From Hybrid Modal Logic to Matching Logic and Back

Abstract

Building on our previous work on hybrid polyadic modal logic we identify modal logic equivalents for Matching Logic, a logic for program specification and verification. This provides a rigorous way to transfer results between the two approaches, which should benefit both systems. * All authors contributed equally to this work. 1 Note that Matching logic was further developed in [6], where techniques from modal logic are employed for the theoretical development. s = φ for any w ∈ W s ; in this case we write M | s = φ . The deductive system of H Σ is recalled in 1 and the completeness theorem is proved in [9]. The hybridization of our many-sorted modal logic is developed using a combination of ideas and techniques from [1,2,3,5,7,8], but for this section we drew our inspiration mainly form [3]. We refer to [10] for some similar proofs of the results presented in this section.Hybrid logic is defined on top of modal logic by adding nominals, states variables and specific operators and binders. Nominals allow us to directly refer the worlds (states) of a model, since they are evaluated in singletons in any model. However, a nominal may refer different worlds in different models. The sorts will be denoted by s, t, . . . and by PROP = {PROP s } s∈S , NOM = {NOM s } s∈S and SVAR = {SVAR s } s∈S we will denote some countable S-sorted sets. The elements of PROP are ordinary propositional variables and they will be denoted p, q,. . .; the elements of NOM are called nominals and they will be denoted by j, k, . . .; the elements of SVAR are called state variables and they are denoted x, y, . . .. We shall assume that for any distinct sorts s = t ∈ S, the corresponding sets of propositional variables, nominals and state variables are distinct. A state symbol is a nominal or a state variable.Definition 1 (H Σ (∀) formulas). For any s ∈ S we define the formulas of sort s: φ s := p | j | y s | ¬φ s | φ s ∨ φ s | σ (φ s 1 , . . . , φ s n ) s | ∀x t φ s

Many-Sorted Hybrid Modal Languages

“…• Deduction rules: (Gen) if | s φ then | s ∀xφ, where φ ∈ F orm s and x ∈ SVAR t for some t ∈ S. In [12] we have proved the soundness and completeness of the H Σ (∀) system.…”

“…Improving over previous work [10,11,12], this paper makes the following contributions: (1) We study the @-only fragment of the more general hybrid modal logic proposed in [11,12], and provide a sound and complete deduction system for it. This logic is important as it is weaker than the full hybrid modal logic and thus it is expected to have better computational properties.…”

“…(test): in our setting, in order to take a decision, we test the top value of the value stack. Consequently, the (13) [c(pgm)](conf ig(vs, mf ) ∧ conf ig(vs, mem ′ )) → [c(pgm)]conf ig(vs ∧ vs, mf ∧ mem ′ ) (UG):( 5), (K σ ), (M P ) (14) [c(pgm)]conf ig(vs∧vs, mf ∧mem ′ ) → [c(pgm)]conf ig(vs∧vs, @ mem ′ mf ) (UG):( 6), (K σ ), (M P ) (15) conf ig(vs, mem) → [c(pgm)]@ mem ′ get(m, 1) PL:(3),( 4),( 13),( 14),( 9), (12) 3 The many-sorted hybrid modal logic H Σ (∀)…”

“…For that, we needed to define the satisfaction operator only on nominals. Furthermore, in [12], in order to establish the connection with Matching logic, we have introduced the H Σ (@ z , ∀) system which allows the satisfaction operators @ z to also range over state variables, not just over nominals.…”

“…Moreover, we provide an example of using this system to axiomatically express operational semantics and to derive proofs for statements concerning program executions. Sections 3 and 4 recall two related hybrid systems introduced in [12]: H Σ (∀), an orthodogal extension of K Σ ; and H Σ (@ z , ∀), a common extension of both H Σ (@ z ) and H Σ (∀). The paper concludes by providing a standard translation from H Σ (@ z , ∀) to first order logic, showing that any many-sorted modal formula corresponds to a firstorder formula from its corresponding first-order language.…”