogonality models for the classical focussed sequent calculus, but now operating at the higher level of abstraction mentioned above. We prove, at that level, the Adequacy Lemma, namely that if a term is of type A, then in the model its denotation is in the (set-theoretic) interpretation of A. This exhibits the fact that the universal quantification involved when taking the orthogonal of a set, reflects in the semantics Zeilberger's universal quantification in the macro rule for the asynchronous phase. The system and its semantics are all formalised in Coq

Guiding SMT solvers with Monte Carlo Tree Search and neural networks

International audienceMonte Carlo Tree Search (MCTS) is a technique to guide search in a large decision space by taking random samples and evaluating their outcome. Frequently, MCTS is employed together with reward heuristics learnt by neural networks. The talk will propose a guidance mechanism for SMT solvers based on a combination of MCTS and neural networks. Machine learning methods gain importance in automated reasoning. A particularly strong trend are neural networks, having produced state-of-the-art results for premise selection [WTWD17, ISA + 16]. Outside of automated reasoning, neural networks have been combined with Monte Carlo Tree Search, treating problems as diverse as finding good strategies to play the game of Go [SHM + 16] and planning of chemical syntheses [SKTW17, SPW17]. In automated reasoning, Monte Carlo Tree Search (MCTS) has been applied to first-order automated theorem proving, using hand-crafted heuristics instead of neural networks [FKU17]. We propose a combination of Monte Carlo Tree Search and neural networks to guide the search performed by an SMT solver. We are exploring the idea of such guidance in the Psyche platform [GL13], which offers a modular architecture for theorem proving. It implements an adaptation, to automated reasoning in general and to SMT solving in particular, of the LCF architecture [Mil79, GMW79]. LCF is mostly used in Interactive Theorem Proving and is particualrly widely implemented in the proof assistants of the HOL family, such as the HOL system [HOL], Isabelle [Isa], etc. The LCF architecture allows theorem proving strategies to be programmable, while guaranteeing the correctness of any claim that a formula is provable. The architecture's kernel component offers an API whose primitives implement basic reasoning inferences, and strategies can be programmed on top of the kernel via the API. The claims of provability are then necessarily correct-by-construction, assuming the correctness of the kernel, but regardless of any potential defects in the design or in the implementation of strategies (or in the user's input, for the case of Interactive Theorem Proving). Psyche embraces this paradigm and, in the case of SMT solving, relates to a position paper by de Moura and Passmore, entitled "The strategy challenge in SMT solving" [dMP13], which promoted the programmability of strategies as compositions of basic reasoning tasks, explicitly referring to the LCF paradigm. This approach opens up the possibilities of extensively experimenting with various strategies, whether they be handcrafted or machine-learned, while never jeopardising the correctness of the solver's output. Guiding the search by techniques such as MCTS and neural networks can be envisaged more easily in provers whose architecture implements this approach. Psyche's architecture does so at a rather fine-grained level, separating the code that implements reasoning inferences from the code that implements search strategies

An MCSAT treatment of Bit-Vectors (preliminary report)

International audienceWe propose a general scheme for treating the theory of bit-vectors (BV) in the MCSAT framework, complementing the approach by Zeljić, Wintersteiger, and Rümmer. MCSAT assigns values to first-order variables. In order to keep track of the set of feasible values for a given bit-vector variable, we propose the use of Binary Decision Diagram. This allows an assignment mechanism that is generic for BV. When a conflict arises, involving some of the constraints and some of the assignments made so far, MCSAT must produce an explanation for the conflict. This mechanism can be specialized according to the constraints involved in the conflict. We propose an explanation mechanism that applies when these constraints are in the core fragment of BV, based on slicing and equality reasoning. We plan to add support for more BV fragments in the future

Special Issue on Computational Logic ( in honor to Roy Dyckhoff) of Journal of Logic and Computation

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