Two simulations about DPLL(T)

The Journal of Symbolic Logic, 1993

We introduce new proof systems for propositional logic, simple deduction Frege systems, general deduction Frege systems and nested deduction Frege systems, which augment Frege systems with variants of the deduction rule. We give upper bounds on the lengths of proofs in Frege proof systems compared to lengths in these new systems. As applications we give near-linear simulations of the propositional Gentzen sequent calculus and the natural deduction calculus by Frege proofs. The length of a proof is the number of lines (or formulas) in the proof.

2013

Abstract. dpll (t) is a central algorithm for Satisfiability Modulo Theories (smt) solvers. The algorithm combines results of reasoning about the Boolean structure of a formula with reasoning about conjunctions of theory facts to decide satisfiability. This architecture enables modern solvers to combine the performance benefits of propositional satisfiability solvers and conjunctive theory solvers. We characterise dpll (t) as an abstract interpretation algorithm that computes a product of two abstractions.

Lecture Notes in Computer Science, 2005

At CAV'04 we presented the DPLL(T ) approach for satisfiability modulo theories T . It is based on a general DPLL(X) engine whose X can be instantiated with different theory solvers Solver T for conjunctions of literals.

2013

In this paper, we introduce two focussed sequent calculi, LK p (T) and LK + (T), that are based on Miller-Liang’s LKF system [LM09] 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 other works [FLM12, FGLM13] to simulate the DPLL(T) procedure [NOT06] as proof-search in the extension of LK p (T) with a cut-rule. In this report we therefore prove cut-elimination in LK p (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 LK p (T) with a relaxed focussing discipline, but ...

In a previously published ENTCS paper (Santos et al. (2016)), we introduced a sequent calculus called $\mathbf{LMT^{\rightarrow}}$ for Minimal Implicational Propositional Logic ($\mathbf{LMT^{\rightarrow}}$). This calculus provides a proof search procedure for $\mathbf{LMT^{\rightarrow}}$ that works in a bottom-up approach. We proved there that $\mathbf{LMT^{\rightarrow}}$ is sound and complete. We also suggested a strategy to guarantee termination of the proof search procedure. In this current paper, we refined this strategy and presented a new strategy for $\mathbf{LMT^{\rightarrow}}$ termination. Considering this new strategy, we also provide a (new) completeness proof for the system, which improves the previous version. Besides that, we present explicit upper bounds on the proof search procedure, derived from this new strategy. We also provide a full soundness proof of the system.

Lecture Notes in Computer Science, 2007

Sequent calculi usually provide a general deductive setting that uniformly embeds other proof-theoretical approaches, such as tableaux methods, resolution techniques, goal-directed proofs, etc. Unfortunately, in temporal logic, existing sequent calculi make use of a kind of inference rules that prevent the effective mechanization of temporal deduction in the general setting. In particular, temporal sequent calculi either need some form of cut, or they make use of invariants, or they include infinitary rules. This is the case even for the simplest kind of temporal logic, propositional linear temporal logic (PLTL). In this paper, we provide a complete finitary sequent calculus for PLTL, called FC, that not only is cut-free but also invariant-free. In particular, we introduce new rules which provide a new style of temporal deduction. We give a detailed proof of completeness.

2007

Abstract. In this work we propose an alternative approach to inference in DL-Lite, based on a reduction to reasoning in an extension of function-free Horn Logic (EHL). We develop a calculus for EHL and prove its soundness and completeness. We also show how to achieve decidability by means of a specific strategy, and how alternative strategies can lead to improved results in specific cases.

This thesis presents a new sequent calculus called LMT → that has the properties to be terminating, sound and complete for Propositional Implicational Minimal Logic (M →). LMT → is aimed to be used for proof search in M → , in a bottom-up approach. Termination of the calculus is guaranteed by a strategy of rule application that forces an ordered way to search for proofs such that all possible combinations are stressed. For an initial formula α, proofs in LMT → has an upper bound of |α| • 2 |α|+1+2•log 2 |α| , which together with the system strategy ensure decidability. System rules are conceived to deal with the necessity of hypothesis repetition and the contextsplitting nature of →-left, avoiding the occurrence of loops and the usage of backtracking. Therefore, LMT → steers the proof search always in a forward, deterministic manner. LMT → has the property to allow extractability of counter-models from failed proof searches (bicompleteness), i.e., the attempt proof tree of an expanded branch produces a Kripke model that falsifies the initial formula. Counter-model generation (using Kripke semantics) is achieved as a consequence of the completeness of the system. LMT → is implemented as an interactive theorem prover based on the calculus proposed here. We compare our calculus with other known deductive systems for M → , especially with Fitting's Tableaux, a method that also has the bicompleteness property. We also proposed here a translation of LMT → to the Dedukti proof checker as a way to evaluate the correctness of the implementation regarding the system specification and to make our system easier to compare to others.

The Journal of Logic and Algebraic Programming, 2009

On one hand, traditional tableau systems for temporal logic (TL) generate an auxiliary graph that must be checked and (possibly) pruned in a second phase of the refutation procedure. On the other hand, traditional sequent calculi for TL make use of a kind of inference rules (mainly, invariant-based rules or infinitary rules) that complicates their automatization. A remarkable consequence of using auxiliary graphs in the tableaux framework and invariants or infinitary rules in the sequents framework is that TL fails to carry out the classical correspondence between tableaux and sequents. In this paper, we first provide a tableau method ttm that does not require auxiliary graphs to decide whether a set of PLTL-formulas is satisfiable. This tableau method ttm is directly associated to a one-sided sequent calculus called ttc. Since ttm is free from all the structural rules that hinder the mechanization of deduction, e.g. weakening and contraction, then the resulting sequent calculus ttc is also free from this kind of structural rules. In particular, ttc is free of any kind of cut, including invariant-based cut. From the deduction system ttc, we obtain a two-sided sequent calculus GTC that preserves all these good freeness properties and is finitary, sound and complete for PLTL. Therefore, we show that the classical correspondence between tableaux and sequent calculi can be extended to TL. Every deduction system is proved to be complete. In addition, we provide illustrative examples of deductions in the different systems.

Arxiv preprint arXiv:1012.3372, 2010