[Paper Review] Completeness for Two Left-Sequential Logics

Daan J. C. Staudt|arXiv (Cornell University)|Jun 9, 2012

Logic, programming, and type systems6 references3 citations

TL;DR

This thesis presents completeness proofs for two left-sequential logics: Fully Evaluated Left-Sequential Logic (FFEL) and Free Short-Circuit Logic (FSCL), using evaluation trees as a semantics. It establishes sound and complete axiomatizations for both logics, showing that FFEL is a sublogic of FSCL and that FSCL can be axiomatized via equational logic, providing a formal foundation for reasoning about side-effecting, left-to-right evaluated propositional terms in programming languages.

ABSTRACT

Left-sequential logics provide a means for reasoning about (closed) propositional terms with atomic propositions that may have side effects and that are evaluated sequentially from left to right. Such propositional terms are commonly used in programming languages to direct the flow of a program. In this thesis we explore two such left-sequential logics. First we discuss Fully Evaluated Left-Sequential Logic, which employs a full evaluation strategy, i.e., to evaluate a term every one of its atomic propositions is evaluated causing its possible side effects to occur. We then turn to Short-Circuit (Left-Sequential) Logic as presented in [BP10b], where the evaluation may be 'short-circuited', thus preventing some, if not all, of the atomic propositions in a term being evaluated. We propose evaluation trees as a natural semantics for both logics and provide axiomatizations for the least identifying variant of each. From this, we define a logic with connectives that prescribe a full evaluation strategy as well as connectives that prescribe a short-circuit evaluation strategy.

Motivation & Objective

• To formalize and prove completeness for Fully Evaluated Left-Sequential Logic (FFEL), which models full left-to-right evaluation of propositional terms with side effects.
• To develop a semantics based on evaluation trees for both FFEL and FSCL, enabling precise modeling of sequential evaluation with state changes.
• To establish a complete equational axiomatization for Free Short-Circuit Logic (FSCL), capturing short-circuit evaluation behavior in programming constructs.
• To show that FFEL is a sublogic of FSCL, demonstrating that full evaluation is a special case of short-circuit evaluation.
• To relate both logics to Proposition Algebra and clarify their connections to classical propositional logic and programming semantics.

Proposed method

• Introduces evaluation trees as a semantic model for both FFEL and FSCL, where internal nodes represent connectives and leaves represent atomic propositions with possible side effects.
• Defines normal forms for FFEL and FSCL terms, enabling systematic analysis of term structure and evaluation behavior.
• Proposes a mapping function $ g^* $ from evaluation trees to logical terms, proving its correctness and surjectivity to establish completeness.
• Uses induction on term complexity to prove that the evaluation tree semantics fully captures the truth values of terms under both full and short-circuit evaluation.
• Applies the concept of 'cd' (conditional disjunction) and 'dd' (disjunctive disjunction) to decompose complex terms and analyze their evaluation paths.
• Establishes that the axiomatization of FSCL via equational logic (EqFSCL) is sound and complete, using the evaluation tree semantics as a foundation.

Experimental results

Research questions

• RQ1 Can a complete and sound axiomatization be developed for Fully Evaluated Left-Sequential Logic (FFEL) using evaluation trees as a semantics?
• RQ2 How can a semantics based on evaluation trees be used to formally capture the behavior of short-circuit evaluation in programming constructs?
• RQ3 Is Free Short-Circuit Logic (FSCL) axiomatizable in a way that fully captures its equational theory under short-circuit evaluation?
• RQ4 What is the logical relationship between FFEL and FSCL, and can FFEL be embedded as a sublogic within FSCL?
• RQ5 How do these logics relate to Proposition Algebra and classical propositional logic, particularly in the context of side effects and evaluation order?

Key findings

• The evaluation tree semantics correctly captures the behavior of both FFEL and FSCL, providing a sound and complete model for left-sequential evaluation with side effects.
• A complete and sound axiomatization for FSCL is established via the equational theory EqFSCL, showing that all valid equalities between terms can be derived from the axioms.
• The mapping $ g^* $ from evaluation trees to terms is shown to be correct and surjective, proving that every term corresponds to a unique evaluation tree under the semantics.
• FFEL is formally proven to be a sublogic of FSCL, meaning that all valid equalities in FFEL hold in FSCL, but not vice versa due to short-circuiting.
• The completeness of the axiomatization for FFEL is established by induction on term complexity, using the normal form and tree structure to verify that all valid equalities are derivable.
• The paper demonstrates that the distinction between full and short-circuit evaluation is logically significant, as commutativity of conjunction and disjunction fails in both logics due to side effects and evaluation order.